hypothesis testing z calculator

Z-test Calculator

What is a z-test, when do i use z-tests, z-test formula, p-value from z-test, two-tailed z-test and one-tailed z-test, z-test critical values & critical regions, how to use the one-sample z-test calculator, z-test examples.

This Z-test calculator is a tool that helps you perform a one-sample Z-test on the population's mean . Two forms of this test - a two-tailed Z-test and a one-tailed Z-tests - exist, and can be used depending on your needs. You can also choose whether the calculator should determine the p-value from Z-test or you'd rather use the critical value approach!

Read on to learn more about Z-test in statistics, and, in particular, when to use Z-tests, what is the Z-test formula, and whether to use Z-test vs. t-test. As a bonus, we give some step-by-step examples of how to perform Z-tests!

Or you may also check our t-statistic calculator , where you can learn the concept of another essential statistic. If you are also interested in F-test, check our F-statistic calculator .

A one sample Z-test is one of the most popular location tests. The null hypothesis is that the population mean value is equal to a given number, μ 0 \mu_0 μ 0 ​ :

We perform a two-tailed Z-test if we want to test whether the population mean is not μ 0 \mu_0 μ 0 ​ :

and a one-tailed Z-test if we want to test whether the population mean is less/greater than μ 0 \mu_0 μ 0 ​ :

Let us now discuss the assumptions of a one-sample Z-test.

You may use a Z-test if your sample consists of independent data points and:

the data is normally distributed , and you know the population variance ;

the sample is large , and data follows a distribution which has a finite mean and variance. You don't need to know the population variance.

The reason these two possibilities exist is that we want the test statistics that follow the standard normal distribution N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) . In the former case, it is an exact standard normal distribution, while in the latter, it is approximately so, thanks to the central limit theorem.

The question remains, "When is my sample considered large?" Well, there's no universal criterion. In general, the more data points you have, the better the approximation works. Statistics textbooks recommend having no fewer than 50 data points, while 30 is considered the bare minimum.

Let x 1 , . . . , x n x_1, ..., x_n x 1 ​ , ... , x n ​ be an independent sample following the normal distribution N ( μ , σ 2 ) \mathrm N(\mu, \sigma^2) N ( μ , σ 2 ) , i.e., with a mean equal to μ \mu μ , and variance equal to σ 2 \sigma ^2 σ 2 .

We pose the null hypothesis, H 0  ⁣  ⁣ :  ⁣  ⁣   μ = μ 0 \mathrm H_0 \!\!:\!\! \mu = \mu_0 H 0 ​ :   μ = μ 0 ​ .

We define the test statistic, Z , as:

x ˉ \bar x x ˉ is the sample mean, i.e., x ˉ = ( x 1 + . . . + x n ) / n \bar x = (x_1 + ... + x_n) / n x ˉ = ( x 1 ​ + ... + x n ​ ) / n ;

μ 0 \mu_0 μ 0 ​ is the mean postulated in H 0 \mathrm H_0 H 0 ​ ;

n n n is sample size; and

σ \sigma σ is the population standard deviation.

In what follows, the uppercase Z Z Z stands for the test statistic (treated as a random variable), while the lowercase z z z will denote an actual value of Z Z Z , computed for a given sample drawn from N(μ,σ²).

If H 0 \mathrm H_0 H 0 ​ holds, then the sum S n = x 1 + . . . + x n S_n = x_1 + ... + x_n S n ​ = x 1 ​ + ... + x n ​ follows the normal distribution, with mean n μ 0 n \mu_0 n μ 0 ​ and variance n 2 σ n^2 \sigma n 2 σ . As Z Z Z is the standardization (z-score) of S n / n S_n/n S n ​ / n , we can conclude that the test statistic Z Z Z follows the standard normal distribution N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , provided that H 0 \mathrm H_0 H 0 ​ is true. By the way, we have the z-score calculator if you want to focus on this value alone.

If our data does not follow a normal distribution, or if the population standard deviation is unknown (and thus in the formula for Z Z Z we substitute the population standard deviation σ \sigma σ with sample standard deviation), then the test statistics Z Z Z is not necessarily normal. However, if the sample is sufficiently large, then the central limit theorem guarantees that Z Z Z is approximately N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) .

In the sections below, we will explain to you how to use the value of the test statistic, z z z , to make a decision , whether or not you should reject the null hypothesis . Two approaches can be used in order to arrive at that decision: the p-value approach, and critical value approach - and we cover both of them! Which one should you use? In the past, the critical value approach was more popular because it was difficult to calculate p-value from Z-test. However, with help of modern computers, we can do it fairly easily, and with decent precision. In general, you are strongly advised to report the p-value of your tests!

Formally, the p-value is the smallest level of significance at which the null hypothesis could be rejected. More intuitively, p-value answers the questions: provided that I live in a world where the null hypothesis holds, how probable is it that the value of the test statistic will be at least as extreme as the z z z - value I've got for my sample? Hence, a small p-value means that your result is very improbable under the null hypothesis, and so there is strong evidence against the null hypothesis - the smaller the p-value, the stronger the evidence.

