zero hypothesis t test

Approximate Hypothesis Tests: the z Test and the t Test

This chapter presents two common tests of the hypothesis that a population mean equals a particular value and of the hypothesis that two population means are equal: the z test and the t test. These tests are approximate : They are based on approximations to the probability distribution of the test statistic when the null hypothesis is true, so their significance levels are not exactly what they claim to be. If the sample size is reasonably large and the population from which the sample is drawn has a nearly normal distribution —a notion defined in this chapter—the nominal significance levels of the tests are close to their actual significance levels. If these conditions are not met, the significance levels of the approximate tests can differ substantially from their nominal values. The z test is based on the normal approximation ; the t test is based on Student's t curve, which approximates some probability histograms better than the normal curve does. The chapter also presents the deep connection between hypothesis tests and confidence intervals, and shows how to compute approximate confidence intervals for the population mean of nearly normal populations using Student's t -curve.

where \(\phi\) is the pooled sample percentage of the two samples. The estimate of \(SE(\phi^{t-c})\) under the null hypothesis is

\[ se = s^*\times(1/n_t + 1/n_c)^{1/2}, \]

where \(n_t\) and \(n_c\) are the sizes of the two samples. If the null hypothesis is true, the Z statistic,

\[ Z=\phi^{t-c}/se, \]

is the original test statistic \(\phi^{t-c}\) in approximately standard units , and Z has a probability histogram that is approximated well by the normal curve , which allowed us to select the rejection region for the approximate test.

This strategy—transforming a test statistic approximately to standard units under the assumption that the null hypothesisis true, and then using the normal approximation to determine the rejection region for the test—works to construct approximate hypothesis tests in many other situations, too. The resulting hypothesis test is called a z test. Suppose that we are testing a null hypothesis using a test statistic \(X\) , and the following conditions hold:

• We have a probability model for how the observations arise, assuming the null hypothesis is true. Typically, the model is that under the null hypothesis, the data are like random draws with or without replacement from a box of numbered tickets.
• Under the null hypothesis, the test statistic \(X\) , converted to standard units, has a probability histogram that can be approximated well by the normal curve.
• Under the null hypothesis, we can find the expected value of the test statistic, \(E(X)\) .
• Under the null hypothesis, either we can find the SE of the test statistic, \(SE(X)\) , or we can estimate \(SE(X)\) accurately enough to ignore the error of the estimate of the SE. Let se denote either the exact SE of \(X\) under the null hypothesis, or the estimated value of \(SE(X)\) under the null hypothesis.

Then, under the null hypothesis, the probability histogram of the Z statistic

\[ Z = (X-E(X))/se \]

is approximated well by the normal curve, and we can use the normal approximation to select the rejection region for the test using \(Z\) as the test statistic. If the null hypothesis is true,

\[ P(Z < z_a) \approx a \]

\[ P(Z > z_{1-a} ) \approx a, \]

\[ P(|Z| > z_{1-a/2} ) \approx a. \]

These three approximations yield three different z tests of the hypothesis that \(\mu = \mu_0\) at approximate significance level \(a\) :

• Reject the null hypothesis whenever \(Z (left-tail z test)
• Reject the null hypothesis whenever \(Z > z_{1-a}\) (right-tail z test)
• Reject the null hypothesis whenever \(|Z|> z_{1-a/2}\) (two-tail z test)

The word "tail" refers to the tails of the normal curve: In a left-tail test, the probability of a Type I error is approximately the area of the left tail of the normal curve, from minus infinity to \(z_a\) . In a right-tail test, the probability of a Type I error is approximately the area of the right tail of the normal curve, from \(z_{1-a}\) to infinity. In a two-tail test, the probability of a Type I error is approximately the sum of the areas of both tails of the normal curve, the left tail from minus infinity to \(z_{a/2}\) and the right tail from \(z_{1-a/2}\) to infinity. All three of these tests are called z tests. The observed value of Z is called the z score .

Which of these three tests, if any, should one use? The answer depends on the probability distribution of Z when the alternative hypothesis is true. As a rule of thumb, if, under the alternative hypothesis, \(E(Z) , use the left-tail test. If, under the alternative hypothesis, \(E(Z) > 0\) , use the right-tail test. If, under the alternative hypothesis, it is possible that \(E(Z) and it is possible that \(E(Z) > 0\) , use the two-tail test. If, under the alternative hypothesis, \(E(Z) = 0\) , consult a statistician. Generally (but not always), this rule of thumb selects the test with the most power for a given significance level.

P values for z tests

Each of the three z tests gives us a family of procedures for testing the null hypothesis at any (approximate) significance level \(a\) between 0 and 100%—we just use the appropriate quantile of the normal curve. This makes it particularly easy to find the P value for a z test. Recall that the P value is the smallest significance level for which we would reject the null hypothesis, among a family of tests of the null hypothesis at different significance levels.

Suppose the z score (the observed value of \(Z\) ) is \(x\) . In a left-tail test, the P value is the area under the normal curve to the left of \(x\) : Had we chosen the significance level \(a\) so that \(z_a=x\) , we would have rejected the null hypothesis, but we would not have rejected it for any smaller value of \(a\) , because for all smaller values of \(a\) , \(z_a . Similarly, for a right-tail z test, the P value is the area under the normal curve to the right of \(x\) : If \(x=z_{1-a}\) we would reject the null hypothesis at approximate significance level \(a\) , but not at smaller significance levels. For a two-tail z test, the P value is the sum of the area under the normal curve to the left of \(-|x|\) and the area under the normal curve to the right of \(|x|\) .

Finding P values and specifying the rejection region for the z test involves the probability distribution of \(Z\) under the assumption that the null hypothesis is true. Rarely is the alternative hypothesis sufficiently detailed to specify the probability distribution of \(Z\) completely, but often the alternative does help us choose intelligently among left-tail, right-tail, and two-tail z tests. This is perhaps the most important issue in deciding which hypothesis to take as the null hypothesis and which as the alternative: We calculate the significance level under the null hypothesis, and that calculation must be tractable.

However, to construct a z test, we need to know the expected value and SE of the test statistic under the null hypothesis. Usually it is easy to determine the expected value, but often the SE must be estimated from the data. Later in this chapter we shall see what to do if the SE cannot be estimated accurately, but the shape of the distribution of the numbers in the population is known. The next section develops z tests for the population percentage and mean, and for the difference between two population means.

Examples of z tests

The central limit theorem assures us that the probability histogram of the sample mean of random draws with replacement from a box of tickets—transformed to standard units—can be approximated increasingly well by a normal curve as the number of draws increases. In the previous section, we learned that the probability histogram of a sum or difference of independent sample means of draws with replacement also can be approximated increasingly well by a normal curve as the two sample sizes increase. We shall use these facts to derive z tests for population means and percentages and differences of population means and percentages.

z Test for a Population Percentage

Suppose we have a population of \(N\) units of which \(G\) are labeled "1" and the rest are labeled "0." Let \(p = G/N\) be the population percentage. Consider testing the null hypothesis that \(p = p_0\) against the alternative hypothesis that \(p \ne p_0\) , using a random sample of \(n\) units drawn with replacement. (We could assume instead that \(N >> n\) and allow the draws to be without replacement.)

Under the null hypothesis, the sample percentage

\[ \phi = \frac{\mbox{# tickets labeled "1" in the sample}}{n} \]

has expected value \(E(\phi) = p_0\) and standard error

\[ SE(\phi) = \sqrt{\frac{p_0 \times (1 - p_0)}{n}}. \]

Let \(Z\) be \(\phi\) transformed to standard units :

\[ Z = (\phi - p_0)/SE(\phi). \]

Provided \(n\) is large and \(p_0\) is not too close to zero or 100% (say \(n \times p > 30\) and \(n \times (1-p) > 30)\) , the probability histogram of \(Z\) will be approximated reasonably well by the normal curve, and we can use it as the Z statistic in a z test. For example, if we reject the null hypothesis when \(|Z| > 1.96\) , the significance level of the test will be about 95%.

z Test for a Population Mean

The approach in the previous subsection applies, mutatis mutandis , to testing the hypothesis that the population mean equals a given value, even when the population contains numbers other than just 0 and 1. However, in contrast to the hypothesis that the population percentage equals a given value, the null hypothesis that a more general population mean equals a given value does not specify the SD of the population, which poses difficulties that are surmountable (by approximation and estimation) if the sample size is large enough. (There are also nonparametric methods that can be used.)

Consider testing the null hypothesis that the population mean \(\mu\) is equal to a specific null value \(\mu_0\) , against the alternative hypothesis that \(\mu , on the basis of a random sample with replacement of size \(n\) . Recall that the sample mean \(M\) of \(n\) random draws with or without replacement from a box of numbered tickets is an unbiased estimator of the population mean \(\mu\) : If

\[ M = \frac{\mbox{sum of sample values}}{n}, \]

\[ E(M) = \mu = \frac{\mbox{sum of population values}}{N}, \]

where \(N\) is the size of the population. The population mean determines the expected value of the sample mean. The SE of the sample mean of a random sample with replacement is

\[ \frac{SD(\mbox{box})}{\sqrt{n}}, \]

where SD(box) is the SD of the list of all the numbers in the box, and \(n\) is the sample size. As a special case, the sample percentage \phi of \(n\) independent random draws from a 0-1 box is an unbiased estimator of the population percentage p , with SE equal to

\[ \sqrt{\frac{p\times(1-p)}{n}}. \]

In testing the null hypothesis that a population percentage \(p\) equals \(p_0\) , the null hypothesis specifies not only the expected value of the sample percentage \phi, it automatically specifies the SE of the sample percentage as well, because the SD of the values in a 0-1 box is determined by the population percentage \(p\) :

\[ SD(box) = \sqrt{p\times(1-p)}. \]

The null hypothesis thus gives us all the information we need to standardize the sample percentage under the null hypothesis. In contrast, the SD of the values in a box of tickets labeled with arbitrary numbers bears no particular relation to the mean of the values, so the null hypothesis that the population mean \(\mu\) of a box of tickets labeled with arbitrary numbers equals a specific value \(\mu_0\) determines the expected value of the sample mean, but not the standard error of the sample mean. To standardize the sample mean to construct a z test for the value of a population mean, we need to estimate the SE of the sample mean under the null hypothesis. When the sample size is large, the sample standard deviation s> is likely to be close to the SD of the population, and

\[ se=\frac{s}{\sqrt{n}} \]

is likely to be an accurate estimate of \(SE(M)\) . The central limit theorem tells us that when the sample size \(n\) is large, the probability histogram of the sample mean, converted to standard units, is approximated well by the normal curve. Under the null hypothesis,

\[ E(M) = \mu_0, \]

and thus when \(n\) is large

\[ Z = \frac{M-\mu_0}{s/\sqrt{n}} \]

has expected value zero, and its probability histogram is approximated well by the normal curve, so we can use \(Z\) as the Z statistic in a z test. If the alternative hypothesis is true, the expected value of \(Z\) could be either greater than zero or less than zero, so it is appropriate to use a two-tail z test. If the alternative hypothesis is \(\mu > \mu_0\) , then under the alternative hypothesis, the expected value of \(Z\) is greater than zero, and it is appropriate to use a right-tail z test. If the alternative hypothesis is \(\mu , then under the alternative hypothesis, the expected value of \(Z\) is less than zero, and it is appropriate to use a left-tail z test.

z Test for a Difference of Population Means

Consider the problem of testing the hypothesis that two population means are equal, using random samples from the two populations. Different sampling designs lead to different hypothesis testing procedures. In this section, we consider two kinds of random samples from the two populations: paired samples and independent samples , and construct z tests appropriate for each.

Paired Samples

Consider a population of \(N\) individuals, each of whom is labeled with two numbers. For example, the \(N\) individuals might be a group of doctors, and the two numbers that label each doctor might be the annual payments to the doctor by an HMO under the terms of the current contract and under the terms of a proposed revision of the contract. Let the two numbers associated with individual \(i\) be \(c_i\) and \(t_i\) . (Think of \(c\) as control and \(t\) as treatment . In this example, control is the current contract, and treatment is the proposed contract.) Let \(\mu_c\) be the population mean of the \(N\) values

\[ \{c_1, c_2, \ldots, c_N \}, \]

and let \(\mu_t\) be the population mean of the \(N\) values

\[ \{t_1, t_2, \ldots, t_N\}. \]

Suppose we want to test the null hypothesis that

\[ \mu = \mu_t - \mu_c = \mu_0 \]

against the alternative hypothesis that \(\mu . With \(\mu_0=\$0\) , this null hypothesis is that the average annual payment to doctors under the proposed revision would be the same as the average payment under the current contract, and the alternative is that on average doctors would be paid less under the new contract than under the current contract. With \(\mu_0=-\$5,000\) , this null hypothesis is that the proposed contract would save the HMO an average of $5,000 per doctor, compared with the current contract; the alternative is that under the proposed contract, the HMO would save even more than that. With \(\mu_0=\$1,000\) , this null hypothesis is that doctors would be paid an average of $1,000 more per year under the new contract than under the old one; the alternative hypothesis is that on average doctors would be paid less than an additional $1,000 per year under the new contract—perhaps even less than they are paid under the current contract. For the remainder of this example, we shall take \(\mu_0=\$1,000\) .

The data on which we shall base the test are observations of both \(c_i\) and \(t_i\) for a sample of \(n\) individuals chosen at random with replacement from the population of \(N\) individuals (or a simple random sample of size \(n ): We select \(n\) doctors at random from the \(N\) doctors under contract to the HMO, record the current annual payments to them, and calculate what the payments to them would be under the terms of the new contract. This is called a paired sample , because the samples from the population of control values and from the population of treatment values come in pairs: one value for control and one for treatment for each individual in the sample. Testing the hypothesis that the difference between two population means is equal to \(\mu_0\) using a paired sample is just the problem of testing the hypothesis that the population mean \(\mu\) of the set of differences

\[ d_i = t_i - c_i, \;\; i= 1, 2, \ldots, N, \]

is equal to \(\mu_0\) . Denote the \(n\) (random) observed values of \(c_i\) and \(t_i\) by \(\{C_1, C_2, \ldots, C_n\}\) and \(\{T_1, T_2, \ldots, T_n \}\) , respectively. The sample mean \(M\) of the differences between the observed values of \(t_i\) and \(c_i\) is the difference of the two sample means:

\[ M = \frac{(T_1-C_1)+(T_2-C_2) + \cdots + (T_n-C_n)}{n} = \frac{T_1+T_2+ \cdots + T_n}{n} - \frac{C_1+C_2+ \cdots + C_n}{n} \]

\[ = (\mbox{sample mean of observed values of } t_i) - (\mbox{sample mean of observed values of } c_i). \]

\(M\) is an unbiased estimator of \(\mu\) , and if n is large, the normal approximation to its probability histogram will be accurate. The SE of \(M\) is the population standard deviation of the \(N\) values \(\{d_1, d_2, \ldots, d_N\}\) , which we shall denote \(SD_d\) , divided by the square root of the sample size, \(n^{1/2}\) . Let \(sd\) denote the sample standard deviation of the \(n\) observed differences \((T_i - C_i), \;\; i=1, 2, \ldots, n\) :

\[ sd = \sqrt{\frac{(T_1-C_1-M)^2 + (T_2-C_2-M)^2 + \cdots + (T_n-C_n-M)^2}{n-1}} \]

(recall that \(M\) is the sample mean of the observed differences). If the sample size \(n\) is large, sd is very likely to be close to SD( d ), and so, under the null hypothesis,

\[ Z = \frac{M-\mu_0}{sd/n^{1/2}} \]

has expected value zero, and when \(n\) is large the probability histogram of \(Z\) can be approximated well by the normal curve. Thus we can use \(Z\) as the Z statistic in a z test of the null hypothesis that \(\mu=\mu_0\) . Under the alternative hypothesis that \(\mu (doctors on the average are paid less than an additional $1,000 per year under the new contract), the expected value of \(Z\) is less than zero, so we should use a left-tail z test. Under the alternative hypothesis \(\mu\ne\mu_0\) (on average, the difference in average annual payments to doctors is not an increase of $1,000, but some other number instead), the expected value of \(Z\) could be positive or negative, so we would use a two-tail z test. Under the alternative hypothesis that \(\mu>\mu_0\) (on average, under the new contract, doctors are paid more than an additional $1,000 per year), the expected value of \(Z\) would be greater than zero, so we should use a right-tail z test.

