z test null hypothesis calculator

Z-Test Calculator

This Z-test calculator computes data for both one-sample and two-sample Z-tests. It also provides a diagram to show the position of the Z-score and the acceptance/rejection regions. When making a two-sample Z-test calculation, the population mean difference, d, represents the difference between the population means of sample one and sample two, which is μ 1 -μ 2 . To use this calculator, simply select the type of calculation from the tab, enter the values, and click the 'Calculate' button.

The Z-test is a statistical procedure used to determine whether there is a significant difference between means, either between a sample mean and a known population mean (one-sample Z-test) or between the means of two independent samples (two-sample Z-test). It assumes that the data is normally distributed and is particularly useful when the sample sizes are large (>30) and the population standard deviations are known. When analyzing data to make informed decisions, statistical hypothesis tests are indispensable tools used to determine if evidence exists to reject a prevailing assumption or theory, known as the null hypothesis. The Z-test is one of these tests.

One-Sample Z-Test

The one-sample Z-test is used when you want to compare the mean of a single sample to a known population mean to see if there is a significant difference. This is particularly common in quality control and other scenarios where the standard deviation of the population is known.

• Null Hypothesis (H 0 ): The sample mean is equal to the population mean (x̅=μ).
• Alternative Hypothesis (H 1 ): The sample mean is not equal to the population mean (x̅≠μ). This can also be one-tailed (x̅>μ or x̅<μ) depending on the direction of interest.

The formula for the Z-statistic in a one-sample Z-test is:

• x̅ is the sample mean
• μ is the population mean
• σ is the population standard deviation
• n is the sample size

Example: Suppose a school administrator knows the national average score for a standardized test is 500 with a standard deviation of 50. A sample of 100 students from a new teaching program scores an average of 520. To determine if this program significantly differs from the national average:

This Z-value would then be compared against a critical value from the Z-distribution table typically at a 0.05 significance level. The critical value for a 0.05 significance level is approximately ±1.96. The Z-value of 4 is greater than 1.96. Therefore, the null hypothesis is rejected and the score of this program is considered significantly different from the national average at the 0.05 significance level.

Two-Sample Z-Test

The two-sample Z-test (or independent samples Z-test) compares the means from two independent groups to determine if there is a statistically significant difference between them.

• Null Hypothesis (H 0 ): The two population means have a difference of d (μ 1 -μ 2 =d). If d is 0, the null hypothesis states that the two population means are equal (μ 1 =μ 2 ).
• Alternative Hypothesis (H 1 ): The difference between two population means is not d (μ 1 -μ 2 ≠d), which can also be directional (μ 1 -μ 2 >d or μ 1 -μ 2 <d). If d is 0, the alternative hypothesis becomes μ 1 ≠μ 2 , or μ 1 >μ 2 or μ 1 <μ 2 if it is directional.

The formula for calculating the Z-statistic in a two-sample Z-test is:

• x̅ 1 and x̅ 2 are the sample means of groups 1 and 2, respectively
• μ 1 and μ 2 are the population means, with μ 1 - μ 2 = d. d is often hypothesized to be zero under the null hypothesis.
• σ 1 and σ 2 are the population standard deviations
• n 1 and n 2 are the sample sizes of the two groups

Example: Consider two groups of employees from different branches of a company undergoing training. Group A has 50 employees with an average score of 80 and a standard deviation of 10, and Group B has 50 employees with an average score of 75 and a standard deviation of 12. To test if there's a significant difference:

This Z-value is then compared to the critical Z-values to assess significance. The critical value of a 0.05 significance level is around ±1.95. The Z-value of 2.26 is more than 1.95. Therefore, the two group has significant difference at 0.05 significance level.

Significance Level

The significance level (α) is a critical concept in hypothesis testing. It represents the probability threshold below which the null hypothesis will be rejected. Common levels are 0.05 (5%) or 0.01 (1%). The choice of α affects the Z-critical value, which is used to determine whether to reject the null hypothesis based on the computed Z-score.