To find the p-value, you have to calculate the probability that the test statistic, Z Z Z , is at least as extreme as the value we've actually observed, z z z , provided that the null hypothesis is true. (The probability of an event calculated under the assumption that H 0 \mathrm H_0 H 0 ​ is true will be denoted as P r ( event ∣ H 0 ) \small \mathrm{Pr}(\text{event} | \mathrm{H_0}) Pr ( event ∣ H 0 ​ ) .) It is the alternative hypothesis which determines what more extreme means :

• Two-tailed Z-test: extreme values are those whose absolute value exceeds ∣ z ∣ |z| ∣ z ∣ , so those smaller than − ∣ z ∣ -|z| − ∣ z ∣ or greater than ∣ z ∣ |z| ∣ z ∣ . Therefore, we have:

The symmetry of the normal distribution gives:

• Left-tailed Z-test: extreme values are those smaller than z z z , so
• Right-tailed Z-test: extreme values are those greater than z z z , so

To compute these probabilities, we can use the cumulative distribution function, (cdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , which for a real number, x x x , is defined as:

Also, p-values can be nicely depicted as the area under the probability density function (pdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , due to:

With all the knowledge you've got from the previous section, you're ready to learn about Z-tests.

• Two-tailed Z-test:

From the fact that Φ ( − z ) = 1 − Φ ( z ) \Phi(-z) = 1 - \Phi(z) Φ ( − z ) = 1 − Φ ( z ) , we deduce that

The p-value is the area under the probability distribution function (pdf) both to the left of − ∣ z ∣ -|z| − ∣ z ∣ , and to the right of ∣ z ∣ |z| ∣ z ∣ :

• Left-tailed Z-test:

The p-value is the area under the pdf to the left of our z z z :

• Right-tailed Z-test:

The p-value is the area under the pdf to the right of z z z :

The decision as to whether or not you should reject the null hypothesis can be now made at any significance level, α \alpha α , you desire!

if the p-value is less than, or equal to, α \alpha α , the null hypothesis is rejected at this significance level; and

if the p-value is greater than α \alpha α , then there is not enough evidence to reject the null hypothesis at this significance level.

The critical value approach involves comparing the value of the test statistic obtained for our sample, z z z , to the so-called critical values . These values constitute the boundaries of regions where the test statistic is highly improbable to lie . Those regions are often referred to as the critical regions , or rejection regions . The decision of whether or not you should reject the null hypothesis is then based on whether or not our z z z belongs to the critical region.

The critical regions depend on a significance level, α \alpha α , of the test, and on the alternative hypothesis. The choice of α \alpha α is arbitrary; in practice, the values of 0.1, 0.05, or 0.01 are most commonly used as α \alpha α .

Once we agree on the value of α \alpha α , we can easily determine the critical regions of the Z-test:

To decide the fate of H 0 \mathrm H_0 H 0 ​ , check whether or not your z z z falls in the critical region:

If yes, then reject H 0 \mathrm H_0 H 0 ​ and accept H 1 \mathrm H_1 H 1 ​ ; and

If no, then there is not enough evidence to reject H 0 \mathrm H_0 H 0 ​ .

As you see, the formulae for the critical values of Z-tests involve the inverse, Φ − 1 \Phi^{-1} Φ − 1 , of the cumulative distribution function (cdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) .

Our calculator reduces all the complicated steps:

Choose the alternative hypothesis: two-tailed or left/right-tailed.

In our Z-test calculator, you can decide whether to use the p-value or critical regions approach. In the latter case, set the significance level, α \alpha α .

Enter the value of the test statistic, z z z . If you don't know it, then you can enter some data that will allow us to calculate your z z z for you:

• sample mean x ˉ \bar x x ˉ (If you have raw data, go to the average calculator to determine the mean);
• tested mean μ 0 \mu_0 μ 0 ​ ;
• sample size n n n ; and
• population standard deviation σ \sigma σ (or sample standard deviation if your sample is large).

Results appear immediately below the calculator.

If you want to find z z z based on p-value , please remember that in the case of two-tailed tests there are two possible values of z z z : one positive and one negative, and they are opposite numbers. This Z-test calculator returns the positive value in such a case. In order to find the other possible value of z z z for a given p-value, just take the number opposite to the value of z z z displayed by the calculator.

To make sure that you've fully understood the essence of Z-test, let's go through some examples:

• A bottle filling machine follows a normal distribution. Its standard deviation, as declared by the manufacturer, is equal to 30 ml. A juice seller claims that the volume poured in each bottle is, on average, one liter, i.e., 1000 ml, but we suspect that in fact the average volume is smaller than that...

Formally, the hypotheses that we set are the following:

H 0  ⁣ :   μ = 1000  ml \mathrm H_0 \! : \mu = 1000 \text{ ml} H 0 ​ :   μ = 1000  ml

H 1  ⁣ :   μ < 1000  ml \mathrm H_1 \! : \mu \lt 1000 \text{ ml} H 1 ​ :   μ < 1000  ml

We went to a shop and bought a sample of 9 bottles. After carefully measuring the volume of juice in each bottle, we've obtained the following sample (in milliliters):

1020 , 970 , 1000 , 980 , 1010 , 930 , 950 , 980 , 980 \small 1020, 970, 1000, 980, 1010, 930, 950, 980, 980 1020 , 970 , 1000 , 980 , 1010 , 930 , 950 , 980 , 980 .

Sample size: n = 9 n = 9 n = 9 ;

Sample mean: x ˉ = 980   m l \bar x = 980 \ \mathrm{ml} x ˉ = 980   ml ;

Population standard deviation: σ = 30   m l \sigma = 30 \ \mathrm{ml} σ = 30   ml ;

And, therefore, p-value = Φ ( − 2 ) ≈ 0.0228 \text{p-value} = \Phi(-2) \approx 0.0228 p-value = Φ ( − 2 ) ≈ 0.0228 .