Independent Samples

Consider two separate populations of numbers, with population means \(\mu_t\) and \(\mu_c\) , respectively. Let \(\mu=\mu_t-\mu_c\) be the difference between the two population means. We would like to test the null hypothesis that \(\mu=\mu_0\) against the alternative hypothesis that \(\mu>0\) . For example, let \(\mu_t\) be the average annual payment by an HMO to doctors in the Los Angeles area, and let \(\mu_c\) be the average annual payment by the same HMO to doctors in the San Francisco area. Then the null hypothesis with \(\mu_0=0\) is that the HMO pays doctors in the two regions the same amount annually, on average; the alternative hypothesis is that the average annual payment by the HMO to doctors differs between the two areas. Suppose we draw a random sample of size \(n_t\) with replacement from the first population, and independently draw a random sample of size \(n_c\) with replacement from the second population. Let \(M_t\) and \(M_c\) be the sample means of the two samples, respectively, and let

\[ M = M_t - M_c \]

be the difference between the two sample means. Because the expected value of \(M_t\) is \(\mu_t\) and the expected value of \(M_c\) is \(\mu_c\) , the expected value of \(M\) is

\[ E(M) = E(M_t - M_c) = E(M_t) - E(M_c) = \mu_t - \mu_c = \mu. \]

Because the two random samples are independent , \(M_t\) and \(-M_c\) are independent random variables, and the SE of their sum is

\[ SE(M) = (SE^2(M_t) + SE^2(M_c))^{1/2}. \]

Let \(s_t\) and \(s_c\) be the sample standard deviations of the two samples, respectively. If \(n_t\) and \(n_c\) are both very large, the two sample standard deviations are likely to be close to the standard deviations of the corresponding populations, and so \(s_t/n_t^{1/2}\) is likely to be close to \(SE(M_t)\) , and \(s_c/n_c^{1/2}\) is likely to be close to \(SE(M_c)\) . Therefore, the pooled estimate of the standard error

\[ se_\mbox{diff} = ( (s_t/n_t^{1/2})^2 + (s_c/n_c^{1/2})^2)^{1/2} = \sqrt{ s_t^2/n_t + s_c^2/n_c} \]

is likely to be close to \(SE(M)\) . Under the null hypothesis, the statistic

\[ Z = \frac{M - \mu_0}{se_\mbox{diff}} = \frac{M_1 - M_2 - \mu_0}{\sqrt{ s_t^2/n_t + s_c^2/n_c}} \]

has expected value zero and its probability histogram is approximated well by the normal curve, so we can use it as the Z statistic in a z test.

Under the alternative hypothesis

\[ \mu = \mu_t - \mu_c > \mu_0, \]

the expected value of \(Z\) is greater than zero, so it is appropriate to use a right-tail z test.

If the alternative hypothesis were \(\mu \ne \mu_0\) , under the alternative the expected value of \(Z\) could be greater than zero or less than zero, so it would be appropriate to use a two-tail z test. If the alternative hypothesis were \(\mu , under the alternative the expected value of \(Z\) would be less than zero, so it would be appropriate to use a left-tail z test.

The following exercises check that you can compute the z test for a population mean or a difference of population means. The exercises are dynamic: the data will tend to change when you reload the page.

For the nominal significance level of the z test for a population mean to be approximately correct, the sample size typically must be large. When the sample size is small, two factors limit the accuracy of the z test: the normal approximation to the probability distribution of the sample mean can be poor, and the sample standard deviation can be an inaccurate estimate of the population standard deviation, so se is not an accurate estimate of the SE of the test statistic Z . For nearly normal populations , defined in the next subsection, the probability distribution of the sample mean is nearly normal even when the sample size is small, and the uncertainty of the sample standard deviation as an estimate of the population standard deviation can be accounted for by using a curve that is broader than the normal curve to approximate the probability distribution of the (approximately) standardized test statistic. The broader curve is Student's t curve . Student's t curve depends on the sample size: The smaller the sample size, the more spread out the curve.

Nearly Normally Distributed Populations

A list of numbers is nearly normally distributed if the fraction of values in any range is close to the area under the normal curve for the corresponding range of standard units—that is, if the list has mean \(\mu\) and standard deviation SD, and for every pair of values \(a < b\) ,

\[ \mbox{ the fraction of numbers in the list between } a \mbox{ and } b \approx \mbox{the area under the normal curve between } (a - \mu)/SD \mbox{ and } (b - \mu)/SD. \]

A list is nearly normally distributed if the normal curve is a good approximation to the histogram of the list transformed to standard units. The histogram of a list that is approximately normally distributed is (nearly) symmetric about some point, and is (nearly) bell-shaped.

No finite population can be exactly normally distributed, because the area under the normal curve between every two distinct values is strictly positive—no matter how large or small the values nor how close together they are. No population that contains only a finite number of distinct values can be exactly normally distributed, for the same reason. In particular, populations that contain only zeros and ones are not approximately normally distributed, so results for the sample mean of samples drawn from nearly normally distributed populations need not apply to the sample percentage of samples drawn from 0-1 boxes. Such results will be more accurate for the sample percentage when the population percentage is close to 50% than when the population percentage is close to 0% or 100%, because then the histogram of population values is more nearly symmetric.

Suppose a population is nearly normally distributed. Then a histogram of the population is approximately symmetric about the mean of the population. The fraction of numbers in the population within ±1 SD of the mean of the population is about 68%, the fraction of numbers within ±2 SD of the mean of the population is about 95%, and the fraction of numbers in the population within ±3 SD of the mean of the population is about 99.7%.

The following exercises check that you understand what it means for a list to be nearly normally distributed. The exercises are dynamic: the data tend to change when you reload the page.

Student's t -curve

Student's t curve is similar to the normal curve, but broader. It is positive, has a single maximum, and is symmetric about zero. The total area under Student's t curve is 100%. Student's t curve approximates some probability histograms more accurately than the normal curve does. There are actually infinitely many Student t curves, one for each positive integer value of the degrees of freedom. As the degrees of freedom increases, the difference between Student's t curve and the normal curve decreases.

Consider a population of \(N\) units labeled with numbers. Let \(\mu\) denote the population mean of the \(N\) numbers, and let SD denote the population standard deviation of the \(N\) numbers. Let \(M\) denote the sample mean of a random sample of size \(n\) drawn with replacement from a population, and let s> denote the sample standard deviation of the sample. The expected value of \(M\) is \(\mu\) , and the SE of \(M\) is \(SD/n^{1/2}\) . Let

\[ Z = (M - \mu)/(SD/n^{1/2}). \]

Then the expected value of \(Z\) is zero, the SE of \(Z\) is 1, and if \(n\) is large enough, the normal curve is a good approximation to the probability histogram of \(Z\) . The closer to normal the distribution of values in the population is, the smaller \(n\) needs to be for the normal curve to be a good approximation to the distribution of \(Z\) . Consider the statistic

\[ T = \frac{M - \mu}{s/n^{1/2}}, \]

which replaces SD by its estimated value (the sample standard deviation \(s\) ). If \(n\) is large enough, \(s\) is very likely to be close to SD, so \(T\) will be close to \(Z\) ; the normal curve will be a good approximation to the probability histogram of \(T\) ; and we can use \(T\) as the Z statistic in a z test of hypotheses about \(\mu\) .

For many populations, when the sample size is small—say less than 25, but the accuracy depends on the population—the normal curve is not a good approximation to the probability histogram of \(T\) . For nearly normally distributed populations, when the sample size is intermediate—say 25–100, but again this depends on the population—the normal curve is a good approximation to the probability histogram of \(Z\) , but not to the probability histogram of \(T\) , because of the variability of the sample standard deviation s> from sample to sample, which tends to broaden the probability distribution of \(T\) (i.e., to make \(SE(T)>1\) ).

When you first load this page, the degrees of freedom will be set to 25, and the region from -1.96 to 1.96 will be hilighted. The area under the normal curve between ±1.96 is 95%, but for Student's t curve with 25 degrees of freedom, the area is about 93.9%: Student's t curve with d.f.=25 is broader than the normal curve. Increase the degrees of freedom to 200; you will see that the Student t curve gets slightly narrower, and the area under the curve between ±1.96 is about 94.9%.

We define quantiles of Student t curves in the same way we defined quantiles of the normal curve: For any number a between 0 and 100%, the a quantile of Student's t curve with \(d.f.=d\) , \(t_{d,a}\) , is the unique value such that the area under the Student t curve with d degrees of freedom from minus infinity to \(t_{d,a}\) is equal to \(a\) . For example, \(t_{d,0.5} = 0\) for all values of \(d\) . Generally, the value of \(t_{d,a}\) depends on the degrees of freedom \(d\) . The probability calculator allows you to find quantiles of Student's t curve.

t test for the Mean of a Nearly Normally Distributed Population

We can use Student's t curve to construct approximate tests of hypotheses about the population mean \(\mu\) when the population standard deviation is unknown, for intermediate values of the sample size \(n\) . The approach is directly analogous to the z test, but instead of using a quantile of the normal curve, we use the corresponding quantile of Student's t curve (with the appropriate number of degrees of freedom). However, for the test to be accurate when \(n\) is small or intermediate, the distribution of values in the population must be nearly normal for the test to have approximately its nominal level. This is a somewhat bizarre restriction: It may require a very large sample to detect that the population is not nearly normal—but if the sample is very large, we can use the z test instead of the t test, so we don't need to rely as much on the assumption. It is my opinion that the t test is over-taught and overused—because its assumptions are not verifiable in the situations where it is potentially useful.

Consider testing the null hypothesis that \(\mu=\mu_0\) using the sample mean \(M\) and sample standard deviation s> of a random sample of size \(n\) drawn with replacement from a population that is known to have a nearly normal distribution. Define

\[ T = \frac{M - \mu_0}{s/n^{1/2}}. \]

Under the null hypothesis, if \(n\) is not too small, Student's t curve with \(n-1\) degrees of freedom will be an accurate approximation to the probability histogram of \(T\) , so

\[ P(T < t_{n-1,a}), \]

\[ P(T > t_{n-1,1-a}), \]

\[ P(|T| > t_{n-1,1-a/2}) \]

all are approximately equal to \(a\) . As we saw earlier in this chapter for the Z statistic, these three approximations give three tests of the null hypothesis \(\mu=\mu_0\) at approximate significance level \(a\) —a left-tail t test, a right-tail t test, and a two-tail t test:

• Reject the null hypothesis if \(T (left-tail)
• Reject the null hypothesis if \(T > t_{n-1,1-a}\) (right-tail)
• Reject the null hypothesis if \(|T| > t_{n-1,1-a/2}\) (two-tail)

To decide which t test to use, we can apply the same rule of thumb we used for the z test:

• Use a left-tail t test if, under the alternative hypothesis, the expected value of \(T\) is less than zero.
• Use a right-tail t test if, under the alternative hypothesis, the expected value of \(T\) is greater than zero.
• Use a two-tail t test if, under the alternative hypothesis, the expected value of \(T\) is not zero, but could be less than or greater than zero.
• Consult a statistician for a more appropriate test if, under the alternative hypothesis, the expected value of \(T\) is zero.

P-values for t tests are computed in much the same way as P-values for z tests. Let t be the observed value of \(T\) (the t score). In a left-tail t test, the P-value is the area under Student's t curve with \(n-1\) degrees of freedom, from minus infinity to \(t\) . In a right-tail t test, the P-value is the area under Student's t curve with \(n-1\) degrees of freedom, from \(t\) to infinity. In a two-tail t test, the P-value is the total area under Student's t curve with \(n-1\) degrees of freedom between minus infinity and \(-|t|\) and between \(|t|\) and infinity.

There are versions of the t test for comparing two means, as well. Just like for the z test, the method depends on how the samples from the two populations are drawn. For example, if the two samples are paired (if we are sampling individuals labeled with two numbers and for each individual in the sample, we observe both numbers), we may base the t test on the sample mean of the paired differences and the sample standard deviation of the paired differences. Let \(\mu_1\) and \(\mu_2\) be the means of the two populations, and let

\[ \mu = \mu_1 - \mu_2. \]

The \(T\) statistic to test the null hypothesis that \(\mu=\mu_0\) is

\[ T = \frac{(\mbox{sample mean of differences}) - \mu_0 }{(\mbox{sample standard deviation of differences})/n^{1/2}}, \]

and the appropriate curve to use to find the rejection region for the test is Student's t curve with \(n-1\) degrees of freedom, where \(n\) is the number of individuals (differences) in the sample.

Two-sample t tests for a difference of means using independent samples depend on additional assumptions, such as equality of the two population standard deviations; we shall not present such tests here. The following exercises check your ability to compute t tests. The exercises are dynamic: the data tend to change when you reload the page.

Hypothesis Tests and Confidence Intervals

There is a deep connection between hypothesis tests about parameters, and confidence intervals for parameters. If we have a procedure for constructing a level \(100\% \times (1-a)\) confidence interval for a parameter \(\mu\) , then the following rule is a two-sided significance level \(a\) test of the null hypothesis that \(\mu = \mu_0\) :

reject the null hypothesis if the confidence interval does not contain \(\mu_0\).

Similarly, suppose we have an hypothesis-testing procedure that lets us test the null hypothesis that \(\mu=\mu_0\) for any value of \(\mu_0\) , at significance level \(a\) . Define

\(A\) = (all values of \(\mu_0\) for which we would not reject the null hypothesis that \(\mu = \mu_0\)).

Then \(A\) is a \(100\% \times (1-a)\) confidence set for \(\mu\) :

\[ P( A \mbox{ contains the true value of } \mu ) = 100\% \times (1-a). \]

(A confidence set is a generalization of the idea of a confidence interval: a \(1-a\) confidence set for the parameter \(\mu\) is a random set that has probability \(1-a\) of containing \(\mu\) . As is the case with confidence intervals, the probability makes sense only before collecting the data.) The set \(A\) might or might not be an interval, depending on the nature of the test. If one starts with a two-tail z test or two-tail t test, one ends up with a confidence interval rather than a more general confidence set.

Confidence Intervals Using Student's t curve

The t test lets us test the hypothesis that the population mean \(\mu\) is equal to \(\mu_0\) at approximate significance level a using a random sample with replacement of size n from a population with a nearly normal distribution. If the sample size n is small, the actual significance level is likely to differ considerably from the nominal significance level. Consider a two-sided t test of the hypothesis \(\mu=\mu_0\) at significance level \(a\) . If the sample mean is \(M\) and the sample standard deviation is \(s\) , we would not reject the null hypothesis at significance level \(a\) if

\[ \frac{|M-\mu_0|}{s/n^{1/2}} \le t_{n-1,1-a/2}. \]

We rearrange this inequality:

\[ -t_{n-1,1-a/2} \le \frac{M-\mu_0}{s/n^{1/2}} \le t_{n-1,1-a/2} \]

\[ -t_{n-1,1-a/2} \times s/n^{1/2} \le M - \mu_0 \le t_{n-1,1-a/2} \times s/n^{1/2} \]

\[ -M - t_{n-1,1-a/2} \times s/n^{1/2} \le - \mu_0 \le -M + t_{n-1,1-a/2} \times s/n^{1/2} \]

\[ M + t_{n-1,1-a/2} \times s/n^{1/2} \le \mu_0 \le M - t_{n-1,1-a/2} \times s/n^{1/2} \]

That is, we would not reject the hypothesis \(\mu = \mu_0\) provided \(\mu_0\) is in the interval

\[ [M - t_{n-1,1-a/2} \times s/n^{1/2}, M + t_{n-1,1-a/2} \times s/n^{1/2}]. \]

Therefore, that interval is a \(100\%-a\) confidence interval for \(\mu\) :

\[ P([M - t_{n-1,1-a/2} \times s/n^{1/2}, M + t_{n-1,1-a/2} \times s/n^{1/2}] \mbox{ contains } \mu) \approx 1-a. \]

The following exercise checks that you can use Student's t curve to construct a confidence interval for a population mean. The exercise is dynamic: the data tend to change when you reload the page.