• Critical Value: This is a point on the Z-distribution that the test statistic must exceed to reject the null hypothesis. For instance, at a 5% significance level in a two-tailed test, the critical values are approximately ±1.96. The significance level (probability) and critical value (Z-score) can be converted with each other the Z-distribution table or use our Z/P converter .

Using the above examples, if the computed Z-scores exceed the respective critical values, the null hypotheses in each case would be rejected, indicating a statistically significant difference as per the alternative hypotheses. These examples demonstrate how the Z-test is applied in different scenarios to test hypotheses concerning population means.

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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

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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Z Score Calculator for 2 Population Proportions

This is a simple z score calculator that calculates the value of z (and associated p value) for two population proportions.

Further Information

The z score test for two population proportions is used when you want to know whether two populations or groups (e.g., males and females; theists and atheists) differ significantly on some single (categorical) characteristic - for example, whether they are vegetarians.

Requirements

• A random sample of each of the population groups to be compared.
• Categorial data

Null Hypothesis

H0: p1 - p2 = 0, where p1 is the proportion from the first population and p2 the proportion from the second.

As above, the null hypothesis tends to be that there is no difference between the two population proportions; or, more formally, that the difference is zero (so, for example, that there is no difference between the proportion of males who are vegetarian and the proportion of females who are vegetarian). 

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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27: Hypothesis Test for a Population Mean Given Statistics Calculator

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hypothesis test for a population mean given statistics calculator

Select if the population standard deviation, \(\sigma\), is known or unknown. Then fill in the standard deviation, the sample mean, \(\bar{x}\), the sample size, \(n\), the hypothesized population mean \(\mu_0\), and indicate if the test is left tailed, <, right tailed, >, or two tailed, \(\neq\).  Then hit "Calculate" and the test statistic and p-Value will be calculated for you.

Critical Value Calculator

How to use critical value calculator, what is a critical value, critical value definition, how to calculate critical values, z critical values, t critical values, chi-square critical values (χ²), f critical values, behind the scenes of the critical value calculator.

Welcome to the critical value calculator! Here you can quickly determine the critical value(s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N(0,1) (that is when you have a Z-score), t-Student, chi-square, and F-distribution .

What is a critical value? And what is the critical value formula? Scroll down – we provide you with the critical value definition and explain how to calculate critical values in order to use them to construct rejection regions (also known as critical regions).

The critical value calculator is your go-to tool for swiftly determining critical values in statistical tests, be it one-tailed or two-tailed. To effectively use the calculator, follow these steps:

In the first field, input the distribution of your test statistic under the null hypothesis: is it a standard normal N (0,1), t-Student, chi-squared, or Snedecor's F? If you are not sure, check the sections below devoted to those distributions, and try to localize the test you need to perform.

In the field What type of test? choose the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

If needed, specify the degrees of freedom of the test statistic's distribution. If you need more clarification, check the description of the test you are performing. You can learn more about the meaning of this quantity in statistics from the degrees of freedom calculator .

Set the significance level, α \alpha α . By default, we pre-set it to the most common value, 0.05, but you can adjust it to your needs.

The critical value calculator will display your critical value(s) and the rejection region(s).

Click the advanced mode if you need to increase the precision with which the critical values are computed.

For example, let's envision a scenario where you are conducting a one-tailed hypothesis test using a t-Student distribution with 15 degrees of freedom. You have opted for a right-tailed test and set a significance level (α) of 0.05. The results indicate that the critical value is 1.7531, and the critical region is (1.7531, ∞). This implies that if your test statistic exceeds 1.7531, you will reject the null hypothesis at the 0.05 significance level.

👩‍🏫 Want to learn more about critical values? Keep reading!

In hypothesis testing, critical values are one of the two approaches which allow you to decide whether to retain or reject the null hypothesis. The other approach is to calculate the p-value (for example, using the p-value calculator ).

The critical value approach consists of checking if the value of the test statistic generated by your sample belongs to the so-called rejection region , or critical region , which is the region where the test statistic is highly improbable to lie . A critical value is a cut-off value (or two cut-off values in the case of a two-tailed test) that constitutes the boundary of the rejection region(s). In other words, critical values divide the scale of your test statistic into the rejection region and the non-rejection region.