As 0.0228 < 0.05 0.0228 \lt 0.05 0.0228 < 0.05 , we conclude that our suspicions aren't groundless; at the most common significance level, 0.05, we would reject the producer's claim, H 0 \mathrm H_0 H 0 ​ , and accept the alternative hypothesis, H 1 \mathrm H_1 H 1 ​ .

We tossed a coin 50 times. We got 20 tails and 30 heads. Is there sufficient evidence to claim that the coin is biased?

Clearly, our data follows Bernoulli distribution, with some success probability p p p and variance σ 2 = p ( 1 − p ) \sigma^2 = p (1-p) σ 2 = p ( 1 − p ) . However, the sample is large, so we can safely perform a Z-test. We adopt the convention that getting tails is a success.

Let us state the null and alternative hypotheses:

H 0  ⁣ :   p = 0.5 \mathrm H_0 \! : p = 0.5 H 0 ​ :   p = 0.5 (the coin is fair - the probability of tails is 0.5 0.5 0.5 )

H 1  ⁣ :   p ≠ 0.5 \mathrm H_1 \! : p \ne 0.5 H 1 ​ :   p  = 0.5 (the coin is biased - the probability of tails differs from 0.5 0.5 0.5 )

In our sample we have 20 successes (denoted by ones) and 30 failures (denoted by zeros), so:

Sample size n = 50 n = 50 n = 50 ;

Sample mean x ˉ = 20 / 50 = 0.4 \bar x = 20/50 = 0.4 x ˉ = 20/50 = 0.4 ;

Population standard deviation is given by σ = 0.5 × 0.5 \sigma = \sqrt{0.5 \times 0.5} σ = 0.5 × 0.5 ​ (because 0.5 0.5 0.5 is the proportion p p p hypothesized in H 0 \mathrm H_0 H 0 ​ ). Hence, σ = 0.5 \sigma = 0.5 σ = 0.5 ;

• And, therefore

Since 0.1573 > 0.1 0.1573 \gt 0.1 0.1573 > 0.1 we don't have enough evidence to reject the claim that the coin is fair , even at such a large significance level as 0.1 0.1 0.1 . In that case, you may safely toss it to your Witcher or use the coin flip probability calculator to find your chances of getting, e.g., 10 heads in a row (which are extremely low!).

What is the difference between Z-test vs t-test?

We use a t-test for testing the population mean of a normally distributed dataset which had an unknown population standard deviation . We get this by replacing the population standard deviation in the Z-test statistic formula by the sample standard deviation, which means that this new test statistic follows (provided that H₀ holds) the t-Student distribution with n-1 degrees of freedom instead of N(0,1) .

When should I use t-test over the Z-test?

For large samples, the t-Student distribution with n degrees of freedom approaches the N(0,1). Hence, as long as there are a sufficient number of data points (at least 30), it does not really matter whether you use the Z-test or the t-test, since the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test instead of Z-test .

How do I calculate the Z test statistic?

To calculate the Z test statistic:

• Compute the arithmetic mean of your sample .
• From this mean subtract the mean postulated in null hypothesis .
• Multiply by the square root of size sample .
• Divide by the population standard deviation .
• That's it, you've just computed the Z test statistic!

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Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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Z-test Calculator

Streamline your statistical calculations with the z-test calculator by newtum.

Discover the power of our Z-test Calculator, expertly developed by Newtum. This essential tool simplifies statistical testing, making it accessible for professionals and students alike. Unveil the mysteries of z-scores and enhance your data analysis skills.

Understanding the Statistical Significance Tool

The Z-test Calculator is a statistical tool designed to determine if there is a significant difference between sample and population means. It's ideal for researchers and students engaged in hypothesis testing and data analysis.

Z-test Calculation Formula Explained

Learn the critical formula used in the Z-test Calculator and its significance in statistical analysis. Understanding this formula is vital for accurate hypothesis testing and research conclusions.

• Define the null and alternative hypotheses.
• Calculate the sample mean (x̄) and population mean (μ).
• Determine the standard deviation (σ) and the sample size (n).
• Compute the standard error of the mean (σ/√n).
• Use the formula Z = (x̄ - μ) / (σ/√n) to calculate the Z-score.

Step-by-Step Guide to Using the Z-test Calculator

Our Z-test Calculator is incredibly user-friendly. Just follow the simple instructions below, and you'll be on your way to obtaining quick and accurate z-score results.

• Enter the sample mean into the designated field.
• Input the population mean.
• Provide the standard deviation of the population.
• Specify the sample size.
• Click 'Calculate' to get your Z-score and p-value.

Discover the Superior Features of Our Z-test Calculator

• User-Friendly Interface: Navigate with ease.
• Instant Results: Get your answers without delays.
• Data Security: Your data remains on your device.
• Accessibility Across Devices: Use on any modern device.
• No Installation Needed: Access directly from your browser.
• Examples for Clarity: Understand with practical scenarios.
• Transparent Process: No hidden steps or calculations.
• Educational Resource: Enhance your statistical knowledge.
• Responsive Customer Support: We're here to assist you.
• Regular Updates: Benefit from the latest features.
• Privacy Assurance: Your data is safe with us.
• Efficient Age Retrieval: Quick and accurate.
• Language Accessibility: Use in your preferred language.
• Engaging and Informative Content: Learn while you use.
• Fun and Interactive Learning: Enjoy the process.
• Shareable Results: Easily export your findings.
• Responsive Design: Works flawlessly on any screen size.
• Educational Platform Integration: Perfect for e-learning environments.
• Comprehensive Documentation: All the information you need.