In hypothesis testing, a Z statistic is a random variable whose probability histogram is approximated well by the normal curve if the null hypothesis is correct: If the null hypothesis is true, the expected value of a Z statistic is zero, the SE of a Z statistic is approximately 1, and the probability that a Z statistic is between \(a\) and \(b\) is approximately the area under the normal curve between \(a\) and \(b\) . Suppose that the random variable \(Z\) is a Z statistic. If, under the alternative hypothesis, \(E(Z) , the appropriate z test to test the null hypothesis at approximate significance level \(a\) is the left-tailed z test: Reject the null hypothesis if \(Z , where \(z_a\) is the \(a\) quantile of the normal curve. If, under the alternative hypothesis, \(E(Z)>0\) , the appropriate z test to test the null hypothesis at approximate significance level \(a\) is the right-tailed z test: Reject the null hypothesis if \(Z>z_{1-a}\) . If, under the alternative hypothesis, \(E(Z)\ne 0 \) but could be greater than 0 or less than 0, the appropriate z test to test the null hypothesis at approximate significance level \(a\) is the two-tailed z test: reject the null hypothesis if \(|Z|>z_{1-a/2}\) . If, under the alternative hypothesis, \(E(Z)=0\) , a z test probably is not appropriate—consult a statistician. The exact significance levels of these tests differ from \(a\) by an amount that depends on how closely the normal curve approximates the probability histogram of \(Z\) .

Z statistics often are constructed from other statistics by transforming approximately to standard units, which requires knowing the expected value and SE of the original statistic on the assumption that the null hypothesis is true. Let \(X\) be a test statistic; let \(E(X)\) be the expected value of \(X\) if the null hypothesis is true, and let \(se\) be approximately equal to the SE of \(X\) if the null hypothesis is true. If \(X\) is a sample sum of a large random sample with replacement, a sample mean of a large random sample with replacement, or a sum or difference of independent sample means of large samples with replacement,

\[ Z = \frac{X-E(X)}{se} \]

is a Z statistic.

Consider testing the null hypothesis that a population percentage \(p\) is equal to the value \(p_0\) on the basis of the sample percentage \phi of a random sample of size \(n\) with replacement. Under the null hypothesis, \(E(\phi)=p_0\) and

\[ SE(\phi) = \sqrt{\frac{p_0\times(1-p_0)}{n}}, \]

and if \(n\) is sufficiently large (say \(n \times p > 30\) and \(n \times (1-p)>30\) , but this depends on the desired accuracy), the normal approximation to

\[ Z = \frac{\phi-p_0}{\sqrt{(p_0 \times (1-p_0))/n}} \]

will be reasonably accurate, so \(Z\) can be used as the Z statistic in a z test of the null hypothesis \(p=p_0\) .

Consider testing the null hypothesis that a population mean \(\mu\) is equal to the value \(\mu_0\) , on the basis of the sample mean \(M\) of a random sample of size \(n\) with replacement. Let \(s\) denote the sample standard deviation. Under the null hypothesis, \(E(M)=\mu_0\) , and if \(n\) is large,

\[ SE(M)=SD/n^{1/2} \approx s/n^{1/2}, \]

and the normal approximation to

\[ Z = \frac{M-\mu_0}{s/n^{1/2}} \]

will be reasonably accurate, so \(Z\) can be used as the Z statistic in a z test of the null hypothesis \(\mu=\mu_0\) .

Consider a population of \(N\) individuals, each labeled with two numbers. The \(i\) th individual is labeled with the numbers \(c_i\) and \(t_i\) , \(i=1, 2, \ldots, N\) . Let \(\mu_c\) be the population mean of the \(N\) values \(\{c_1, \ldots, c_N\}\) and let \(\mu_t\) be the population mean of the \(N\) values \(\{t_1, \ldots, t_N \}\) . Let \(\mu=\mu_t-\mu_c\) be the difference between the two population means. Consider testing the null hypothesis that \(\mu=\mu_0\) on the basis of a paired random sample of size \(n\) with replacement from the population: that is, a random sample of size \(n\) is drawn with replacement from the population, and for each individual \(i\) in the sample, \(c_i\) and \(t_i\) are observed. This is equivalent to testing the hypothesis that the population mean of the \(N\) values \(\{(t_1-c_1), \ldots, (t_N-c_N)\}\) is equal to \(\mu_0\) , on the basis of the random sample of size \(n\) drawn with replacement from those \(N\) values. Let \(M_t\) be the sample mean of the \(n\) observed values of \(t_i\) and let \(M_c\) be the sample mean of the \(n\) observed values of \(c_i\) . Let \(sd\) denote the sample standard deviation of the \(n\) observed differences \(\{(t_i-c_i)\}\) . Under the null hypothesis, the expected value of \(M_t-M_c\) is \(\mu_0\) , and if \(n\) is large,

\[ SE(M_t-M_c) \approx sd/n^{1/2}, \]

and the normal approximation to the probability histogram of

\[ Z = \frac{M_t-M_c-\mu_0}{sd/n^{1/2}} \]

will be reasonably accurate, so \(Z\) can be used as the Z statistic in a z test of the null hypothesis that \(\mu_t-\mu_c=\mu_0\) .

Consider testing the hypothesis that the difference ( \(\mu_t-\mu_c\) ) between two population means, \(\mu_c\) and \(\mu_t\) , is equal to \(\mu_0\) , on the basis of the difference ( \(M_t-M_c\) ) between the sample mean \(M_c\) of a random sample of size \(n_c\) with replacement from the first population and the sample mean \(M_t\) of an independent random sample of size \(n_t\) with replacement from the second population. Let \(s_c\) denote the sample standard deviation of the sample of size \(n_c\) from the first population and let \(s_t\) denote the sample standard deviation of the sample of size \(n_t\) from the second population. If the null hypothesis is true,

\[ E(M_t-M_c)=\mu_0, \]

and if \(n_c\) and \(n_t\) are both large,

\[ SE(M_t-M_c) \approx \sqrt{s_t^2/n_t + s_c^2/n_c} \]

\[ Z = \frac{M_t-M_c-\mu_0}{\sqrt{s_t^2/n_t + s_c^2/n_c}} \]

A list of numbers is nearly normally distributed if the fraction of numbers between any pair of values, \(a , is approximately equal to the area under the normal curve between \((a-\mu)/SD\) and \((b-\mu)/SD\) , where \(\mu\) is the mean of the list and SD is the standard deviation of the list.

Student's t curve with \(d\) degrees of freedom is symmetric about 0, has a single bump centered at 0, and is broader and flatter than the normal curve. The total area under Student's t curve is 1, no matter what \(d\) is; as \(d\) increases, Student's t curve gets narrower, its peak gets higher, and it becomes closer and closer to the normal curve.

Let \(M\) be the sample mean of a random sample of size \(n\) with replacement from a population with mean \(\mu\) and a nearly normal distribution, and let \(s\) be the sample standard deviation of the random sample. For moderate values of \(n\) ( \(n or so), Student's t curve approximates the probability histogram of \((M-\mu)/(s/n^{1/2})\) better than the normal curve does, which can lead to an approximate hypothesis test about \(\mu\) that is more accurate than the z test.

Consider testing the null hypothesis that the mean \(\mu\) of a population with a nearly normal distribution is equal to \(\mu_0\) from a random sample of size \(n\) with replacement. Let

\[ T=\frac{M-\mu_0}{s/n^{1/2}}, \]

where \(M\) is the sample mean and \(s\) is the sample standard deviation. The tests that reject the null hypothesis if \(T (left-tail t test), if \(T>t_{n-1,1-a}\) (right-tail t test), or if \(|T|>t_{n-1,1-a/2}\) (two-tail t test) all have approximate significance level \(a\) . How close the nominal significance level \(a\) is to the true significance level depends on the distribution of the numbers in the population, the sample size \(n\) , and \(a\) . The same rule of thumb for selecting whether to use a left, right, or two-tailed z test (or not to use a z test at all) works to select whether to use a left, right, or two-tailed t test: If, under the alternative hypothesis, \(E(T) , use a left-tail test. If, under the alternative hypothesis, \(E(T) > 0 \) , use a right-tail test. If, under the alternative hypothesis, \(E(T)\) could be less than zero or greater than zero, use a two-tail test. If, under the alternative hypothesis, \(E(T) = 0 \) , consult an expert. Because the t test differs from the z test only when the sample size is small, and from a small sample it is not possible to tell whether the population has a nearly normal distribution, the t test should be used with caution.

A \(1-a\) confidence set for a parameter \(\mu\) is like a \(1-a\) confidence interval for a parameter \(\mu\) : It is a random set of values that has probability \(1-a\) of containing the true value of \(\mu\) . The difference is that the set need not be an interval.

There is a deep duality between hypothesis tests about a parameter \(\mu\) and confidence sets for \(\mu\) . Given a procedure for constructing a \(1-a\) confidence set for \(\mu\) , the rule reject the null hypothesis that \(\mu=\mu_0\) if the confidence set does not contain \(\mu\) is a significance level \(a\) test of the null hypothesis that \(\mu=\mu_0\) . Conversely, given a family of significance level \(a\) hypothesis tests that allow one to test the hypothesis that \(\mu=\mu_0\) for any value of \(\mu_0\) , the set of all values \(\mu_0\) for which the test does not reject the null hypothesis that \(\mu=\mu_0\) is a \(1-a\) confidence set for \(\mu\) .

• alternative hypothesis
• central limit theorem
• confidence interval
• confidence set
• expected value
• independent
• independent random variable
• mutatis mutandis
• nearly normal distribution
• normal approximation
• normal curve
• null hypothesis
• pooled bootstrap estimate of the population SD
• pooled bootstrap estimate of the SE
• population mean
• population percentage
• population standard deviation
• probability
• probability distribution
• probability histogram
• random sample
• random variable
• rejection region
• sample mean
• sample percentage
• sample size
• sample standard deviation
• significance level
• simple random sample
• standard deviation (SD)
• standard error (SE)
• standard unit
• Student's t curve
• test statistic
• two-tailed test
• Type I error
• Z statistic

• Quality Improvement
• Talk To Minitab

Understanding t-Tests: t-values and t-distributions

Topics: Hypothesis Testing , Data Analysis

T-tests are handy hypothesis tests in statistics when you want to compare means. You can compare a sample mean to a hypothesized or target value using a one-sample t-test. You can compare the means of two groups with a two-sample t-test. If you have two groups with paired observations (e.g., before and after measurements), use the paired t-test.

How do t-tests work? How do t-values fit in? In this series of posts, I’ll answer these questions by focusing on concepts and graphs rather than equations and numbers. After all, a key reason to use statistical software like Minitab  is so you don’t get bogged down in the calculations and can instead focus on understanding your results.

In this post, I will explain t-values, t-distributions, and how t-tests use them to calculate probabilities and assess hypotheses.

What Are t-Values?

T-tests are called t-tests because the test results are all based on t-values. T-values are an example of what statisticians call test statistics. A test statistic is a standardized value that is calculated from sample data during a hypothesis test. The procedure that calculates the test statistic compares your data to what is expected under the null hypothesis .

Each type of t-test uses a specific procedure to boil all of your sample data down to one value, the t-value. The calculations behind t-values compare your sample mean(s) to the null hypothesis and incorporates both the sample size and the variability in the data. A t-value of 0 indicates that the sample results exactly equal the null hypothesis. As the difference between the sample data and the null hypothesis increases, the absolute value of the t-value increases.

Assume that we perform a t-test and it calculates a t-value of 2 for our sample data. What does that even mean? I might as well have told you that our data equal 2 fizbins! We don’t know if that’s common or rare when the null hypothesis is true.

By itself, a t-value of 2 doesn’t really tell us anything. T-values are not in the units of the original data, or anything else we’d be familiar with. We need a larger context in which we can place individual t-values before we can interpret them. This is where t-distributions come in.

What Are t-Distributions?

When you perform a t-test for a single study, you obtain a single t-value. However, if we drew multiple random samples of the same size from the same population and performed the same t-test, we would obtain many t-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution .

Fortunately, the properties of t-distributions are well understood in statistics, so we can plot them without having to collect many samples! A specific t-distribution is defined by its degrees of freedom (DF) , a value closely related to sample size. Therefore, different t-distributions exist for every sample size.  You can graph t-distributions u sing Minitab’s probability distribution plots .

T-distributions assume that you draw repeated random samples from a population where the null hypothesis is true. You place the t-value from your study in the t-distribution to determine how consistent your results are with the null hypothesis.

The graph above shows a t-distribution that has 20 degrees of freedom, which corresponds to a sample size of 21 in a one-sample t-test. It is a symmetric, bell-shaped distribution that is similar to the normal distribution, but with thicker tails. This graph plots the probability density function (PDF), which describes the likelihood of each t-value.

The peak of the graph is right at zero, which indicates that obtaining a sample value close to the null hypothesis is the most likely. That makes sense because t-distributions assume that the null hypothesis is true. T-values become less likely as you get further away from zero in either direction. In other words, when the null hypothesis is true, you are less likely to obtain a sample that is very different from the null hypothesis.

Our t-value of 2 indicates a positive difference between our sample data and the null hypothesis. The graph shows that there is a reasonable probability of obtaining a t-value from -2 to +2 when the null hypothesis is true. Our t-value of 2 is an unusual value, but we don’t know exactly how unusual. Our ultimate goal is to determine whether our t-value is unusual enough to warrant rejecting the null hypothesis. To do that, we'll need to calculate the probability.

Ready for a demo of Minitab Statistical Software? Just ask! 

Using t-Values and t-Distributions to Calculate Probabilities

The foundation behind any hypothesis test is being able to take the test statistic from a specific sample and place it within the context of a known probability distribution. For t-tests, if you take a t-value and place it in the context of the correct t-distribution, you can calculate the probabilities associated with that t-value.

A probability allows us to determine how common or rare our t-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that the effect observed in our sample is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

Before we calculate the probability associated with our t-value of 2, there are two important details to address.

First, we’ll actually use the t-values of +2 and -2 because we’ll perform a two-tailed test. A two-tailed test is one that can test for differences in both directions. For example, a two-tailed 2-sample t-test can determine whether the difference between group 1 and group 2 is statistically significant in either the positive or negative direction. A one-tailed test can only assess one of those directions.

Second, we can only calculate a non-zero probability for a range of t-values. As you’ll see in the graph below, a range of t-values corresponds to a proportion of the total area under the distribution curve, which is the probability. The probability for any specific point value is zero because it does not produce an area under the curve.

With these points in mind, we’ll shade the area of the curve that has t-values greater than 2 and t-values less than -2.

The graph displays the probability for observing a difference from the null hypothesis that is at least as extreme as the difference present in our sample data while assuming that the null hypothesis is actually true. Each of the shaded regions has a probability of 0.02963, which sums to a total probability of 0.05926. When the null hypothesis is true, the t-value falls within these regions nearly 6% of the time.

This probability has a name that you might have heard of—it’s called the p-value!  While the probability of our t-value falling within these regions is fairly low, it’s not low enough to reject the null hypothesis using the common significance level of 0.05.

Learn how to correctly interpret the p-value.

t-Distributions and Sample Size

As mentioned above, t-distributions are defined by the DF, which are closely associated with sample size. As the DF increases, the probability density in the tails decreases and the distribution becomes more tightly clustered around the central value. The graph below depicts t-distributions with 5 and 30 degrees of freedom.

The t-distribution with fewer degrees of freedom has thicker tails. This occurs because the t-distribution is designed to reflect the added uncertainty associated with analyzing small samples. In other words, if you have a small sample, the probability that the sample statistic will be further away from the null hypothesis is greater even when the null hypothesis is true.