Once you have found the rejection region, check if the value of the test statistic generated by your sample belongs to it :

• If so, it means that you can reject the null hypothesis and accept the alternative hypothesis; and
• If not, then there is not enough evidence to reject H 0 .

But how to calculate critical values? First of all, you need to set a significance level , α \alpha α , which quantifies the probability of rejecting the null hypothesis when it is actually correct. The choice of α is arbitrary; in practice, we most often use a value of 0.05 or 0.01. Critical values also depend on the alternative hypothesis you choose for your test , elucidated in the next section .

To determine critical values, you need to know the distribution of your test statistic under the assumption that the null hypothesis holds. Critical values are then points with the property that the probability of your test statistic assuming values at least as extreme at those critical values is equal to the significance level α . Wow, quite a definition, isn't it? Don't worry, we'll explain what it all means.

First, let us point out it is the alternative hypothesis that determines what "extreme" means. In particular, if the test is one-sided, then there will be just one critical value; if it is two-sided, then there will be two of them: one to the left and the other to the right of the median value of the distribution.

Critical values can be conveniently depicted as the points with the property that the area under the density curve of the test statistic from those points to the tails is equal to α \alpha α :

Left-tailed test: the area under the density curve from the critical value to the left is equal to α \alpha α ;

Right-tailed test: the area under the density curve from the critical value to the right is equal to α \alpha α ; and

Two-tailed test: the area under the density curve from the left critical value to the left is equal to α / 2 \alpha/2 α /2 , and the area under the curve from the right critical value to the right is equal to α / 2 \alpha/2 α /2 as well; thus, total area equals α \alpha α .

As you can see, finding the critical values for a two-tailed test with significance α \alpha α boils down to finding both one-tailed critical values with a significance level of α / 2 \alpha/2 α /2 .

The formulae for the critical values involve the quantile function , Q Q Q , which is the inverse of the cumulative distribution function ( c d f \mathrm{cdf} cdf ) for the test statistic distribution (calculated under the assumption that H 0 holds!): Q = c d f − 1 Q = \mathrm{cdf}^{-1} Q = cdf − 1 .

Once we have agreed upon the value of α \alpha α , the critical value formulae are the following:

• Left-tailed test :
• Right-tailed test :
• Two-tailed test :

In the case of a distribution symmetric about 0 , the critical values for the two-tailed test are symmetric as well:

Unfortunately, the probability distributions that are the most widespread in hypothesis testing have somewhat complicated c d f \mathrm{cdf} cdf formulae. To find critical values by hand, you would need to use specialized software or statistical tables. In these cases, the best option is, of course, our critical value calculator! 😁

Use the Z (standard normal) option if your test statistic follows (at least approximately) the standard normal distribution N(0,1) .

In the formulae below, u u u denotes the quantile function of the standard normal distribution N(0,1):

Left-tailed Z critical value: u ( α ) u(\alpha) u ( α )

Right-tailed Z critical value: u ( 1 − α ) u(1-\alpha) u ( 1 − α )

Two-tailed Z critical value: ± u ( 1 − α / 2 ) \pm u(1- \alpha/2) ± u ( 1 − α /2 )

Check out Z-test calculator to learn more about the most common Z-test used on the population mean. There are also Z-tests for the difference between two population means, in particular, one between two proportions.

Use the t-Student option if your test statistic follows the t-Student distribution . This distribution is similar to N(0,1) , but its tails are fatter – the exact shape depends on the number of degrees of freedom . If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). Check our t-statistic calculator to compute the related test statistic.