Applications and Use Cases for the Z-test Calculator

• Analyze the difference between sample means and population means.
• Validate research findings with statistical significance.
• Enhance academic projects with precise hypothesis testing.
• Apply in various scientific and market research studies.
• Utilize in quality control processes for product consistency.

Illustrating the Z-test Calculator with Practical Examples

Example 1: Suppose your sample mean (x) is 105, the population mean (y) is 100, the population standard deviation is 15, and the sample size is 30. Plugging these into the Z-test formula, we get a Z-score, which we then compare against the standard normal distribution.

Example 2: If your sample mean is 130, the population mean is 120, the standard deviation is 20, and the sample size is 50, the Z-test Calculator will give you a Z-score indicating the probability of this difference occurring by chance.

Ensuring Data Security with the Z-test Calculator

Our Z-test Calculator not only provides precise statistical analysis but also guarantees the utmost data security. As the calculations are performed entirely within your browser, your data never leaves your computer, ensuring complete confidentiality. This tool is a crucial asset for users who prioritize privacy while seeking reliable statistical solutions. Rest assured that with our Z-test Calculator, your data is processed securely without any risk of server-side exposure.

Frequently Asked Questions About Z-test Calculator

• What is a Z-test Calculator used for?
• How accurate is the Z-test Calculator?
• Can I use the Z-test Calculator for any sample size?
• Is there a cost associated with using the Z-test Calculator?
• How does the Z-test Calculator ensure the privacy of my data?

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One Sample Z-Test Calculator

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Single Sample Z Score Calculator

This tool calculates the z -score of the mean of a single sample. It can be used to make a judgement about whether the sample differs significantly on some axis from the population from which it was originally drawn.

By default, this tool works on the assumption that you already know the mean value of your sample scores and the number of individuals in your sample. If you want the tool to calculate these for you from raw values, please select the checkbox below.

To use this calculator, just input your population mean, population variance, sample mean and the number of individuals in the sample into the text boxes below. You also need to select a significance level and whether your hypothesis is one or two-tailed. Hit the calculate button (below) when you're ready.

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Z-Test for Statistical Hypothesis Testing Explained

The Z-test is a statistical hypothesis test used to determine where the distribution of the test statistic we are measuring, like the mean , is part of the normal distribution .

There are multiple types of Z-tests, however, we’ll focus on the easiest and most well known one, the one sample mean test. This is used to determine if the difference between the mean of a sample and the mean of a population is statistically significant.

What Is a Z-Test?

A Z-test is a type of statistical hypothesis test where the test-statistic follows a normal distribution.  

The name Z-test comes from the Z-score of the normal distribution. This is a measure of how many standard deviations away a raw score or sample statistics is from the populations’ mean.

Z-tests are the most common statistical tests conducted in fields such as healthcare and data science . Therefore, it’s an essential concept to understand.

Requirements for a Z-Test

In order to conduct a Z-test, your statistics need to meet a few requirements, including:

• A Sample size that’s greater than 30. This is because we want to ensure our sample mean comes from a distribution that is normal. As stated by the c entral limit theorem , any distribution can be approximated as normally distributed if it contains more than 30 data points.
• The standard deviation and mean of the population is known .
• The sample data is collected/acquired randomly .

More on Data Science:   What Is Bootstrapping Statistics?

Z-Test Steps

There are four steps to complete a Z-test. Let’s examine each one.

4 Steps to a Z-Test

• State the null hypothesis.
• State the alternate hypothesis.
• Choose your critical value.
• Calculate your Z-test statistics. 

1. State the Null Hypothesis

The first step in a Z-test is to state the null hypothesis, H_0 . This what you believe to be true from the population, which could be the mean of the population, μ_0 :

2. State the Alternate Hypothesis

Next, state the alternate hypothesis, H_1 . This is what you observe from your sample. If the sample mean is different from the population’s mean, then we say the mean is not equal to μ_0:

3. Choose Your Critical Value

Then, choose your critical value, α , which determines whether you accept or reject the null hypothesis. Typically for a Z-test we would use a statistical significance of 5 percent which is z = +/- 1.96 standard deviations from the population’s mean in the normal distribution:

This critical value is based on confidence intervals.

4. Calculate Your Z-Test Statistic

Compute the Z-test Statistic using the sample mean, μ_1 , the population mean, μ_0 , the number of data points in the sample, n and the population’s standard deviation, σ :

If the test statistic is greater (or lower depending on the test we are conducting) than the critical value, then the alternate hypothesis is true because the sample’s mean is statistically significant enough from the population mean.

Another way to think about this is if the sample mean is so far away from the population mean, the alternate hypothesis has to be true or the sample is a complete anomaly.

More on Data Science: Basic Probability Theory and Statistics Terms to Know

Z-Test Example

Let’s go through an example to fully understand the one-sample mean Z-test.

A school says that its pupils are, on average, smarter than other schools. It takes a sample of 50 students whose average IQ measures to be 110. The population, or the rest of the schools, has an average IQ of 100 and standard deviation of 20. Is the school’s claim correct?

The null and alternate hypotheses are:

Where we are saying that our sample, the school, has a higher mean IQ than the population mean.

Now, this is what’s called a right-sided, one-tailed test as our sample mean is greater than the population’s mean. So, choosing a critical value of 5 percent, which equals a Z-score of 1.96 , we can only reject the null hypothesis if our Z-test statistic is greater than 1.96.