Small samples are more likely to be unusual. This affects the probability associated with any given t-value. For 5 and 30 degrees of freedom, a t-value of 2 in a two-tailed test has p-values of 10.2% and 5.4%, respectively. Large samples are better!

I’ve showed how t-values and t-distributions work together to produce probabilities. To see how each type of t-test works and actually calculates the t-values, read the other post in this series, Understanding t-Tests: 1-sample, 2-sample, and Paired t-Tests .

If you'd like to learn how the ANOVA F-test works, read my post, Understanding Analysis of Variance (ANOVA) and the F-test .

You Might Also Like

• Trust Center

© 2023 Minitab, LLC. All Rights Reserved.

• Terms of Use
• Privacy Policy
• Cookies Settings

t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

.css-slt4t3.css-slt4t3{color:#2B3148;background-color:transparent;font-family:"Roboto","Helvetica","Arial",sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-slt4t3.css-slt4t3:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-slt4t3 .js-external-link-button.link-like,.css-slt4t3 .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-slt4t3 .js-external-link-button.link-like:hover,.css-slt4t3 .js-external-link-anchor:hover,.css-slt4t3 .js-external-link-button.link-like:active,.css-slt4t3 .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-slt4t3 .js-external-link-button.link-like:focus-visible,.css-slt4t3 .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-slt4t3 p,.css-slt4t3 div{margin:0px;display:block;}.css-slt4t3 pre{margin:0px;display:block;}.css-slt4t3 pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-slt4t3 pre:not(:first-child){padding-top:8px;}.css-slt4t3 ul,.css-slt4t3 ol{display:block margin:0px;padding-left:20px;}.css-slt4t3 ul li,.css-slt4t3 ol li{padding-top:8px;}.css-slt4t3 ul ul,.css-slt4t3 ol ul,.css-slt4t3 ul ol,.css-slt4t3 ol ol{padding-top:0px;}.css-slt4t3 ul:not(:first-child),.css-slt4t3 ol:not(:first-child){padding-top:4px;} .css-4okk7a{margin:auto;background-color:white;overflow:auto;overflow-wrap:break-word;word-break:break-word;}.css-4okk7a code,.css-4okk7a kbd,.css-4okk7a pre,.css-4okk7a samp{font-family:monospace;}.css-4okk7a code{padding:2px 4px;color:#444;background:#ddd;border-radius:4px;}.css-4okk7a figcaption,.css-4okk7a caption{text-align:center;}.css-4okk7a figcaption{font-size:12px;font-style:italic;overflow:hidden;}.css-4okk7a h3{font-size:1.75rem;}.css-4okk7a h4{font-size:1.5rem;}.css-4okk7a .mathBlock{font-size:24px;-webkit-padding-start:4px;padding-inline-start:4px;}.css-4okk7a .mathBlock .katex{font-size:24px;text-align:left;}.css-4okk7a .math-inline{background-color:#f0f0f0;display:inline-block;font-size:inherit;padding:0 3px;}.css-4okk7a .videoBlock,.css-4okk7a .imageBlock{margin-bottom:16px;}.css-4okk7a .imageBlock__image-align--left,.css-4okk7a .videoBlock__video-align--left{float:left;}.css-4okk7a .imageBlock__image-align--right,.css-4okk7a .videoBlock__video-align--right{float:right;}.css-4okk7a .imageBlock__image-align--center,.css-4okk7a .videoBlock__video-align--center{display:block;margin-left:auto;margin-right:auto;clear:both;}.css-4okk7a .imageBlock__image-align--none,.css-4okk7a .videoBlock__video-align--none{clear:both;margin-left:0;margin-right:0;}.css-4okk7a .videoBlock__video--wrapper{position:relative;padding-bottom:56.25%;height:0;}.css-4okk7a .videoBlock__video--wrapper iframe{position:absolute;top:0;left:0;width:100%;height:100%;}.css-4okk7a .videoBlock__caption{text-align:left;}@font-face{font-family:'KaTeX_AMS';src:url(/katex-fonts/KaTeX_AMS-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_AMS-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_AMS-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Script';src:url(/katex-fonts/KaTeX_Script-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Script-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Script-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size1';src:url(/katex-fonts/KaTeX_Size1-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size1-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size1-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size2';src:url(/katex-fonts/KaTeX_Size2-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size2-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size2-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size3';src:url(/katex-fonts/KaTeX_Size3-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size3-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size3-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size4';src:url(/katex-fonts/KaTeX_Size4-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size4-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size4-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Typewriter';src:url(/katex-fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Typewriter-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}.css-4okk7a .katex{font:normal 1.21em KaTeX_Main,Times New Roman,serif;line-height:1.2;text-indent:0;text-rendering:auto;}.css-4okk7a .katex *{-ms-high-contrast-adjust:none!important;border-color:currentColor;}.css-4okk7a .katex .katex-version::after{content:'0.13.13';}.css-4okk7a .katex .katex-mathml{position:absolute;clip:rect(1px, 1px, 1px, 1px);padding:0;border:0;height:1px;width:1px;overflow:hidden;}.css-4okk7a .katex .katex-html>.newline{display:block;}.css-4okk7a .katex .base{position:relative;display:inline-block;white-space:nowrap;width:-webkit-min-content;width:-moz-min-content;width:-webkit-min-content;width:-moz-min-content;width:min-content;}.css-4okk7a .katex .strut{display:inline-block;}.css-4okk7a .katex .textbf{font-weight:bold;}.css-4okk7a .katex .textit{font-style:italic;}.css-4okk7a .katex .textrm{font-family:KaTeX_Main;}.css-4okk7a .katex .textsf{font-family:KaTeX_SansSerif;}.css-4okk7a .katex .texttt{font-family:KaTeX_Typewriter;}.css-4okk7a .katex .mathnormal{font-family:KaTeX_Math;font-style:italic;}.css-4okk7a .katex .mathit{font-family:KaTeX_Main;font-style:italic;}.css-4okk7a .katex .mathrm{font-style:normal;}.css-4okk7a .katex .mathbf{font-family:KaTeX_Main;font-weight:bold;}.css-4okk7a .katex .boldsymbol{font-family:KaTeX_Math;font-weight:bold;font-style:italic;}.css-4okk7a .katex .amsrm{font-family:KaTeX_AMS;}.css-4okk7a .katex .mathbb,.css-4okk7a .katex .textbb{font-family:KaTeX_AMS;}.css-4okk7a .katex .mathcal{font-family:KaTeX_Caligraphic;}.css-4okk7a .katex .mathfrak,.css-4okk7a .katex .textfrak{font-family:KaTeX_Fraktur;}.css-4okk7a .katex .mathtt{font-family:KaTeX_Typewriter;}.css-4okk7a .katex .mathscr,.css-4okk7a .katex .textscr{font-family:KaTeX_Script;}.css-4okk7a .katex .mathsf,.css-4okk7a .katex .textsf{font-family:KaTeX_SansSerif;}.css-4okk7a .katex .mathboldsf,.css-4okk7a .katex .textboldsf{font-family:KaTeX_SansSerif;font-weight:bold;}.css-4okk7a .katex .mathitsf,.css-4okk7a .katex .textitsf{font-family:KaTeX_SansSerif;font-style:italic;}.css-4okk7a .katex .mainrm{font-family:KaTeX_Main;font-style:normal;}.css-4okk7a .katex .vlist-t{display:inline-table;table-layout:fixed;border-collapse:collapse;}.css-4okk7a .katex .vlist-r{display:table-row;}.css-4okk7a .katex .vlist{display:table-cell;vertical-align:bottom;position:relative;}.css-4okk7a .katex .vlist>span{display:block;height:0;position:relative;}.css-4okk7a .katex .vlist>span>span{display:inline-block;}.css-4okk7a .katex .vlist>span>.pstrut{overflow:hidden;width:0;}.css-4okk7a .katex .vlist-t2{margin-right:-2px;}.css-4okk7a .katex .vlist-s{display:table-cell;vertical-align:bottom;font-size:1px;width:2px;min-width:2px;}.css-4okk7a .katex .vbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-align-items:baseline;-webkit-box-align:baseline;-ms-flex-align:baseline;align-items:baseline;}.css-4okk7a .katex .hbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:100%;}.css-4okk7a .katex .thinbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:0;max-width:0;}.css-4okk7a .katex .msupsub{text-align:left;}.css-4okk7a .katex .mfrac>span>span{text-align:center;}.css-4okk7a .katex .mfrac .frac-line{display:inline-block;width:100%;border-bottom-style:solid;}.css-4okk7a .katex .mfrac .frac-line,.css-4okk7a .katex .overline .overline-line,.css-4okk7a .katex .underline .underline-line,.css-4okk7a .katex .hline,.css-4okk7a .katex .hdashline,.css-4okk7a .katex .rule{min-height:1px;}.css-4okk7a .katex .mspace{display:inline-block;}.css-4okk7a .katex .llap,.css-4okk7a .katex .rlap,.css-4okk7a .katex .clap{width:0;position:relative;}.css-4okk7a .katex .llap>.inner,.css-4okk7a .katex .rlap>.inner,.css-4okk7a .katex .clap>.inner{position:absolute;}.css-4okk7a .katex .llap>.fix,.css-4okk7a .katex .rlap>.fix,.css-4okk7a .katex .clap>.fix{display:inline-block;}.css-4okk7a .katex .llap>.inner{right:0;}.css-4okk7a .katex .rlap>.inner,.css-4okk7a .katex .clap>.inner{left:0;}.css-4okk7a .katex .clap>.inner>span{margin-left:-50%;margin-right:50%;}.css-4okk7a .katex .rule{display:inline-block;border:solid 0;position:relative;}.css-4okk7a .katex .overline .overline-line,.css-4okk7a .katex .underline .underline-line,.css-4okk7a .katex .hline{display:inline-block;width:100%;border-bottom-style:solid;}.css-4okk7a .katex .hdashline{display:inline-block;width:100%;border-bottom-style:dashed;}.css-4okk7a .katex .sqrt>.root{margin-left:0.27777778em;margin-right:-0.55555556em;}.css-4okk7a .katex .sizing.reset-size1.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size1{font-size:1em;}.css-4okk7a .katex .sizing.reset-size1.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size2{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size1.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size3{font-size:1.4em;}.css-4okk7a .katex .sizing.reset-size1.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size4{font-size:1.6em;}.css-4okk7a .katex .sizing.reset-size1.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size5{font-size:1.8em;}.css-4okk7a .katex .sizing.reset-size1.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size6{font-size:2em;}.css-4okk7a .katex .sizing.reset-size1.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size7{font-size:2.4em;}.css-4okk7a .katex .sizing.reset-size1.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size8{font-size:2.88em;}.css-4okk7a .katex .sizing.reset-size1.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size9{font-size:3.456em;}.css-4okk7a .katex .sizing.reset-size1.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size10{font-size:4.148em;}.css-4okk7a .katex .sizing.reset-size1.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size1.size11{font-size:4.976em;}.css-4okk7a .katex .sizing.reset-size2.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size1{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size2.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size2{font-size:1em;}.css-4okk7a .katex .sizing.reset-size2.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size3{font-size:1.16666667em;}.css-4okk7a .katex .sizing.reset-size2.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size4{font-size:1.33333333em;}.css-4okk7a .katex .sizing.reset-size2.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size5{font-size:1.5em;}.css-4okk7a .katex .sizing.reset-size2.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size6{font-size:1.66666667em;}.css-4okk7a .katex .sizing.reset-size2.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size7{font-size:2em;}.css-4okk7a .katex .sizing.reset-size2.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size8{font-size:2.4em;}.css-4okk7a .katex .sizing.reset-size2.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size9{font-size:2.88em;}.css-4okk7a .katex .sizing.reset-size2.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size10{font-size:3.45666667em;}.css-4okk7a .katex .sizing.reset-size2.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size2.size11{font-size:4.14666667em;}.css-4okk7a .katex .sizing.reset-size3.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size1{font-size:0.71428571em;}.css-4okk7a .katex .sizing.reset-size3.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size2{font-size:0.85714286em;}.css-4okk7a .katex .sizing.reset-size3.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size3{font-size:1em;}.css-4okk7a .katex .sizing.reset-size3.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size4{font-size:1.14285714em;}.css-4okk7a .katex .sizing.reset-size3.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size5{font-size:1.28571429em;}.css-4okk7a .katex .sizing.reset-size3.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size6{font-size:1.42857143em;}.css-4okk7a .katex .sizing.reset-size3.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size7{font-size:1.71428571em;}.css-4okk7a .katex .sizing.reset-size3.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size8{font-size:2.05714286em;}.css-4okk7a .katex .sizing.reset-size3.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size9{font-size:2.46857143em;}.css-4okk7a .katex .sizing.reset-size3.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size10{font-size:2.96285714em;}.css-4okk7a .katex .sizing.reset-size3.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size3.size11{font-size:3.55428571em;}.css-4okk7a .katex .sizing.reset-size4.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size1{font-size:0.625em;}.css-4okk7a .katex .sizing.reset-size4.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size2{font-size:0.75em;}.css-4okk7a .katex .sizing.reset-size4.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size3{font-size:0.875em;}.css-4okk7a .katex .sizing.reset-size4.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size4{font-size:1em;}.css-4okk7a .katex .sizing.reset-size4.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size5{font-size:1.125em;}.css-4okk7a .katex .sizing.reset-size4.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size6{font-size:1.25em;}.css-4okk7a .katex .sizing.reset-size4.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size7{font-size:1.5em;}.css-4okk7a .katex .sizing.reset-size4.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size8{font-size:1.8em;}.css-4okk7a .katex .sizing.reset-size4.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size9{font-size:2.16em;}.css-4okk7a .katex .sizing.reset-size4.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size10{font-size:2.5925em;}.css-4okk7a .katex .sizing.reset-size4.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size4.size11{font-size:3.11em;}.css-4okk7a .katex .sizing.reset-size5.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size1{font-size:0.55555556em;}.css-4okk7a .katex .sizing.reset-size5.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size2{font-size:0.66666667em;}.css-4okk7a .katex .sizing.reset-size5.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size3{font-size:0.77777778em;}.css-4okk7a .katex .sizing.reset-size5.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size4{font-size:0.88888889em;}.css-4okk7a .katex .sizing.reset-size5.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size5{font-size:1em;}.css-4okk7a .katex .sizing.reset-size5.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size6{font-size:1.11111111em;}.css-4okk7a .katex .sizing.reset-size5.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size7{font-size:1.33333333em;}.css-4okk7a .katex .sizing.reset-size5.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size8{font-size:1.6em;}.css-4okk7a .katex .sizing.reset-size5.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size9{font-size:1.92em;}.css-4okk7a .katex .sizing.reset-size5.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size10{font-size:2.30444444em;}.css-4okk7a .katex .sizing.reset-size5.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size5.size11{font-size:2.76444444em;}.css-4okk7a .katex .sizing.reset-size6.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size1{font-size:0.5em;}.css-4okk7a .katex .sizing.reset-size6.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size2{font-size:0.6em;}.css-4okk7a .katex .sizing.reset-size6.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size3{font-size:0.7em;}.css-4okk7a .katex .sizing.reset-size6.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size4{font-size:0.8em;}.css-4okk7a .katex .sizing.reset-size6.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size5{font-size:0.9em;}.css-4okk7a .katex .sizing.reset-size6.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size6{font-size:1em;}.css-4okk7a .katex .sizing.reset-size6.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size7{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size6.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size8{font-size:1.44em;}.css-4okk7a .katex .sizing.reset-size6.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size9{font-size:1.728em;}.css-4okk7a .katex .sizing.reset-size6.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size10{font-size:2.074em;}.css-4okk7a .katex .sizing.reset-size6.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size6.size11{font-size:2.488em;}.css-4okk7a .katex .sizing.reset-size7.