In the formulae below, Q t , d Q_{\text{t}, d} Q t , d ​ is the quantile function of the t-Student distribution with d d d degrees of freedom:

Left-tailed t critical value: Q t , d ( α ) Q_{\text{t}, d}(\alpha) Q t , d ​ ( α )

Right-tailed t critical value: Q t , d ( 1 − α ) Q_{\text{t}, d}(1 - \alpha) Q t , d ​ ( 1 − α )

Two-tailed t critical values: ± Q t , d ( 1 − α / 2 ) \pm Q_{\text{t}, d}(1 - \alpha/2) ± Q t , d ​ ( 1 − α /2 )

Visit the t-test calculator to learn more about various t-tests: the one for a population mean with an unknown population standard deviation , those for the difference between the means of two populations (with either equal or unequal population standard deviations), as well as about the t-test for paired samples .

Use the χ² (chi-square) option when performing a test in which the test statistic follows the χ²-distribution .

You need to determine the number of degrees of freedom of the χ²-distribution of your test statistic – below, we list them for the most commonly used χ²-tests.

Here we give the formulae for chi square critical values; Q χ 2 , d Q_{\chi^2, d} Q χ 2 , d ​ is the quantile function of the χ²-distribution with d d d degrees of freedom:

Left-tailed χ² critical value: Q χ 2 , d ( α ) Q_{\chi^2, d}(\alpha) Q χ 2 , d ​ ( α )

Right-tailed χ² critical value: Q χ 2 , d ( 1 − α ) Q_{\chi^2, d}(1 - \alpha) Q χ 2 , d ​ ( 1 − α )

Two-tailed χ² critical values: Q χ 2 , d ( α / 2 ) Q_{\chi^2, d}(\alpha/2) Q χ 2 , d ​ ( α /2 ) and Q χ 2 , d ( 1 − α / 2 ) Q_{\chi^2, d}(1 - \alpha/2) Q χ 2 , d ​ ( 1 − α /2 )

Several different tests lead to a χ²-score:

Goodness-of-fit test : does the empirical distribution agree with the expected distribution?

This test is right-tailed . Its test statistic follows the χ²-distribution with k − 1 k - 1 k − 1 degrees of freedom, where k k k is the number of classes into which the sample is divided.

Independence test : is there a statistically significant relationship between two variables?

This test is also right-tailed , and its test statistic is computed from the contingency table. There are ( r − 1 ) ( c − 1 ) (r - 1)(c - 1) ( r − 1 ) ( c − 1 ) degrees of freedom, where r r r is the number of rows, and c c c is the number of columns in the contingency table.

Test for the variance of normally distributed data : does this variance have some pre-determined value?

This test can be one- or two-tailed! Its test statistic has the χ²-distribution with n − 1 n - 1 n − 1 degrees of freedom, where n n n is the sample size.

Finally, choose F (Fisher-Snedecor) if your test statistic follows the F-distribution . This distribution has a pair of degrees of freedom .

Let us see how those degrees of freedom arise. Assume that you have two independent random variables, X X X and Y Y Y , that follow χ²-distributions with d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ degrees of freedom, respectively. If you now consider the ratio ( X d 1 ) : ( Y d 2 ) (\frac{X}{d_1}):(\frac{Y}{d_2}) ( d 1 ​ X ​ ) : ( d 2 ​ Y ​ ) , it turns out it follows the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom. That's the reason why we call d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ the numerator and denominator degrees of freedom , respectively.

In the formulae below, Q F , d 1 , d 2 Q_{\text{F}, d_1, d_2} Q F , d 1 ​ , d 2 ​ ​ stands for the quantile function of the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom:

Left-tailed F critical value: Q F , d 1 , d 2 ( α ) Q_{\text{F}, d_1, d_2}(\alpha) Q F , d 1 ​ , d 2 ​ ​ ( α )

Right-tailed F critical value: Q F , d 1 , d 2 ( 1 − α ) Q_{\text{F}, d_1, d_2}(1 - \alpha) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α )

Two-tailed F critical values: Q F , d 1 , d 2 ( α / 2 ) Q_{\text{F}, d_1, d_2}(\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( α /2 ) and Q F , d 1 , d 2 ( 1 − α / 2 ) Q_{\text{F}, d_1, d_2}(1 -\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α /2 )

Here we list the most important tests that produce F-scores: each of them is right-tailed .