If the school claimed its students’ IQs were an average of 90, then we would use a left-tailed test, as shown in the figure above. We would then only reject the null hypothesis if our Z-test statistic is less than -1.96.

Computing our Z-test statistic, we see:

Therefore, we have sufficient evidence to reject the null hypothesis, and the school’s claim is right.

Hope you enjoyed this article on Z-tests. In this post, we only addressed the most simple case, the one-sample mean test. However, there are other types of tests, but they all follow the same process just with some small nuances.  

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• Z-Test Calculator

Z-Test (Z 0 , Z e & H 0 ) Calculator

getcalc.com's Z-test calculator to estimate Z-statistic (Z 0 ), critical value (Z e ) from normal distribution table for given degrees of freedom (ν) & hypothesis test (H 0 ) to conduct the test of significance for mean, difference between two means , proportion & difference between two proportions for large sample mean, proportion or difference between two means or proportions in statistical surveys & experiments. In addition, users can generate the complete work with steps for any corresponding input values to solve grade school Z-test worksheet problems. Users also use the population standard deviation & sample mean calculators seperately to calculate the corresponding values.

Z-Test (Large Samples) & its Applications

Z-test is the statistical technique which represents how many standard deviation or standard error the sample mean or proportion (p value) is away from the population mean or success proportions (p value) to check if the test of hypothesis (significance) is accepted in statistical experiments. The Z-statistic (Z 0 ) value used to test the validity of assumptions in the test of significance. Here the difference between the hypothesized & actual value of the sample data is being analyzed to determine if the difference is significant or not. Z-statistic is applicable for the test of significance for proportion, difference between two proportions, mean or difference between two means. The probability is higher for the hypothesized value for mean or proportion to be correct, if the difference between the hypothesized & actual value is smaller. The probability is smaller for the hypothesized value for mean or proportion to be correct, if the difference between the hypothesized & actual value is higher. The sample values should be large enough or approximately equal to the population parameters and all the sampling distributions follow normal asymptotically to conduct the test of significance for large samples using z-test.

Test of Significance for Large Samples (Z 0 , Z e & H 0 ) & Formulas

Work with steps for Z-Test (Z 0 , Z e & H 0 )

This Z-test calculator for test of significance featured to generate the complete work with steps for any corresponding input values for test of significance for mean (using standard deviation), test of significance for difference between two means (using standard deviation), test of significance for proportion (using p value) and test of significance for difference between two proportions (using p value) to solve the grade school Z-statistic workout problems. Supply the input values, click on "CALCULATE" button and then "Generate Workout" to generate the complete work with steps for the input given to this calculator. The below is the solved examples for Z-statistic calculation by using standard deviation & without using standard deviation.

• Test of significance for sample mean
• Test of significance for 2 sample proportion with unknown P values
• Test of significance for difference between 2 proportions with known P values
• Z-test for difference between 2 proportions with unknown P Values
• Z-test example for difference between 2 samples with different σ 1 & σ 2
• Z-test example for difference between 2 samples with common σ
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Z Score Calculator

• Enter the Raw Score, Mean (μ), and Standard Deviation (σ) for your data.
• Click "Calculate Z-Score" to calculate the Z-Score and related values.
• Results, including the Z-Score, p-values, and confidence level, will be displayed below.
• Calculation steps will also be shown to explain how the Z-Score was computed.
• A chart visualizes the Z-Score in the context of the normal distribution.
• You can clear the entries, copy the results, and view calculation history.

What is Z Score

A z-score, also known as a standard score, is a statistical measure that indicates how many standard deviations a specific data point is away from the mean of the entire dataset . In simpler terms, it tells you how unusual or typical a particular value is compared to the rest of the data.

Key Aspects of Z Score

1. Standardization:

• Z-scores transform raw data points into a standardized scale with a mean of 0 and a standard deviation of 1.
• This allows for meaningful comparisons of data points from different datasets, even if they have different units or scales.

2. Relative Position:

• The z-score directly indicates how far a data point is from the mean in terms of standard deviations.
• A positive z-score means the point is above the mean, while a negative z-score means it’s below the mean.
• The further the z-score is from zero, the more unusual the value is within the dataset.

3. Normal Distribution:

• Z-scores are particularly useful when working with normally distributed data.
• In a normal distribution, approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
• By knowing the z-score, you can quickly determine the percentile rank of a data point within a normal distribution.

4. Outlier Detection:

• Z-scores can be used to identify outliers, which are significantly different values from the rest of the data.
• A common rule of thumb is that data points with z-scores greater than 3 or less than -3 are considered potential outliers.

5. Hypothesis Testing:

• Z-scores play a crucial role in hypothesis testing, where they are used to assess the likelihood of observed results occurring by chance.
• They are used to calculate p-values, which indicate the statistical significance of findings.

6. Data Transformation:

• Z-scores can be used to transform non-normal distributions into a normal distribution, which is required for statistical analyses that assume normality.

7. Standard Normal Distribution:

• The standard normal distribution, also known as the “Z-distribution,” is a special normal distribution with a mean of 0 and a standard deviation of 1.
• It’s a valuable reference for interpreting z-scores and understanding probabilities associated with different z-score values.

All Formulae Related to Z Score

Here are the key formulae related to z-scores:

1. Calculating the z-score:

• z = the z-score
• x = the raw data point
• μ = the mean of the dataset
• σ = the standard deviation of the dataset

2. Finding the raw data point from a z-score:

3. Calculating percentile rank in a normal distribution:

where the cumulative area under the z-curve can be found using a z-table or statistical software.