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size1{font-size:0.41666667em;}.css-4okk7a .katex .sizing.reset-size7.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size2{font-size:0.5em;}.css-4okk7a .katex .sizing.reset-size7.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size3{font-size:0.58333333em;}.css-4okk7a .katex .sizing.reset-size7.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size4{font-size:0.66666667em;}.css-4okk7a .katex .sizing.reset-size7.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size5{font-size:0.75em;}.css-4okk7a .katex .sizing.reset-size7.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size6{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size7.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size7{font-size:1em;}.css-4okk7a .katex .sizing.reset-size7.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size8{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size7.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size9{font-size:1.44em;}.css-4okk7a .katex .sizing.reset-size7.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size10{font-size:1.72833333em;}.css-4okk7a .katex .sizing.reset-size7.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size7.size11{font-size:2.07333333em;}.css-4okk7a .katex .sizing.reset-size8.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size1{font-size:0.34722222em;}.css-4okk7a .katex .sizing.reset-size8.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size2{font-size:0.41666667em;}.css-4okk7a .katex .sizing.reset-size8.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size3{font-size:0.48611111em;}.css-4okk7a .katex .sizing.reset-size8.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size4{font-size:0.55555556em;}.css-4okk7a .katex .sizing.reset-size8.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size5{font-size:0.625em;}.css-4okk7a .katex .sizing.reset-size8.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size6{font-size:0.69444444em;}.css-4okk7a .katex .sizing.reset-size8.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size7{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size8.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size8{font-size:1em;}.css-4okk7a .katex .sizing.reset-size8.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size9{font-size:1.2em;}.css-4okk7a .katex .sizing.reset-size8.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size10{font-size:1.44027778em;}.css-4okk7a .katex .sizing.reset-size8.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size8.size11{font-size:1.72777778em;}.css-4okk7a .katex .sizing.reset-size9.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size1{font-size:0.28935185em;}.css-4okk7a .katex .sizing.reset-size9.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size2{font-size:0.34722222em;}.css-4okk7a .katex .sizing.reset-size9.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size3{font-size:0.40509259em;}.css-4okk7a .katex .sizing.reset-size9.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size4{font-size:0.46296296em;}.css-4okk7a .katex .sizing.reset-size9.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size5{font-size:0.52083333em;}.css-4okk7a .katex .sizing.reset-size9.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size6{font-size:0.5787037em;}.css-4okk7a .katex .sizing.reset-size9.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size7{font-size:0.69444444em;}.css-4okk7a .katex .sizing.reset-size9.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size8{font-size:0.83333333em;}.css-4okk7a .katex .sizing.reset-size9.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size9{font-size:1em;}.css-4okk7a .katex .sizing.reset-size9.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size10{font-size:1.20023148em;}.css-4okk7a .katex .sizing.reset-size9.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size9.size11{font-size:1.43981481em;}.css-4okk7a .katex .sizing.reset-size10.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size1{font-size:0.24108004em;}.css-4okk7a .katex .sizing.reset-size10.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size2{font-size:0.28929605em;}.css-4okk7a .katex .sizing.reset-size10.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size3{font-size:0.33751205em;}.css-4okk7a .katex .sizing.reset-size10.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size4{font-size:0.38572806em;}.css-4okk7a .katex .sizing.reset-size10.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size5{font-size:0.43394407em;}.css-4okk7a .katex .sizing.reset-size10.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size6{font-size:0.48216008em;}.css-4okk7a .katex .sizing.reset-size10.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size7{font-size:0.57859209em;}.css-4okk7a .katex .sizing.reset-size10.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size8{font-size:0.69431051em;}.css-4okk7a .katex .sizing.reset-size10.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size9{font-size:0.83317261em;}.css-4okk7a .katex .sizing.reset-size10.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size10{font-size:1em;}.css-4okk7a .katex .sizing.reset-size10.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size10.size11{font-size:1.19961427em;}.css-4okk7a .katex .sizing.reset-size11.size1,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size1{font-size:0.20096463em;}.css-4okk7a .katex .sizing.reset-size11.size2,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size2{font-size:0.24115756em;}.css-4okk7a .katex .sizing.reset-size11.size3,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size3{font-size:0.28135048em;}.css-4okk7a .katex .sizing.reset-size11.size4,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size4{font-size:0.32154341em;}.css-4okk7a .katex .sizing.reset-size11.size5,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size5{font-size:0.36173633em;}.css-4okk7a .katex .sizing.reset-size11.size6,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size6{font-size:0.40192926em;}.css-4okk7a .katex .sizing.reset-size11.size7,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size7{font-size:0.48231511em;}.css-4okk7a .katex .sizing.reset-size11.size8,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size8{font-size:0.57877814em;}.css-4okk7a .katex .sizing.reset-size11.size9,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size9{font-size:0.69453376em;}.css-4okk7a .katex .sizing.reset-size11.size10,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size10{font-size:0.83360129em;}.css-4okk7a .katex .sizing.reset-size11.size11,.css-4okk7a .katex .fontsize-ensurer.reset-size11.size11{font-size:1em;}.css-4okk7a .katex .delimsizing.size1{font-family:KaTeX_Size1;}.css-4okk7a .katex .delimsizing.size2{font-family:KaTeX_Size2;}.css-4okk7a .katex .delimsizing.size3{font-family:KaTeX_Size3;}.css-4okk7a .katex .delimsizing.size4{font-family:KaTeX_Size4;}.css-4okk7a .katex .delimsizing.mult .delim-size1>span{font-family:KaTeX_Size1;}.css-4okk7a .katex .delimsizing.mult .delim-size4>span{font-family:KaTeX_Size4;}.css-4okk7a .katex .nulldelimiter{display:inline-block;width:0.12em;}.css-4okk7a .katex .delimcenter{position:relative;}.css-4okk7a .katex .op-symbol{position:relative;}.css-4okk7a .katex .op-symbol.small-op{font-family:KaTeX_Size1;}.css-4okk7a .katex .op-symbol.large-op{font-family:KaTeX_Size2;}.css-4okk7a .katex .op-limits>.vlist-t{text-align:center;}.css-4okk7a .katex .accent>.vlist-t{text-align:center;}.css-4okk7a .katex .accent .accent-body{position:relative;}.css-4okk7a .katex .accent .accent-body:not(.accent-full){width:0;}.css-4okk7a .katex .overlay{display:block;}.css-4okk7a .katex .mtable .vertical-separator{display:inline-block;min-width:1px;}.css-4okk7a .katex .mtable .arraycolsep{display:inline-block;}.css-4okk7a .katex .mtable .col-align-c>.vlist-t{text-align:center;}.css-4okk7a .katex .mtable .col-align-l>.vlist-t{text-align:left;}.css-4okk7a .katex .mtable .col-align-r>.vlist-t{text-align:right;}.css-4okk7a .katex .svg-align{text-align:left;}.css-4okk7a .katex svg{display:block;position:absolute;width:100%;height:inherit;fill:currentColor;stroke:currentColor;fill-rule:nonzero;fill-opacity:1;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;}.css-4okk7a .katex svg path{stroke:none;}.css-4okk7a .katex img{border-style:none;min-width:0;min-height:0;max-width:none;max-height:none;}.css-4okk7a .katex .stretchy{width:100%;display:block;position:relative;overflow:hidden;}.css-4okk7a .katex .stretchy::before,.css-4okk7a .katex .stretchy::after{content:'';}.css-4okk7a .katex .hide-tail{width:100%;position:relative;overflow:hidden;}.css-4okk7a .katex .halfarrow-left{position:absolute;left:0;width:50.2%;overflow:hidden;}.css-4okk7a .katex .halfarrow-right{position:absolute;right:0;width:50.2%;overflow:hidden;}.css-4okk7a .katex .brace-left{position:absolute;left:0;width:25.1%;overflow:hidden;}.css-4okk7a .katex .brace-center{position:absolute;left:25%;width:50%;overflow:hidden;}.css-4okk7a .katex .brace-right{position:absolute;right:0;width:25.1%;overflow:hidden;}.css-4okk7a .katex .x-arrow-pad{padding:0 0.5em;}.css-4okk7a .katex .cd-arrow-pad{padding:0 0.55556em 0 0.27778em;}.css-4okk7a .katex .x-arrow,.css-4okk7a .katex .mover,.css-4okk7a .katex .munder{text-align:center;}.css-4okk7a .katex .boxpad{padding:0 0.3em 0 0.3em;}.css-4okk7a .katex .fbox,.css-4okk7a .katex .fcolorbox{box-sizing:border-box;border:0.04em solid;}.css-4okk7a .katex .cancel-pad{padding:0 0.2em 0 0.2em;}.css-4okk7a .katex .cancel-lap{margin-left:-0.2em;margin-right:-0.2em;}.css-4okk7a .katex .sout{border-bottom-style:solid;border-bottom-width:0.08em;}.css-4okk7a .katex .angl{box-sizing:border-box;border-top:0.049em solid;border-right:0.049em solid;margin-right:0.03889em;}.css-4okk7a .katex .anglpad{padding:0 0.03889em 0 0.03889em;}.css-4okk7a .katex .eqn-num::before{counter-increment:katexEqnNo;content:'(' counter(katexEqnNo) ')';}.css-4okk7a .katex .mml-eqn-num::before{counter-increment:mmlEqnNo;content:'(' counter(mmlEqnNo) ')';}.css-4okk7a .katex .mtr-glue{width:50%;}.css-4okk7a .katex .cd-vert-arrow{display:inline-block;position:relative;}.css-4okk7a .katex .cd-label-left{display:inline-block;position:absolute;right:calc(50% + 0.3em);text-align:left;}.css-4okk7a .katex .cd-label-right{display:inline-block;position:absolute;left:calc(50% + 0.3em);text-align:right;}.css-4okk7a .katex-display{display:block;margin:1em 0;text-align:center;}.css-4okk7a .katex-display>.katex{display:block;white-space:nowrap;}.css-4okk7a .katex-display>.katex>.katex-html{display:block;position:relative;}.css-4okk7a .katex-display>.katex>.katex-html>.tag{position:absolute;right:0;}.css-4okk7a .katex-display.leqno>.katex>.katex-html>.tag{left:0;right:auto;}.css-4okk7a .katex-display.fleqn>.katex{text-align:left;padding-left:2em;}.css-4okk7a body{counter-reset:katexEqnNo mmlEqnNo;}.css-4okk7a table{width:-webkit-max-content;width:-moz-max-content;width:max-content;}.css-4okk7a .tableBlock{max-width:100%;margin-bottom:1rem;overflow-y:scroll;}.css-4okk7a .tableBlock thead,.css-4okk7a .tableBlock thead th{border-bottom:1px solid #333!important;}.css-4okk7a .tableBlock th,.css-4okk7a .tableBlock td{padding:10px;text-align:left;}.css-4okk7a .tableBlock th{font-weight:bold!important;}.css-4okk7a .tableBlock caption{caption-side:bottom;color:#555;font-size:12px;font-style:italic;text-align:center;}.css-4okk7a .tableBlock caption>p{margin:0;}.css-4okk7a .tableBlock th>p,.css-4okk7a .tableBlock td>p{margin:0;}.css-4okk7a .tableBlock [data-background-color='aliceblue']{background-color:#f0f8ff;color:#000;}.css-4okk7a .tableBlock [data-background-color='black']{background-color:#000;color:#fff;}.css-4okk7a .tableBlock [data-background-color='chocolate']{background-color:#d2691e;color:#fff;}.css-4okk7a .tableBlock [data-background-color='cornflowerblue']{background-color:#6495ed;color:#fff;}.css-4okk7a .tableBlock [data-background-color='crimson']{background-color:#dc143c;color:#fff;}.css-4okk7a .tableBlock [data-background-color='darkblue']{background-color:#00008b;color:#fff;}.css-4okk7a .tableBlock [data-background-color='darkseagreen']{background-color:#8fbc8f;color:#000;}.css-4okk7a .tableBlock [data-background-color='deepskyblue']{background-color:#00bfff;color:#000;}.css-4okk7a .tableBlock [data-background-color='gainsboro']{background-color:#dcdcdc;color:#000;}.css-4okk7a .tableBlock [data-background-color='grey']{background-color:#808080;color:#fff;}.css-4okk7a .tableBlock [data-background-color='lemonchiffon']{background-color:#fffacd;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightpink']{background-color:#ffb6c1;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightsalmon']{background-color:#ffa07a;color:#000;}.css-4okk7a .tableBlock [data-background-color='lightskyblue']{background-color:#87cefa;color:#000;}.css-4okk7a .tableBlock [data-background-color='mediumblue']{background-color:#0000cd;color:#fff;}.css-4okk7a .tableBlock [data-background-color='omnigrey']{background-color:#f0f0f0;color:#000;}.css-4okk7a .tableBlock [data-background-color='white']{background-color:#fff;color:#000;}.css-4okk7a .tableBlock [data-text-align='center']{text-align:center;}.css-4okk7a .tableBlock [data-text-align='left']{text-align:left;}.css-4okk7a .tableBlock [data-text-align='right']{text-align:right;}.css-4okk7a .tableBlock [data-vertical-align='bottom']{vertical-align:bottom;}.css-4okk7a .tableBlock [data-vertical-align='middle']{vertical-align:middle;}.css-4okk7a .tableBlock [data-vertical-align='top']{vertical-align:top;}.css-4okk7a .tableBlock__font-size--xxsmall{font-size:10px;}.css-4okk7a .tableBlock__font-size--xsmall{font-size:12px;}.css-4okk7a .tableBlock__font-size--small{font-size:14px;}.css-4okk7a .tableBlock__font-size--large{font-size:18px;}.css-4okk7a .tableBlock__border--some tbody tr:not(:last-child){border-bottom:1px solid #e2e5e7;}.css-4okk7a .tableBlock__border--bordered td,.css-4okk7a .tableBlock__border--bordered th{border:1px solid #e2e5e7;}.css-4okk7a .tableBlock__border--borderless tbody+tbody,.css-4okk7a .tableBlock__border--borderless td,.css-4okk7a .tableBlock__border--borderless th,.css-4okk7a .tableBlock__border--borderless tr,.css-4okk7a .tableBlock__border--borderless thead,.css-4okk7a .tableBlock__border--borderless thead th{border:0!important;}.css-4okk7a .tableBlock:not(.tableBlock__table-striped) tbody tr{background-color:unset!important;}.css-4okk7a .tableBlock__table-striped tbody tr:nth-of-type(odd){background-color:#f9fafc!important;}.css-4okk7a .tableBlock__table-compactl th,.css-4okk7a .tableBlock__table-compact td{padding:3px!important;}.css-4okk7a .tableBlock__full-size{width:100%;}.css-4okk7a .textBlock{margin-bottom:16px;}.css-4okk7a .textBlock__text-formatting--finePrint{font-size:12px;}.css-4okk7a .textBlock__text-infoBox{padding:0.75rem 1.25rem;margin-bottom:1rem;border:1px solid transparent;border-radius:0.25rem;}.css-4okk7a .textBlock__text-infoBox p{margin:0;}.css-4okk7a .textBlock__text-infoBox--primary{background-color:#cce5ff;border-color:#b8daff;color:#004085;}.css-4okk7a .textBlock__text-infoBox--secondary{background-color:#e2e3e5;border-color:#d6d8db;color:#383d41;}.css-4okk7a .textBlock__text-infoBox--success{background-color:#d4edda;border-color:#c3e6cb;color:#155724;}.css-4okk7a .textBlock__text-infoBox--danger{background-color:#f8d7da;border-color:#f5c6cb;color:#721c24;}.css-4okk7a .textBlock__text-infoBox--warning{background-color:#fff3cd;border-color:#ffeeba;color:#856404;}.css-4okk7a .textBlock__text-infoBox--info{background-color:#d1ecf1;border-color:#bee5eb;color:#0c5460;}.css-4okk7a .textBlock__text-infoBox--dark{background-color:#d6d8d9;border-color:#c6c8ca;color:#1b1e21;}.css-4okk7a .text-overline{-webkit-text-decoration:overline;text-decoration:overline;}.css-4okk7a.css-4okk7a{color:#2B3148;background-color:transparent;font-family:"Roboto","Helvetica","Arial",sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-4okk7a.css-4okk7a:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-4okk7a .js-external-link-button.link-like,.css-4okk7a .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-4okk7a .js-external-link-button.link-like:hover,.css-4okk7a .js-external-link-anchor:hover,.css-4okk7a .js-external-link-button.link-like:active,.css-4okk7a .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-4okk7a .js-external-link-button.link-like:focus-visible,.css-4okk7a .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-4okk7a p,.css-4okk7a div{margin:0px;display:block;}.css-4okk7a pre{margin:0px;display:block;}.css-4okk7a pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-4okk7a pre:not(:first-child){padding-top:8px;}.css-4okk7a ul,.css-4okk7a ol{display:block margin:0px;padding-left:20px;}.css-4okk7a ul li,.css-4okk7a ol li{padding-top:8px;}.css-4okk7a ul ul,.css-4okk7a ol ul,.css-4okk7a ul ol,.css-4okk7a ol ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

Statistics Made Easy

Understanding the t-Test in Linear Regression

Linear regression is used to quantify the relationship between a predictor variable and a response variable.