ANOVA : tests the equality of means in three or more groups that come from normally distributed populations with equal variances. There are ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where k k k is the number of groups, and n n n is the total sample size (across every group).

Overall significance in regression analysis . The test statistic has ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where n n n is the sample size, and k k k is the number of variables (including the intercept).

Compare two nested regression models . The test statistic follows the F-distribution with ( k 2 − k 1 , n − k 2 ) (k_2 - k_1, n - k_2) ( k 2 ​ − k 1 ​ , n − k 2 ​ ) degrees of freedom, where k 1 k_1 k 1 ​ and k 2 k_2 k 2 ​ are the number of variables in the smaller and bigger models, respectively, and n n n is the sample size.

The equality of variances in two normally distributed populations . There are ( n − 1 , m − 1 ) (n - 1, m - 1) ( n − 1 , m − 1 ) degrees of freedom, where n n n and m m m are the respective sample sizes.

I'm Anna, the mastermind behind the critical value calculator and a PhD in mathematics from Jagiellonian University .

The idea for creating the tool originated from my experiences in teaching and research. Recognizing the need for a tool that simplifies the critical value determination process across various statistical distributions, I built a user-friendly calculator accessible to both students and professionals. After publishing the tool, I soon found myself using the calculator in my research and as a teaching aid.

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What is a Z critical value?

A Z critical value is the value that defines the critical region in hypothesis testing when the test statistic follows the standard normal distribution . If the value of the test statistic falls into the critical region, you should reject the null hypothesis and accept the alternative hypothesis.

How do I calculate Z critical value?

To find a Z critical value for a given confidence level α :

Check if you perform a one- or two-tailed test .

For a one-tailed test:

Left -tailed: critical value is the α -th quantile of the standard normal distribution N(0,1).

Right -tailed: critical value is the (1-α) -th quantile.

Two-tailed test: critical value equals ±(1-α/2) -th quantile of N(0,1).

No quantile tables ? Use CDF tables! (The quantile function is the inverse of the CDF.)

Verify your answer with an online critical value calculator.

Is a t critical value the same as Z critical value?

In theory, no . In practice, very often, yes . The t-Student distribution is similar to the standard normal distribution, but it is not the same . However, if the number of degrees of freedom (which is, roughly speaking, the size of your sample) is large enough (>30), then the two distributions are practically indistinguishable , and so the t critical value has practically the same value as the Z critical value.

What is the Z critical value for 95% confidence?

The Z critical value for a 95% confidence interval is:

• 1.96 for a two-tailed test;
• 1.64 for a right-tailed test; and
• -1.64 for a left-tailed test.
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Decision Rule Calculator

One-tailed or two-tailed hypothesis?

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Z-statistic or t-statistic?

Decision Rule: fail to reject the null hypothesis

Explanation:

The p-value for a Z-statistic of 1.34 for a two-tailed test is 0.18025 . Since this p-value is greater than 0.05 , we fail to reject the null hypothesis .

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Z-test: One Population Proportion

Instructions: This calculator conducts a Z-test for one population proportion (p). Please select the null and alternative hypotheses, type the hypothesized population proportion \(p_0\), the significance level \(\alpha\), the sample proportion or number o favorable cases, and the sample size, and the results of the z-test for one proportion will be displayed for you:

Z-Test for One Population Proportion

More about the z-test for one population proportion so you can better interpret the results obtained by this solver: A z-test for one proportion is a hypothesis test that attempts to make a claim about the population proportion (p) for a certain population attribute (proportion of males, proportion of people underage). The test has two non-overlapping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population proportion, which corresponds to the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for one population proportion 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 sampling distribution used to construct the test statistics is approximately normal
• 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

The formula for a z-statistic is

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).

This one proportion z test calculator will allow you to compute the critical values are p-values for this one sample proportion test, that will help you decide whether or not the sample data provides enough evidence to reject the null hypothesis. If instead, what you want to do is to compare two sample proportions, you can use this z-test for two proportions calculator , which will help you assess whether the two sample proportions differ significantly.

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