4. Converting a z-score to a probability:

5. Converting a probability to a z-score:

This requires using a z-table or statistical software to find the inverse of the cumulative normal distribution function.

Additional formulae for specific applications:

• Hypothesis testing:  Z-tests involve calculating a test statistic ( z-score ) and comparing it to a critical value to determine statistical significance.
• Confidence intervals:  The margin of error for a confidence interval can be calculated using z-scores and the standard error of the estimate.
• Correlation analysis:  The correlation coefficient (r) can be converted to a z-score to test its statistical significance.

Practical Uses of Z-Score Calculator

Here are some practical uses of z-score calculators:

1. Comparing Scores from Different Tests:

• Scenario:  You took two exams in different subjects, each with different scales and means. A z-score calculator can help you compare your performance on both exams on a standardized scale.
• How it works:  Calculate the z-scores for your scores on each exam. The exam with the higher z-score indicates a better relative performance, regardless of the original scores or scales.

2. Evaluating Individual Performance in a Group:

• Scenario:  You received a score of 85 on a class exam where the mean was 75 and the standard deviation was 5. A z-score calculator can tell you how well you performed compared to your classmates.
• How it works:  Calculate your z-score (z = (85 – 75) / 5 = 2). A z-score of 2 means you scored 2 standard deviations above the average, indicating excellent performance relative to the group.

3. Determining Normality of Data Distribution:

• Scenario:  You have a dataset of customer purchase amounts and want to check if it follows a normal distribution. A z-score calculator can help you assess normality.
• How it works:  Calculate the z-scores for all data points. If the distribution of z-scores is approximately bell-shaped and symmetric, it suggests a normal distribution.

4. Detecting Outliers:

• Scenario:  You’re analyzing website traffic data and need to identify unusual spikes or drops. A z-score calculator can help you detect potential outliers.
• How it works:  Calculate the z-scores for daily traffic numbers. Data points with z-scores significantly higher or lower than the rest might be considered outliers, warranting further investigation.

5. Calculating Percentile Ranks:

• Scenario:  You scored 1500 on the SAT, and you want to know what percentage of test-takers scored lower than you. A z-score calculator can estimate your percentile rank.
• How it works:  Assuming SAT scores are normally distributed, use a z-table or calculator to find the area under the standard normal curve to the left of your z-score. This area represents the percentile rank.

6. Making Predictions in Finance:

• Scenario:  You’re analyzing stock prices and want to assess the probability of a certain stock price movement. Z-scores can help you make predictions based on historical volatility.
• How it works:  Calculate the z-score for a potential price change, and use a z-table to find the corresponding probability. This can inform investment decisions based on risk assessment.

Applications of Z Score in Various Fields

Here are some applications of z-scores in various fields:

1. Statistics:

• Comparing data from different datasets:  Z-scores allow for standardized comparisons even when scales and means differ, enabling meaningful analysis across diverse studies or populations.
• Hypothesis testing:  Z-tests are foundational for determining whether observed results are statistically significant or likely due to chance.
• Outlier detection:  Z-scores help identify unusual values that might warrant further investigation or exclusion from analysis to ensure data integrity.

2. Finance:

• Risk assessment and investment analysis:  Z-scores measure volatility and potential returns of investments, aiding in portfolio optimization and risk management strategies.
• Statistical arbitrage:  Z-scores form the basis of statistical arbitrage models, which seek to exploit pricing inefficiencies in financial markets.
• Credit scoring:  Z-scores are used in credit scoring models to assess the creditworthiness of individuals and businesses, influencing loan approval decisions and interest rates.

3. Education:

• Standardized testing:  Z-scores enable comparisons of student performance across different tests and grade levels, facilitating fair assessments and tracking progress.
• Identifying gifted and talented students:  Z-scores help identify students with exceptional abilities or those requiring additional support, promoting tailored educational approaches.
• Evaluating educational programs and interventions:  Z-scores can measure the effectiveness of instructional methods or interventions by comparing student outcomes before and after implementation.

4. Psychology:

• Psychological assessment:  Z-scores are used to standardize scores on psychological tests and questionnaires, enabling comparisons with norms and identifying potential clinical concerns.
• Research studies:  Z-scores are used to analyze experimental data, assess the effectiveness of interventions, and draw meaningful conclusions from research findings.

5. Healthcare:

• Monitoring patient health:  Z-scores track vital signs, lab results, and other health indicators over time, aiding in early detection of health issues and evaluating treatment effectiveness.
• Epidemiology:  Z-scores are used to identify risk factors for diseases and track disease outbreaks, informing public health policies and interventions.

6. Manufacturing and Quality Control:

• Process control:  Z-scores monitor manufacturing processes to detect deviations from expected quality standards, ensuring consistent product quality and reducing defects.
• Statistical process control (SPC):  Z-scores are central to SPC techniques, which aim to prevent defects and improve process efficiency.

7. Science and Engineering:

• Experiment design and analysis:  Z-scores guide sample size calculations, ensure experimental validity, and assess the significance of findings in various scientific fields.
• Signal processing:  Z-scores are used in signal processing techniques to detect anomalies and extract meaningful information from noisy data.

Benefits of Using the Z-Score Calculator

• Efficiency : Quickly computes z-scores without manual calculations.
• Accuracy : Reduces the risk of human error in computation.
• Ease of Use : Simplifies the process for those unfamiliar with statistical formulas.
• Practical Learning Tool : Helps students and professionals understand and apply z-scores.