Whenever we perform linear regression, we want to know if there is a statistically significant relationship between the predictor variable and the response variable.

We test for significance by performing a t-test for the regression slope. We use the following null and alternative hypothesis for this t-test:

• H 0 : β 1 = 0 (the slope is equal to zero)
• H A : β 1 ≠ 0 (the slope is not equal to zero)

We then calculate the test statistic as follows:

• t = b  / SE b
• b : coefficient estimate
• SE b : standard error of the coefficient estimate

If the p-value that corresponds to t is less than some threshold (e.g. α = .05) then we reject the null hypothesis and conclude that there is a statistically significant relationship between the predictor variable and the response variable.

The following example shows how to perform a t-test for a linear regression model in practice.

Example: Performing a t-Test for Linear Regression

Suppose a professor wants to analyze the relationship between hours studied and exam score received for 40 of his students.

He performs simple linear regression using hours studied as the predictor variable and exam score received as the response variable.

The following table shows the results of the regression model:

To determine if hours studied has a statistically significant relationship with final exam score, we can perform a t-test.

We use the following null and alternative hypothesis for this t-test:

• H 0 : β 1 = 0 (the slope for hours studied is equal to zero)
• H A : β 1 ≠ 0 (the slope for hours studied is not equal to zero)
• t = 1.117 / 1.025

The p-value that corresponds to t = 1.089 with df = n-2 = 40 – 2 = 38 is 0.283 .

Note that we can also use the T Score to P Value Calculator to calculate this p-value:

Since this p-value is not less than .05, we fail to reject the null hypothesis.

This means that hours studied does not have a statistically significant relationship between final exam score.

Additional Resources

The following tutorials offer additional information about linear regression:

Introduction to Simple Linear Regression Introduction to Multiple Linear Regression How to Interpret Regression Coefficients How to Interpret the F-Test of Overall Significance in Regression

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

One Reply to “Understanding the t-Test in Linear Regression”

if I t-test a regression vs another regression instead vs zero, ie a slope vs another slope, do I need to compare against a t crit of n – 4?

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Join the Statology Community

Sign up to receive Statology's exclusive study resource: 100 practice problems with step-by-step solutions. Plus, get our latest insights, tutorials, and data analysis tips straight to your inbox!

By subscribing you accept Statology's Privacy Policy.

Z-test vs T-test: the differences and when to use each

What is hypothesis testing, what is a z-test, examples of a z-test, what is a t-test, examples of a t-test, how to know when to use z-test vs t-test, difference between z-test and t-test: a comparative table.

The EPAM Anywhere Editorial Team is an international collective of senior software engineers, managers and communications professionals who create, review and share their insights on technology, career, remote work, and the daily life here at Anywhere.

Testing is how you determine effectiveness. Whether you work as a data scientist , statistician, or software developer, to ensure quality, you must measure performance. Without tests, you could deploy flawed code, features, or data points.

With that in mind, the use cases of testing are endless. Machine learning models need statistical tests. Data analysis involves statistical tests to validate assumptions. Optimization of any kind requires evaluation. You even need to test the strength of your hypothesis before you begin an inquiry.

Let's explore two inferential statistics: the Z-test vs the T-test. That way you can understand their differences, their unique purposes, and when to use a Z-test vs T-test.

To start, imagine you have a good idea. At the moment of inception, you have no data to back up your idea. It is an unformed thought. But the idea is an excellent starting point that can launch a full investigation. We consider this starting point a hypothesis.

But what if your hypothesis is off-base? You don’t want to dive into a full-scale search if it is a pointless chase with no reward. That is a waste of resources. You need to determine if you have a workable hypothesis.

Enter hypothesis testing. It is a statistical act used to assess the viability of a hypothesis. The method discovers whether there is sufficient data to support your idea. If there is next to no significance, you do not have a very plausible hypothesis.

To confirm the validity of a hypothesis, you compare it against the status quo (also known as the null hypothesis). Your idea is something new, opposite from normal conditions (also known as the alternative hypothesis). It is zero sum: only one hypothesis between the null and alternate hypothesis can be true in a given set of parameters.

In such a comparison test, you can now determine validity. You can compare and contrast conditions to find meaningful conclusions. Whichever conditions become statistically apparent determines which hypothesis is plausible.

A Z-test is a test statistic. It works with two groups: the sample mean and the population mean. It will test whether these two groups are different.

With a Z-test, you know the population standard deviation. That is to ensure statistical accuracy as you compare one group (the sample mean) vs the second group (the population mean). In other words, you can minimize external confounding factors with a normal distribution. In addition, a defining characteristic of a Z-test is that it works with large sample sizes (typically more than 30, so we achieve normal distribution as defined by the central limit theorem). These are two crucial criteria for using a Z-test.

Within hypothesis testing, your null hypothesis states there is no difference between the two groups your Z-test will compare. Your alternative hypothesis will state there is a difference that your Z-test will expose.

How to perform a Z-test

A Z-test occurs in the following standard format:

• Formulate your hypothesis: First, define the parameters of your alternative and null hypothesis.
• Choose a critical value: Second, determine what you consider a viable difference between your two groups. This threshold determines when you can say the null hypothesis should be rejected. Common levels are 0.01 (1%) or 0.05 (5%) , values found to best balance Type I and Type II errors .
• Collect samples: Obtain the needed data. The data must be large enough and random.
• Calculate for a Z-score: Input your data into the standard Z-test statistics formula, shown below, where Z = standard score, x = observed value, mu = mean of the sample, sigma = standard deviation of the sample .
• Compare: If the statistical test is greater than the critical value, you have achieved statistical significance. The sample mean is so different so you can reject the null hypothesis. Your alternative hypothesis (something other than the status quo) is at work, and that's worth investigating.

There are different variations of a Z-test. Let's explore examples of one-sample and two-sample Z-tests.

One-sample Z-test

A one-sample Z-test looks for either an increase or a decrease. There is one sample group involved, taken from a population. We want to see if there is a difference between those two means.

For example, consider a school principal who believes their students' IQ is higher than the state average. The state average is 100 points (population mean), give or take 20 (the population standard deviation). To prove this hypothesis, the principal takes 50 students (the sample size) and finds their IQ scores. To their delight, they earn an average of 110.

But does the difference offer any statistical value? The principal then plugs the numbers into a Z-test. Any Z-score greater than the critical value would state there is sufficient significance. The claim that the students have an above-average IQ is valid.

Two-sample Z-test

A two-sample test involves comparing the average of two sample groups against the population means. It is to determine a difference between two independent samples.

For example, our principal wants to compare their students' IQ scores to the school across the street. They believe their students' average IQ is higher. They don’t need to know the exact numerical increase or decrease. All they want is proof that their student's average scores are higher than the other group.

To confirm the validity of this hypothesis, the principal will search for statistical significance. They can take a 50-student sample size from their school and a 50-student sample size from the rival school. Now in possession of both sample group's average IQ (and the sample standard deviation), they hope to find a number value that is not equal. And they need them to be unequal by a significant amount.

If the test statistic comes in less than the critical value, the differences are negligible. There is not enough evidence to say the hypothesis is worth exploring, the null hypothesis is maintained. He would not have enough proof that the IQ levels between the two schools are different.

Read full story

A T-test performs the same crucial function as a Z-test: determine if there is a difference between the means of two groups. If there is a significant difference, you have achieved statistical validity for your hypothesis.

However, a T-test involves a different set of factors. Most importantly, a T-test applies when you do not know the sample variance of your values. You must generalize the normal distribution (or T-distribution). Plus, there is an expectation that you do not possess all the data in a given scenario.

These conditions better match reality, as it is often hard to collect data from entire populations or always obtain a standard normal distribution. That is why T-tests are more widely applicable than Z-tests, though they operate with less precision.

How to perform a T-test

A T-test occurs in the following standard format:

• Formulate your hypothesis: First, define the parameters of your null and alternative hypothesis.
• Choose a critical value: Like a Z-test, determine what you consider a viable difference between your two groups.
• Collect data: Obtain the needed data. One of the key differences is degrees of freedom in the samples of a T-test, so try to define the typical values and range of values in each group.
• Calculate your T-score: Input your data into the T-test formula you chose. Here is a one-sample formula:
• Compare: If the statistical test is greater than the critical value, you have achieved statistical significance. The sample mean is so far from the population mean that you likely have a useful hypothesis.

There are several different kinds of T-tests as well. Let's go through the standard one-sample and two-sample T-tests.

One-sample T-test

A one-sample T-test looks for an increase or decrease compared to a population mean.

For example, your company just went through sales training. Now, the manager wants to know if the training helped improve sales.

Previous company sales data shows an average of $100 on each transaction from all workers. The null hypothesis would be no change. The alternative hypothesis (which you hope is significant), is that there is an improvement.

To test if there is significance, you take the sales average of 20 salesmen. That is the only available data, and you have no other data from nationwide stores. The average of that sample of salesmen in the past month is $130. We will also assume that the standard deviation is approximately equal .

With this set of factors, you can calculate your T-score with a T-test. You compare the sample result to the critical value. In addition, you assess it against the number of degrees of freedom. Since we know with smaller sample data sizes there is greater uncertainty, we allow more room for our data to vary.

After comparing, we may find a lot of significance. That means the data possesses enough strength to support our hypothesis that sales training likely impacted sales. Of course, this is an estimate, as we only assessed one factor with a small group. Sales could have risen for numerous other reasons. But with our set of assumptions, our hypothesis is valid.

Two-sample T-test

A two-sample T-test occurs the same as a two-sample Z-test and compares if two groups are equal when compared to a defined population parameter.

For example, consider English and non-native speakers. We want to see the effect of maternal language on test scores inside a country. To do that, we will offer both groups a reading test and compare those scores to the average.

Of course, finding the mean of an entire population of language speakers is impossible to procure. Still, we can make some assumptions and compare them with a smaller size. We take 15 English speakers and 15 non-native speakers and collect their results. We can decide on a critical score value on the reading test as well. If the average score on the test is not crucially different or outside the population standard deviation, our assumption failed. There is no significant difference between the groups, so the impact of maternal language is not worth investigating.

Both a Z-test and a T-test validate a hypothesis. Both are parametric tests that rely on assumptions. The key difference between Z-test and T-test is in their assumptions (e.g. population variance).

Key differences about the data used result in different applications. You want to use the appropriate tool, otherwise you won’t draw valid conclusions from your data.

So when should you use a Z-test vs a T-test? Here are some factors to consider:

• Sample size: If the available sample size is small, opt for a T-test. Small sample sizes are more variable, so the greater spread of distribution and estimation of error involved with T-tests is ideal.
• Knowledge of the population standard deviation: Z-tests are more precise and often simpler to execute. So if you know the standard deviation, use a Z-test.
• Test purpose: If you are assessing the validity of a mean, a T-test is the best choice. If you are working with a hypothesized population proportion, go for a Z-test.
• Assumption of normality: A Z-test assumes a normal distribution. This does not apply to all real-world scenarios. If you hope to validate a hypothesis that is not well-defined, opt for a T-test instead.
• Type of data: You can only work within the constraints of the available data. The more information the better, but that is often not possible given testing and collecting conditions. If you have limited data describing means between groups, opt for a T-test. If you have large data sets comparing means between populations, you can use a Z-test.

Knowing the key differences with each statistical test makes selecting the right tool far easier. Here is a table that can help you compare:

Statistical testing lets you determine the validity of a hypothesis. You discover validity by determining if there is a significant difference between your hypothesis and the status quo. If there is, you have a possible idea worth exploring.

That process has numerous applications in the field of computer science and data analysis . You might want to determine the performance of an app with an A/B test. Or you might need to test if an application fits within the defined limits and compare performance metrics. Z-tests and T-tests can depict whether there is significant evidence in each of these scenarios. With that information, you can take the appropriate measures to fix bugs or optimize processes.

Z-test and T-test are helpful tools, especially for hypothesis testing. For data engineers of the future, knowledge of statistical testing will only help your work and overall career trajectory.

Are you a data scientist looking for a job? Check out our remote data scientist jobs available .

Explore our Editorial Policy to learn more about our standards for content creation.

step-by-step guide to creating, building, and showcasing your data analyst portfolio projects

C# introduction: basics, introduction, and beginner projects

what is SQL?

red team vs blue team in cybersecurity: key roles & responsibilities

what is .NET?

shift-left strategy in accessible product development

10 most popular programming languages for 2020 and beyond

how to write effective usability test scripts

how to become a senior software engineer: a guide

creative Salesforce portfolio project ideas

rebase vs. merge: a comprehensive guide

the future of Agile: what to expect in 2024 and beyond

best React projects for a portfolio: from ideas to standout examples

Ruby on Rails interview questions

data scientist interview questions and answers

24 unique JavaScript projects for portfolio

• Calculators
• Descriptive Statistics
• Merchandise
• Which Statistics Test?

T-Test Calculator for 2 Independent Means

This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation.

Further Information

A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females).

Requirements

• Two independent samples
• Data should be normally distributed
• The two samples should have the same variance

Null Hypothesis

H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second.

As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of males and females).

• Math Formulas
• T Test Formula

T-Test Formula

The t-test is any statistical hypothesis test in which the test statistic follows a Student’s t-distribution under the null hypothesis. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known.

T-test uses means and standard deviations of two samples to make a comparison. The formula for T-test is given below:

\begin{array}{l}\qquad t=\frac{\bar{X}_{1}-\bar{X}_{2}}{s_{\bar{\Delta}}} \\ \text { where } \\ \qquad s_{\bar{\Delta}}=\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}} \\ \end{array}

Where, \(\begin{array}{l}\overline{x}\end{array} \) = Mean of first set of values \(\begin{array}{l}\overline{x}_{2}\end{array} \)  = Mean of second set of values \(\begin{array}{l}S_{1}\end{array} \)   = Standard deviation of first set of values \(\begin{array}{l}S_{2}\end{array} \)   = Standard deviation of second set of values \(\begin{array}{l}n_{1}\end{array} \)   = Total number of values in first set \(\begin{array}{l}n_{2}\end{array} \)   = Total number of values in second set.

The formula for standard deviation is given by:

Where, x = Values given \(\begin{array}{l}\overline{x}\end{array} \) = Mean n = Total number of values.

T-Test Solved Examples

Question 1: Find the t-test value for the following two sets of values: 7, 2, 9, 8 and 1, 2, 3, 4?

Formula for standard deviation:  \(\begin{array}{l}S=\sqrt{\frac{\sum\left(x-\overline{x}\right)^{2}}{n-1}}\end{array} \)

Number of terms in first set:  \(\begin{array}{l}n_{1}\end{array} \) = 4

Mean for first set of data: \(\begin{array}{l}\overline{x}_{1}\end{array} \) = 6.5

Construct the following table for standard deviation:

Standard deviation for the first set of data: S 1 = 3.11

Number of terms in second set: n 2 = 4

Standard deviation for first set of data: \(\begin{array}{l}S_{2}\end{array} \) = 1.29

Formula for t-test value:

t = 2.3764 = 2.36 (approx)

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

How to Do a T Test in Excel (2 Ways with Interpretation of Results)

The article will show you how to do a T Test in Excel. T-Tests are hypothesis tests that evaluate one or two groups’ means. Hypothesis tests employ sample data to infer population traits. In this lesson, we will look at the different types of T-Tests , and how to run T-Tests in Excel. We’ll go over both paired and two sample T-Tests , with detailed instructions on how to prepare your data, run the test, and interpret the findings.