The Z-Score Calculator is an invaluable tool for statisticians, researchers, students, and professionals across various fields. It simplifies the process of calculating and understanding z-scores, facilitating data normalization, comparison, and interpretation.

By providing an easy and accurate means to calculate z-scores, this tool aids in the analysis of data sets, ensuring that the nuances and insights embedded in statistical data are accessible to a broader audience. Whether used for academic purposes, professional analysis, or personal interest, the Z-Score Calculator is a testament to the importance of statistical tools in the age of data-driven decision-making.

• NIST/SEMATECH e-Handbook of Statistical Methods – https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
• Z-score – Wikipedia – https://en.wikipedia.org/wiki/Standard_score

Last Updated : 27 February, 2024

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Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields, including database systems, computer networks, and programming. You can read more about him on his bio page.

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20 thoughts on “z score calculator”.

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I agree. Incorporating visualizations could further elucidate the concepts and engage readers more effectively.

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Input 1 Sample

Formula and output, when do you use one sample z-test.

A Z-test assumes the data is normally distributed which according to the central limit theorem is when the sample size is large, usually when n > 30. You use the Z-test for e means when you want to see if there is any different between the means of two normally distributed samples. One sample Z test is used to test whether the mean of a population is less than, greater than or equal to a specific value μ. Example, Morel suspects there are less than 210 candies in an M&Ms bag. Morel buys 60 M&Ms bags and count all the candies in each bag. After some calculations Morel found out the mean value were 203 M&Ms in each bag and the variance was 10. What is the p-value of this test? Was Morel right is there less than 210 M&Ms in the bag? H0: There is 210 M&Ms in each bag H1: There is less than 210 M&Ms in each bag

One Sample Z-Test Calculator

About one sample z-test calculator (formula).

A One Sample Z-Test is a statistical test used to determine whether the mean of a single sample differs significantly from a known population mean or a hypothesized mean. This test is commonly used in hypothesis testing when you have a single set of data points and want to determine if it’s representative of a larger population or if there’s a significant difference between the sample and the population.

Here’s the formula for a One Sample Z-Test:

Z = (X̄ – μ) / (σ / √(n))

• Z is the Z-statistic.
• X̄ (pronounced as “X-bar”) is the sample mean.
• μ (pronounced as “mu”) is the population mean (the known mean or the hypothesized mean).
• σ (pronounced as “sigma”) is the population standard deviation (if known).
• n is the sample size.

The steps to perform a One Sample Z-Test are as follows:

• H0: The sample mean is equal to the population mean (μ).
• Ha: The sample mean is not equal to the population mean (μ), indicating a two-tailed test. Alternatively, you can use a one-tailed test if you have a specific direction in mind (greater than or less than).
• Collect your sample data and calculate the sample mean (X̄) and, if possible, the population standard deviation (σ).
• Determine the significance level (α), which represents the probability of making a Type I error (rejecting the null hypothesis when it is true). Common choices for α include 0.05 and 0.01.
• Calculate the Z-statistic using the formula mentioned above.
• Compare the calculated Z-statistic to the critical Z-value(s) from the standard normal distribution table or use a statistical calculator. The critical value(s) correspond to your chosen significance level (α) and the type of test (two-tailed or one-tailed).
• If |Z| > critical value: Reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha).
• If |Z| ≤ critical value: Fail to reject the null hypothesis (H0).
• Draw a conclusion based on your decision and report the results.

This test helps you determine whether the observed difference between your sample mean and the population mean is statistically significant or if it could have occurred due to random sampling variation.

Keep in mind that for practical purposes, it’s often recommended to use statistical software or calculators to perform One Sample Z-Tests because they can handle the calculations and critical value lookup efficiently.

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• Knowledge Base

Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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Bevans, R. (2023, June 22). Hypothesis Testing | A Step-by-Step Guide with Easy Examples. Scribbr. Retrieved April 15, 2024, from https://www.scribbr.com/statistics/hypothesis-testing/

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Z-test for Two Proportions

Instructions: This calculator conducts a Z-test for two population proportions (\(p_1\) and \(p_2\)), Please select the null and alternative hypotheses, type the significance level, the sample sizes, the number of favorable cases (or the sample proportions) and the results of the z-test will be displayed for you:

When Do You Use a Z-test for Two Proportions?

More about the z-test for two proportions so you can better understand the results yielded by this solver: A z-test for two proportions is a hypothesis test that attempts to make a claim about the population proportions p 1 and p 2 . Specifically, we are interested in assessing whether or not it is reasonable to claim that p 1 = p 2 , using sample information. The Z-test for two proportions has two non-overlapping hypotheses, the null and the alternative hypothesis.

What are the null and alternative hypotheses for the z-test for two proportions?

The null hypothesis is a statement about the population parameter which indicates no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for two population proportions are:

• Depending on our knowledge about the "no effect" situation, the z-test can be two-tailed, left-tailed or right-tailed
• The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
• The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
• In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

What is the z-test formula in this case?

The formula for a z-statistic for two population proportions is

where \(\bar p = \frac{X_1+X_2}{n_1+n_2}\) corresponds to the pooled proportion (Notice that in the above z test for proportions formula, we get in the denominator something like our "best guess" of what the population proportion is from information from the two samples, assuming that the null hypothesis of equality of proportions is true). The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

The Case for one population proportion

In case you only have one sample proportion (so you are testing for one population proportion), you should use our z-test for one proportion calculator , which specifically addresses that case.

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• Hypothesis Testing using the Z-Test on the TI-83 Plus, TI-84 Plus, TI-89, and Voyage 200

The TI-83 Plus and TI-84 Plus are optimized for performing many tasks in statistics, and one of their most powerful features is the ability to perform a variety of tests of statistical significance. With the statistics package installed, the TI-89, TI-92 Plus, and Voyage 200 also have much of this capability. This tutorial demonstrates how to use your graphing calculator to solve basic hypothesis testing problems such as the following using the Z-Test:

A researcher designs an experiment where a random sample of n = 50 high school seniors are given a pill to improve their concentration and problem solving skills. After being administered the pill, subjects take the SAT, and their scores on the SAT Math section are tabulated. The average score of student who took the pill is x̄ = 540. Given that the average score of all high school seniors on the SAT is μ = 510 with standard deviation σ = 100, is there statistically significant evidence that students who took the pill scored higher?

Before beginning the calculations, it is necessary to come up with specific hypotheses for the tests and choose a level of significance. In inferential statistics, there are two hypothesis, the null hypothesis, and the alternative hypothesis. The null hypothesis, denoted H₀, is always that the statistic measures of the treated group (in this case students given a pill) is the same as that for the general population. Since we are only interested in whether or not the pill has a positive effect, we are doing a one-tailed Z-Test, and our null hypothesis is:

H₀: μ <= μ₀

Where μ is the true mean (as opposed to sample mean) of scores of students in the treatment group. μ₀ refers to the known population mean, in this case 510. The alternative hypothesis H 1 is what we expect if the treatment does have an effect on the population, and is always the opposite of the alternative hypothesis. Our alternative hypothesis is:

H₁: μ > μ₀

Finally, we have to choose a level of significance (α) for our test. It is possible that even if the treatment has no effect, we could get a mean score of 540. This seems unlikely and the chances of this happening goes down with the more subjects in the study, but the purpose of hypothesis testing is first of all to avoid coming to the wrong conclusion. The level of significance is a threshold probability below which we say that we have found statistical evidence. It is considered good practice to choose this beforehand so that the statistician doesn’t change α after wards in order to “find” statistical evidence where there is none. For most problems, a level of significance is:

α = .05

This means that if we find there is less than a 5% chance that the sample mean is higher than 540 by chance alone, we will conclude statistical significance.

Performing a Z-Test on the TI-83 Plus and TI-84 Plus

From the home screen, press STAT ▶ ▶ to select the TESTS menu. “Z-Test” should already be selected, so press ENTER to be taken to the Z-Test menu.

Now select the desired settings and values. While it is possible to use a list to store a set of scores from which your calculator can determine the sample data, this problem doesn’t give individual scores, so make sure STATS is selected and press ENTER .

Enter the data given in the problem, μ₀ = 510, σ = 100, x̄ = 540, and n = 50. Finally, make sure to select >μ₀ for the alternative hypothesis.

There are now two options for the output of the Z-Test: “Calculate” displays the z-score (the number of standard deviations x̄ is above or below the mean) and then the corresponding p-value, the probability of getting such a sample by luck alone.

“Draw” draws a normal distribution graph and displays the z-score and p-value at the bottom of the screen.

We have z = 2.12 and p = .017 , which means that there is a 1.7% chance of seeing such a variation in sample mean by chance alone. Since p<α, we can conclude that there is significant evidence that the treatment group is different from the general population. Assuming good experimental practices, this implies (but does not prove) that taking the pill improves students' Math SAT scores. Note that this does not necessarily mean the pill improves concentration and problem solving skills as claimed-although these may be skills important for scoring higher on the Math SAT, this is a separate claim.

Performing a Z-Test on the TI-89, TI-92 Plus, and Voyage 200

Before you begin, it is necessary to have the proper software on your device. If you have a TI-89 Titanium or other newer calculator, then you should have a Stats/List Editor icon on your Apps screen. Otherwise, you should have a Stats/List Editor application in your Flash Apps folder. (Reached by pressing APPS then ENTER ). If you don’t have this software or you aren’t sure, you can download it here .

Once you are in the Stats/List Editor app, press 2nd F1 (F6) to enter the tests menu. Z-Test should already be selected, so press ENTER to confirm. You will be prompted for the data input method. Data uses a list containing the of scores from which your calculator can determine the sample data, this problem doesn’t give individual scores, so make sure STATS is selected and press ENTER .

Enter the data given in the problem, μ₀ = 510, σ = 100, x̄ = 540, and n = 50. Finally, make sure to select μ > μ₀ for the alternative hypothesis.

There are two options for the output of the Z-Test. Selecting “Results: Calculate” displays the z-score (the number of standard deviations x̄ is above or below the mean) and then the corresponding p-value, the probability of getting such a sample by luck alone.

“Results: Draw” draws a normal distribution graph and displays the z-score and p-value at the bottom of the screen.

We have z = 2.12 and p = .017 , which means that there is a 1.7% chance of seeing such a variation in sample mean by chance alone. Since p<α, we can conclude that there is significant evidence that the treatment group is different from the general population. As before, this implies (but does not prove) that taking the pill improves students' Math SAT scores.

You might also like:

• How to Enter Logarithms on Your Graphing Calculator
• Evaluating Integrals on your TI-89, TI-92+, or Voyage 200
• Video: Using Variables on Your TI Graphing Calculator

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