Understanding how to use the T.TEST function in Excel will improve your ability to draw significant insights and make data-driven decisions, whether you’re a student, researcher, business analyst, or anybody else who works with data. Let’s say, you’re doing education research to assess the efficacy between traditional and new approaches. T-tests will guide you through providing the mean scores of students based on the approaches that they were taught. So that, you can make a decision based on the students’ performance.

Download Practice Workbook

T Test.xlsx

T Test Type

There are basically two types of t-tests. They are:

• One-tailed t-test
• Two-tailed t-test

Each of them has 3 types. They are:

• Two sample equal variance
• Two sample unequal variance

We will show you the application of some of these types. The procedure of getting the results for all types of t-tests in Excel are the same. Let’s dig into some details and see how it can be done.

How to Do a T Test in Excel: 2 Effective Ways

1. using excel t.test or ttest function to do t test.

Here, we are going to show you how to determine the T Test result by using formulas. Excel has T.TEST and TTEST functions to operate t-test on different variables. Both functions work similarly. First, we will cover how to determine the t-test value of two sample variables with equal variance.

1.1 Two Sample Equal Variance T Test

In the dataset, you will see the prices of different laptops and smartphones. Here is a formula that performs a T Test on the prices of these products and returns the t-test result.

=T.TEST(B5:B14,C5:C14,2,2)

We set the 3rd argument of the function to 2 as we are doing a two tailed t-test on the dataset. The 4th argument should be 2 for a two sample equal variance t-test.

1.2 Paired T Test

Now, we are going to apply another formula to calculate the Paired T-Test . The following dataset shows the performance mark of some employees in two different criteria.

=T.TEST(C5:C13,D5:D13,2,1)

Note: The explanation of the results is described in the following sections.

2. Using Analysis Toolpak

The above tasks can be done with the Analysis Toolpak Add-in too. The Analysis Toolpak Add-in is not available in the ribbon by default. To initiate it,

• Go to the Options window first.
• Next, select Add-ins and click on the Go button beside the Manage section.
• After that, click OK .

• Thereafter, the Add-ins window will appear. Select Analysis Toolpak >> click OK again.

This Add-in will be added to the ribbon of the Data tab.

2.1 Two Sample Equal Variance T Test

We will do a two sample equal variance t-test using the Analysis Toolpak here. We used the dataset that contains the prices of laptops and smartphones. For this purpose,

• Click on the Data Analysis button from the ribbon of the Data tab.
• The Data Analysis features will appear. Select t-Test: Two Sample Assuming Equal Variances and click OK .

• After that, you need to set up the parameters for the t-test operation. Insert the Laptop and Smartphone prices as Variable 1 Range and Variable 2 Range Include the headings in the range and check Labels.
• Next, set the value of Hypothesized Mean Difference to 0 .
• Finally, select an Output option of your preference and click OK .

As we have chosen a New Worksheet for the outputs, we will see the results in a new sheet.

Now, let’s get to the discussion on the results.

Comments on Results

The output shows that the mean values for Laptops and Smartphones are 1608.85 and 1409.164 respectively. We can see from the Variances row that they are not precisely equal, but they are close enough to be assumed to have equal variances. The most relevant metric is the p-value .

The difference between means is statistically significant if the p-value is less than your significance level. Excel calculates p-values for one- and two-tailed T Tests .

One-tailed T Tests can detect only one direction of difference between means. A one-tailed test, for example, might only evaluate whether Smartphones have higher prices than Laptops . Two-tailed tests can reveal differences that are larger or smaller than. There are some other disadvantages to utilizing one-tailed testing, so I’ll continue with the conventional two-tailed results.

For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.095639932 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. The mean price of Laptops is greater than the mean price of Smartphones’ .

The Analysis Toolpak operation also returns results for one-tailed t-test . Here, the one-tailed P value of two sample equal variance t-test is 1.734 .

2.2 Paired T Test

Similarly, you can find out the Paired t-Test result for the dataset containing employee performances. Just select the t-Test: Paired Two Samples for Mean when you open the Data Analysis window.

The result shows that the mean for the Workpace is 104 and the mean for the Efficiency is 96.56 .

The difference between means is statistically significant if the p-value is less than your significance level. For our results, we’ll utilize P(T=t) two-tail, which is the p-value for the t-test’s two-tailed version. We cannot reject the null hypothesis because our p-value ( 0.188 ) is greater than the conventional significance level of 0.05 . The hypothesis that the population means differ is supported by our sample data. In particular, the Workpace mean exceeds the Efficiency mean.

How to Interpret t-Test Results in Excel

Although we explained the results of the t-Test earlier, we didn’t show the proper interpretation. So here, I’ll show you the interpretation of the two sample equal variance t-test.

Let’s bring out the results again first.

• The mean of laptop prices = 1608.85
• The mean of smartphone prices = 1409.164

ii. Variance

• The variance of laptop prices = 77622.597
• The variance of smartphone prices = 51313.7904

iii. Observations

The number of observations for both laptops and smartphones are 10 .

iv. Pooled Variance

The samples’ average variance, calculated by pooling the variances of each sample.

The mathematical formula for this parameter is:

((No of observations of Sample 1-1)*(Variance of Sample 1) + (No of observations of Sample 2-1)*(Variance of Sample 2))/(No of observations of Sample 1 + No of observations of Sample 2 – 2)

So it becomes: ((10-1)*77622.59676+(10-1)*51313.7904)/(10+10-2) = 64468.19358

v. Hypothesized Mean Difference

We “hypothesize” that the number is the difference between the two population means. In this situation, we chose 0 because we want to see if the difference between the means of the two populations is zero.

It indicates the value of the Degrees of Freedom. Formula for this parameter is:

No of observations of Sample 1 + No of observations of Sample 2 – 2 = 10 + 10 – 2 = 18

vii. t Stat

The test statistic value of the t-Test operation.

The formula for this parameter is given below.

(Mean of Sample 1 – Mean of Sample 2)/(Square root of (Pooling Variance* (1/No of observations of Sample 1 + 1/No of observations of Sample 2)))

So it becomes: (1608.85 – 1409.164)/Sqrt(64468.19358 * (1/10 + 1/10)) = 1.758570846

viii. P(T<=t) two-tail

A two-tailed t-test’s p-value. This value can be found by entering t = 1.758570846 with 18 degrees of freedom into any T Score to P Value Calculator.

In this situation, the value of p is 0.095639932 . Because this is greater than 0.05 , we cannot reject the null hypothesis. This suggests that we lack adequate evidence to conclude that the two population means differ.

ix. t Critical two-tail

This is the test’s crucial value. A t Critical value Calculator with 18 degrees of freedom and a 95% confidence level can be used to calculate this number.

In this instance, the critical value is 2.10092204 . We cannot reject the null hypothesis because our test statistic t is less than this number. Again, we lack adequate information to conclude that the two population means are distinct.

Things to Remember

• Excel demands that your data be arranged in columns, with data from each group in a separate column. The first row should have labels or headers.
• Clearly state your null hypothesis (usually that there is no significant difference between the group means) and your alternative hypothesis (the opposite of the null hypothesis).
• As a result of the t-test, Excel returns the p-value. A little p-value (usually less than the specified alpha level) indicates that the null hypothesis may be rejected and that there is a substantial difference between the group means.

Frequently Asked Questions

1. Can I perform a t-test on unequal sample sizes in Excel?

Answer: Yes, you can use the T.TEST function to do a t-test on unequal sample sizes. When calculating the test statistic, Excel automatically accounts for unequal sample sizes.

2. What is the difference between a one-tailed and a two-tailed t-test?

Answer: A one-tailed t-test determines if the means of the two groups differ substantially in a given direction (e.g., greater or smaller). A two-tailed t-test looks for any significant difference, regardless of direction.

3. Can I calculate effect size in Excel for t-tests?

Answer: While there is no built-in tool in Excel to calculate effect size, you may manually compute Cohen’s d for independent t-tests and paired sample correlations for paired t-tests using Excel’s basic mathematical operations.

In the end, we can conclude that you will learn some basic ideas on how to do a t Test in Excel. If you have any questions or feedback regarding this article, please share them in the comment section. Your valuable ideas will enrich my Excel expertise and hence the content of my upcoming articles.

<< Go Back to Statistical Significance in Excel | Excel for Statistics  |  Learn Excel

What is ExcelDemy?

Tags: Statistical Significance in Excel

Md. Meraz Al Nahian has worked with the ExcelDemy project for over 1.5 years. He wrote 140+ articles for ExcelDemy. He also solved a lot of user problems and worked on dashboards. He is interested in data analysis, advanced Excel, statistics, and dashboards. He also likes to explore various Excel and VBA applications. He completed his graduation in Electrical & Electronic Engineering from Bangladesh University of Engineering & Technology (BUET). He enjoys exploring Excel-related features to gain efficiency... Read Full Bio

Leave a reply Cancel reply

ExcelDemy is a place where you can learn Excel, and get solutions to your Excel & Excel VBA-related problems, Data Analysis with Excel, etc. We provide tips, how to guide, provide online training, and also provide Excel solutions to your business problems.

Contact  |  Privacy Policy  |  TOS

• User Reviews
• List of Services
• Service Pricing

• Create Basic Excel Pivot Tables
• Excel Formulas and Functions
• Excel Charts and SmartArt Graphics
• Advanced Excel Training
• Data Analysis Excel for Beginners

Advanced Excel Exercises with Solutions PDF

• Python for Machine Learning
• Machine Learning with R
• Machine Learning Algorithms
• Math for Machine Learning
• Machine Learning Interview Questions
• ML Projects
• Deep Learning
• Computer vision
• Data Science
• Artificial Intelligence
• Machine Learning Mathematics

Linear Algebra and Matrix

• Scalar and Vector
• Python Program to Add Two Matrices
• Python program to multiply two matrices
• Vector Operations
• Product of Vectors
• Scalar Product of Vectors
• Dot and Cross Products on Vectors
• Transpose a matrix in Single line in Python
• Transpose of a Matrix
• Adjoint and Inverse of a Matrix
• How to inverse a matrix using NumPy
• Determinant of a Matrix
• Program to find Normal and Trace of a matrix
• Data Science | Solving Linear Equations
• Data Science - Solving Linear Equations with Python
• System of Linear Equations
• System of Linear Equations in three variables using Cramer's Rule
• Eigenvalues
• Applications of Eigenvalues and Eigenvectors
• How to compute the eigenvalues and right eigenvectors of a given square array using NumPY?

Statistics for Machine Learning

• Descriptive Statistic
• Measures of Central Tendency
• Measures of Dispersion | Types, Formula and Examples
• Mean, Variance and Standard Deviation
• Calculate the average, variance and standard deviation in Python using NumPy
• Random Variables
• Difference between Parametric and Non-Parametric Methods
• Probability Distribution
• Confidence Interval
• Mathematics | Covariance and Correlation
• Program to find correlation coefficient
• Robust Correlation
• Normal Probability Plot
• Quantile Quantile plots
• True Error vs Sample Error
• Bias-Variance Trade Off - Machine Learning
• Understanding Hypothesis Testing
• Paired T-Test - A Detailed Overview
• P-value in Machine Learning
• F-Test in Statistics
• Residual Leverage Plot (Regression Diagnostic)
• Difference between Null and Alternate Hypothesis
• Mann and Whitney U test
• Wilcoxon Signed Rank Test
• Kruskal Wallis Test
• Friedman Test
• Mathematics | Probability

Probability and Probability Distributions

• Mathematics - Law of Total Probability
• Bayes's Theorem for Conditional Probability
• Mathematics | Probability Distributions Set 1 (Uniform Distribution)
• Mathematics | Probability Distributions Set 4 (Binomial Distribution)
• Mathematics | Probability Distributions Set 5 (Poisson Distribution)
• Uniform Distribution Formula
• Mathematics | Probability Distributions Set 2 (Exponential Distribution)
• Mathematics | Probability Distributions Set 3 (Normal Distribution)
• Mathematics | Beta Distribution Model
• Gamma Distribution Model in Mathematics
• Chi-Square Test for Feature Selection - Mathematical Explanation
• Student's t-distribution in Statistics
• Python - Central Limit Theorem
• Mathematics | Limits, Continuity and Differentiability
• Implicit Differentiation

Calculus for Machine Learning

• Engineering Mathematics - Partial Derivatives
• Advanced Differentiation
• How to find Gradient of a Function using Python?
• Optimization techniques for Gradient Descent
• Higher Order Derivatives
• Taylor Series
• Application of Derivative - Maxima and Minima | Mathematics
• Absolute Minima and Maxima
• Optimization for Data Science
• Unconstrained Multivariate Optimization
• Lagrange Multipliers
• Lagrange's Interpolation
• Linear Regression in Machine learning
• Ordinary Least Squares (OLS) using statsmodels

Regression in Machine Learning

In statistics, various tests are used to compare different samples or groups and draw conclusions about populations. These tests, known as statistical tests, focus on analyzing the likelihood or probability of obtaining the observed data under specific assumptions or hypotheses. They provide a framework for assessing evidence in support of or against a particular hypothesis.

A statistical test begins by formulating a null hypothesis (H 0 ) and an alternative hypothesis (H a ). The null hypothesis represents the default assumption, typically stating no effect or no difference, while the alternative hypothesis suggests a specific relationship or effect.

There are different statistical tests like Z-test , T-test, Chi-squared tests , ANOVA , Z-test , and F-test , etc. which are used to compute the p-value. In this article, we will learn about the T-test.

Table of Content

What is T-Test?

Assumptions in t-test, prerequisites for t-test, types of t-tests, one sample t-test, independent sample t-test, paired two-sample t-test, frequently asked questions on t-test.

The t-test is named after William Sealy Gosset’s Student’s t-distribution, created while he was writing under the pen name “Student.”

A t-test is a type of inferential statistic test used to determine if there is a significant difference between the means of two groups. It is often used when data is normally distributed and population variance is unknown.

The t-test is used in hypothesis testing to assess whether the observed difference between the means of the two groups is statistically significant or just due to random variation.

• Independence : The observations within each group must be independent of each other. This means that the value of one observation should not influence the value of another observation. Violations of independence can occur with repeated measures, paired data, or clustered data.
• Normality : The data within each group should be approximately normally distributed i.e the distribution of the data within each group being compared should resemble a normal (bell-shaped) distribution. This assumption is crucial for small sample sizes (n < 30).
• Homogeneity of Variances (for independent samples t-test) : The variances of the two groups being compared should be equal. This assumption ensures that the groups have a similar spread of values. Unequal variances can affect the standard error of the difference between means and, consequently, the t-statistic.
• Absence of Outliers: There should be no extreme outliers in the data as outliers can disproportionately influence the results, especially when sample sizes are small.

Let’s quickly review some related terms before digging deeper into the specifics of the t-test.

A t-test is a statistical method used to compare the means of two groups to determine if there is a significant difference between them. The t-test is a parametric test, meaning it makes certain assumptions about the data. Here are the key prerequisites for conducting a t-test.

Hypothesis Testing :

Hypothesis testing is a statistical method used to make inferences about a population based on a sample of data.

The p-value is the probability of observing a test statistic (or something more extreme) given that the null hypothesis is true.

• A small p-value (typically less than the chosen significance level) suggests that the observed data is unlikely to have occurred by random chance alone, leading to the rejection of the null hypothesis.
• A large p-value suggests that the observed data is likely to have occurred by random chance, and there is not enough evidence to reject the null hypothesis.

Degree of freedom (df):

Significance Level :

The significance level is the predetermined threshold that is used to decide whether to reject the null hypothesis. Commonly used significance levels are 0.05, 0.01, or 0.10. A significance level of 0.05 indicates that the researcher is willing to accept a 5% chance of making a Type I error (incorrectly rejecting a true null hypothesis).

T-statistic :

The t-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

• If the t-value is large => the two groups belong to different groups. 
• If the t-value is small => the two groups belong to the same group.

T-Distribution

The t-distribution , commonly known as the Student’s t-distribution, is a probability distribution with tails that are thicker than those of the normal distribution.

Statistical Significance

Statistical significance is determined by comparing the p-value to the chosen significance level.

• If the p-value is less than or equal to the significance level, the result is considered statistically significant, and the null hypothesis is rejected.
• If the p-value is greater than the significance level, the result is not statistically significant, and there is insufficient evidence to reject the null hypothesis.

In the context of a t-test, these concepts are applied to compare means between two groups. The t-test assesses whether the means are significantly different from each other, taking into account the variability within the groups. The p-value from the t-test is then compared to the significance level to make a decision about the null hypothesis.

A t-table, or a t-distribution table, is a reference table that provides critical values for the t-test. The table is organized by degrees of freedom and significance levels (usually 0.05 or 0.01). The t-table is used to find the critical t-value corresponding to their specific degrees of freedom and chosen significance level. If the calculated t-value is greater than the critical value from the table, it suggests that the observed difference is statistically significant.

There are three types of t-tests, and they are categorized as dependent and independent t-tests.

• One sample t-test test: The mean of a single group against a known mean.
• Independent samples t-test: compares the means for two groups.
• Paired sample t-test: compares means from the same group at different times (say, one year apart).

One sample t-test is one of the widely used t-tests for comparison of the sample mean of the data to a particularly given value. Used for comparing the sample mean to the true/population mean.

We can use this when the sample size is small. (under 30) data is collected randomly and it is approximately normally distributed. It can be calculated as:

• t = t-value
• x_bar = sample mean
• μ = true/population mean
• σ = standard deviation
• n = sample size

Example Problem

Consider the following example. The weights of 25 obese people were taken before enrolling them into the nutrition camp. The population mean weight is found to be 45 kg before starting the camp. After finishing the camp, for the same 25 people, the sample mean was found to be 75 with a standard deviation of 25. Did the fitness camp work?

One-Sample T-test in Python

The T-value of 6.0 is significantly greater than the critical t-value, leading to rejection of the null hypothesis therefore, we can conclude there is a significant difference in weight before and after the fitness camp. The fitness camp had an effect on the weights of the participants.

The results strongly suggest that the fitness camp was effective in producing a statistically significant change in weight for the participants.

• The T-value and p-value both provide consistent evidence for rejecting the null hypothesis.
• The practical significance should also be considered to understand the real-world impact of this weight change.

An Independent sample t-test, commonly known as an unpaired sample t-test is used to find out if the differences found between two groups is actually significant or just a random occurrence. 

We can use this when:

• the population mean or standard deviation is unknown. (information about the population is unknown)
• the two samples are separate/independent. For eg. boys and girls (the two are independent of each other)

It can be calculated using:

Researchers are investigating whether there is a significant difference in the exam scores of two different teaching methods, A and B. Two independent samples, each representing a different teaching method, have been collected. The objective is to determine if there is enough evidence to suggest that one teaching method leads to higher exam scores compared to the other. Suppose, two independent sample data A and B are given, with the following values. We have to perform the Independent samples t-test for this data.

Two-Sample t-test in Python (Independent)

With T-Value, of 0.989 is less than the critical t-value of 2.1009. Therefore, No significant difference is found between the exam scores of Teaching Method A and Teaching Method B based on the T-value.

With P-Value, of 0.336 is greater than the significance level of 0.05. There is no evidence to reject the null hypothesis, indicating no significant difference between the two teaching methods based on the P-value.

In conclusion, The results suggest that, statistically, there is no significant difference in exam scores between Teaching Method A and Teaching Method B. Therefore, based on this analysis, there is no clear evidence to suggest that one teaching method leads to higher exam scores compared to the other.

Paired sample t-test, commonly known as dependent sample t-test is used to find out if the difference in the mean of two samples is 0. The test is done on dependent samples, usually focusing on a particular group of people or things. In this, each entity is measured twice, resulting in a pair of observations. 

We can use this when :

• Two similar (twin like) samples are given. [Eg, Scores obtained in English and Math (both subjects)]
• The dependent variable (data) is continuous.
• The observations are independent of one another.
• The dependent variable is approximately normally distributed.

It can be calculated using,

• (s_d) is the standard deviation of the differences.
• (n) is the number of paired observations.

Consider the following example. Scores (out of 25) of the subjects Math1 and Math2 are taken for a sample of 10 students. We have to perform the paired sample t-test for this data. 

Paired Two-Sample T-test in Python

The paired sample t-test suggests that there is a statistically significant difference in scores between Math1 and Math2 as T-value of -4.95 is less than the critical t-value of -2.2622 and P-value of 0.00079 is less than the significance level of 0.05. Therefore, based on this analysis, it can be concluded that there is evidence to support the claim that the two sets of scores are different, and the difference is not due to random chance.

The above-discussed types of t-tests are widely used in the fields of research in hospitals by experts to gain important information about the medical data given to them about the effects of various medicines and drugs on the population and help them draw out important inferences regarding the same. However, it is the responsibility of the person to see to it that which t-test would bring out the best results and that all the assumptions of that t-test are adhered to. For any doubt/query, comment below. 

In conclusion, t-test, play a crucial role in hypothesis testing, comparing means, and drawing conclusions about populations. The test can be one-sample, independent two-sample, or paired two-sample, each with specific use cases and assumptions. Interpretation of results involves considering T-values, P-values, and critical values.

These tests aid researchers in making informed decisions based on statistical evidence.

Q. What is the t-test for mean in Python?

The t-test for mean in Python is a statistical method used to determine if there is a significant difference between the means of two groups.

Q. What is the t-test function?

The t-test function is a statistical tool used to compare means and assess the significance of differences between groups, considering factors like sample size and variability.

Q. What is the p-value in t-test Python?

The p-value in a t-test Python indicates the probability of observing the data or more extreme results assuming the null hypothesis is true. A small p-value suggests evidence against the null hypothesis.

Q. Why is it called t-test?

The t-test is named after William Sealy Gosset, who published under the pseudonym “Student.” The name “t” refers to the t-distribution used in the test, particularly applicable for small sample sizes.

Please Login to comment...

Similar reads.

• Machine Learning
• Mathematical

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

• Mastering Hypothesis Testing in Excel: A Practical Guide for Students

Excel for Hypothesis Testing: A Practical Approach for Students

Hypothesis testing lies at the heart of statistical inference, serving as a cornerstone for drawing meaningful conclusions from data. It's a methodical process used to evaluate assumptions about a population parameter, typically based on sample data. The fundamental idea behind hypothesis testing is to assess whether observed differences or relationships in the sample are statistically significant enough to warrant generalizations to the larger population. This process involves formulating null and alternative hypotheses, selecting an appropriate statistical test, collecting sample data, and interpreting the results to make informed decisions. In the realm of statistical software, SAS stands out as a robust and widely used tool for data analysis in various fields such as academia, industry, and research. Its extensive capabilities make it particularly favored for complex analyses, large datasets, and advanced modeling techniques. However, despite its versatility and power, SAS can have a steep learning curve, especially for students who are just beginning their journey into statistics. The intricacies of programming syntax, data manipulation, and interpreting output may pose challenges for novice users, potentially hindering their understanding of statistical concepts like hypothesis testing. If you need assistance with your Excel homework , understanding hypothesis testing is essential for performing statistical analyses and drawing meaningful conclusions from data using Excel's built-in functions and tools.

Enter Excel, a ubiquitous spreadsheet software that most students are already familiar with to some extent. While Excel may not offer the same level of sophistication as SAS in terms of advanced statistical procedures, it remains a valuable tool, particularly for introductory and intermediate-level analyses. Its intuitive interface, user-friendly features, and widespread accessibility make it an attractive option for students seeking a practical approach to learning statistics. By leveraging Excel's built-in functions, data visualization tools, and straightforward formulas, students can gain hands-on experience with hypothesis testing in a familiar environment. In this blog post, we aim to bridge the gap between theoretical concepts and practical application by demonstrating how Excel can serve as a valuable companion for students tackling hypothesis testing problems, including those typically encountered in SAS assignments. We will focus on demystifying the process of hypothesis testing, breaking it down into manageable steps, and showcasing Excel's capabilities for conducting various tests commonly encountered in introductory statistics courses.

Understanding the Basics

Hypothesis testing is a fundamental concept in statistics that allows researchers to draw conclusions about a population based on sample data. At its core, hypothesis testing involves making a decision about whether a statement regarding a population parameter is likely to be true. This decision is based on the analysis of sample data and is guided by two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents the status quo or the absence of an effect. It suggests that any observed differences or relationships in the sample data are due to random variation or chance. On the other hand, the alternative hypothesis contradicts the null hypothesis and suggests the presence of an effect or difference in the population. It reflects the researcher's belief or the hypothesis they aim to support with their analysis.

Formulating Hypotheses

In Excel, students can easily formulate hypotheses using simple formulas and logical operators. For instance, suppose a researcher wants to test whether the mean of a sample is equal to a specified value. They can use the AVERAGE function in Excel to calculate the sample mean and then compare it to the specified value using logical operators like "=" for equality. If the calculated mean is equal to the specified value, it supports the null hypothesis; otherwise, it supports the alternative hypothesis.

Excel's flexibility allows students to customize their hypotheses based on the specific parameters they are testing. Whether it's comparing means, proportions, variances, or other population parameters, Excel provides a user-friendly interface for formulating hypotheses and conducting statistical analysis.

Selecting the Appropriate Test

Excel offers a plethora of functions and tools for conducting various types of hypothesis tests, including t-tests, z-tests, chi-square tests, and ANOVA (analysis of variance). However, selecting the appropriate test requires careful consideration of the assumptions and conditions associated with each test. Students should familiarize themselves with the assumptions underlying each hypothesis test and assess whether their data meets those assumptions. For example, t-tests assume that the data follow a normal distribution, while chi-square tests require categorical data and independence between observations.

Furthermore, students should consider the nature of their research question and the type of data they are analyzing. Are they comparing means of two independent groups or assessing the association between categorical variables? By understanding the characteristics of their data and the requirements of each test, students can confidently choose the appropriate hypothesis test in Excel.

T-tests are statistical tests commonly used to compare the means of two independent samples or to compare the mean of a single sample to a known value. These tests are valuable in various fields, including psychology, biology, economics, and more. In Excel, students can employ the T.TEST function to conduct t-tests, providing them with a practical and accessible way to analyze their data and draw conclusions about population parameters based on sample statistics.

Independent Samples T-Test

The independent samples t-test, also known as the unpaired t-test, is utilized when comparing the means of two independent groups. This test is often employed in experimental and observational studies to assess whether there is a significant difference between the means of the two groups. In Excel, students can easily organize their data into separate columns representing the two groups, calculate the sample means and standard deviations for each group, and then use the T.TEST function to obtain the p-value. The p-value obtained from the T.TEST function represents the probability of observing the sample data if the null hypothesis, which typically states that there is no difference between the means of the two groups, is true.

A small p-value (typically less than the chosen significance level, commonly 0.05) indicates that there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, suggesting a significant difference between the group means. By conducting an independent samples t-test in Excel, students can not only assess the significance of differences between two groups but also gain valuable experience in data analysis and hypothesis testing, which are essential skills in various academic and professional settings.

Paired Samples T-Test

The paired samples t-test, also known as the dependent t-test or matched pairs t-test, is employed when comparing the means of two related groups. This test is often used in studies where participants are measured before and after an intervention or when each observation in one group is matched or paired with a specific observation in the other group. Examples include comparing pre-test and post-test scores, analyzing the performance of individuals under different conditions, and assessing the effectiveness of a treatment or intervention. In Excel, students can perform a paired samples t-test by first calculating the differences between paired observations (e.g., subtracting the before-measurement from the after-measurement). Next, they can use the one-sample t-test function, specifying the calculated differences as the sample data. This approach allows students to determine whether the mean difference between paired observations is statistically significant, indicating whether there is a meaningful change or effect between the two related groups.

Interpreting the results of a paired samples t-test involves assessing the obtained p-value in relation to the chosen significance level. A small p-value suggests that there is sufficient evidence to reject the null hypothesis, indicating a significant difference between the paired observations. This information can help students draw meaningful conclusions from their data and make informed decisions based on statistical evidence. By conducting paired samples t-tests in Excel, students can not only analyze the relationship between related groups but also develop critical thinking skills and gain practical experience in hypothesis testing, which are valuable assets in both academic and professional contexts. Additionally, mastering the application of statistical tests in Excel can enhance students' data analysis skills and prepare them for future research endeavors and real-world challenges.

Chi-Square Test

The chi-square test is a versatile statistical tool used to assess the association between two categorical variables. In essence, it helps determine whether the observed frequencies in a dataset significantly deviate from what would be expected under certain assumptions. Excel provides a straightforward means to perform chi-square tests using the CHISQ.TEST function, which calculates the probability associated with the chi-square statistic.

Goodness-of-Fit Test

One application of the chi-square test is the goodness-of-fit test, which evaluates how well the observed frequencies in a single categorical variable align with the expected frequencies dictated by a theoretical distribution. This test is particularly useful when researchers wish to ascertain whether their data conforms to a specific probability distribution. In Excel, students can organize their data into a frequency table, listing the categories of the variable of interest along with their corresponding observed frequencies. They can then specify the expected frequencies based on the theoretical distribution they are testing against. For example, if analyzing the outcomes of a six-sided die roll, where each face is expected to occur with equal probability, the expected frequency for each category would be the total number of observations divided by six.

Once the observed and expected frequencies are determined, students can employ the CHISQ.TEST function in Excel to calculate the chi-square statistic and its associated p-value. The p-value represents the probability of obtaining a chi-square statistic as extreme or more extreme than the observed value under the assumption that the null hypothesis is true (i.e., the observed frequencies match the expected frequencies). Interpreting the results of the goodness-of-fit test involves comparing the calculated p-value to a predetermined significance level (commonly denoted as α). If the p-value is less than α (e.g., α = 0.05), there is sufficient evidence to reject the null hypothesis, indicating that the observed frequencies significantly differ from the expected frequencies specified by the theoretical distribution. Conversely, if the p-value is greater than α, there is insufficient evidence to reject the null hypothesis, suggesting that the observed frequencies align well with the expected frequencies.

Test of Independence

Another important application of the chi-square test in Excel is the test of independence, which evaluates whether there is a significant association between two categorical variables in a contingency table. This test is employed when researchers seek to determine whether the occurrence of one variable is related to the occurrence of another. To conduct a test of independence in Excel, students first create a contingency table that cross-tabulates the two categorical variables of interest. Each cell in the table represents the frequency of occurrences for a specific combination of categories from the two variables.

Similar to the goodness-of-fit test, students then calculate the expected frequencies for each cell under the assumption of independence between the variables. Using the CHISQ.TEST function in Excel, students can calculate the chi-square statistic and its associated p-value based on the observed and expected frequencies in the contingency table. The interpretation of the test results follows a similar procedure to that of the goodness-of-fit test, with the p-value indicating whether there is sufficient evidence to reject the null hypothesis of independence between the two variables.

Excel, despite being commonly associated with spreadsheet tasks, offers a plethora of features that make it a versatile and powerful tool for statistical analysis, especially for students diving into the intricacies of hypothesis testing. Its widespread availability and user-friendly interface make it accessible to students at various levels of statistical proficiency. However, the true value of Excel lies not just in its accessibility but also in its ability to facilitate a hands-on learning experience that reinforces theoretical concepts.

At the core of utilizing Excel for hypothesis testing is a solid understanding of the fundamental principles of statistical inference. Students need to grasp concepts such as the null and alternative hypotheses, significance levels, p-values, and test statistics. Excel provides a practical platform for students to apply these concepts in a real-world context. Through hands-on experimentation with sample datasets, students can observe how changes in data inputs and statistical parameters affect the outcome of hypothesis tests, thus deepening their understanding of statistical theory.

Post a comment...

Mastering hypothesis testing in excel: a practical guide for students submit your homework, attached files.