Z-test Calculator

Table of contents

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 .

What is a Z-test?

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.

When do I use Z-tests?

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.

Z-test formula

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!

p-value from Z-test

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:

Two-tailed Z-test and one-tailed Z-test

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 ∣ :

two-tailed p value

  • Left-tailed Z-test:

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

left-tailed p value

  • Right-tailed Z-test:

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

right-tailed p value

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.

Z-test critical values & critical regions

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 ) .

How to use the one-sample Z-test calculator?

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.

Z-test examples

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!

Here, we perform a Z-test for population mean μ. Null hypothesis H₀: μ = μ₀.

Alternative hypothesis H₁

Significance level α

The probability that we reject the true hypothesis H₀ (type I error).

Hypothesis Testing Calculator

$H_o$:
$H_a$: μ μ₀
$n$ =   $\bar{x}$ =   =
$\text{Test Statistic: }$ =
$\text{Degrees of Freedom: } $ $df$ =
$ \text{Level of Significance: } $ $\alpha$ =

Type II Error

$H_o$: $\mu$
$H_a$: $\mu$ $\mu_0$
$n$ =   σ =   $\mu$ =
$\text{Level of Significance: }$ $\alpha$ =

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.

$\sigma$ Known $\sigma$ Unknown
Test Statistic $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $

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.

Lower Tail Test Upper Tail Test Two-Tailed Test
$H_0 \colon \mu \geq \mu_0$ $H_0 \colon \mu \leq \mu_0$ $H_0 \colon \mu = \mu_0$
$H_a \colon \mu $H_a \colon \mu \neq \mu_0$

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.

Lower Tail Test Upper Tail Test Two-Tailed Test
If $z \leq -z_\alpha$, reject $H_0$. If $z \geq z_\alpha$, reject $H_0$. If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$.
If $t \leq -t_\alpha$, reject $H_0$. If $t \geq t_\alpha$, reject $H_0$. If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$.

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.

Condition
$H_0$ True $H_a$ True
Conclusion Accept $H_0$ Correct Type II Error
Reject $H_0$ Type I Error Correct

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.

Z-Test Calculator

Sample size, n
Population mean, μ
Population standard deviation, σ
Alternative hypothesis H :
Sample 1
Sample mean, x̅
Sample size, n
Population standard deviation, σ
Sample 2
Sample mean, x̅
Sample size, n
Population standard deviation, σ
 
Population mean difference, d
Alternative hypothesis H : -μ ≠d -μ <d -μ >d

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:

Z =
  x̅ - μ  
σ
n
  • 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:

Z =
  520 - 500  
50
100
  20  
5

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:

Z =
  (x̅ - x̅ ) - (μ - μ )  
σ
n
σ
n
  • 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:

Z =
  (80 - 75) - 0  
10
50
12
50
  5  
2.21

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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Statistics By Jim

Making statistics intuitive

Z Test: Uses, Formula & Examples

By Jim Frost Leave a Comment

What is a Z Test?

Use a Z test when you need to compare group means. Use the 1-sample analysis to determine whether a population mean is different from a hypothesized value. Or use the 2-sample version to determine whether two population means differ.

A Z test is a form of inferential statistics . It uses samples to draw conclusions about populations.

For example, use Z tests to assess the following:

  • One sample : Do students in an honors program have an average IQ score different than a hypothesized value of 100?
  • Two sample : Do two IQ boosting programs have different mean scores?

In this post, learn about when to use a Z test vs T test. Then we’ll review the Z test’s hypotheses, assumptions, interpretation, and formula. Finally, we’ll use the formula in a worked example.

Related post : Difference between Descriptive and Inferential Statistics

Z test vs T test

Z tests and t tests are similar. They both assess the means of one or two groups, have similar assumptions, and allow you to draw the same conclusions about population means.

However, there is one critical difference.

Z tests require you to know the population standard deviation, while t tests use a sample estimate of the standard deviation. Learn more about Population Parameters vs. Sample Statistics .

In practice, analysts rarely use Z tests because it’s rare that they’ll know the population standard deviation. It’s even rarer that they’ll know it and yet need to assess an unknown population mean!

A Z test is often the first hypothesis test students learn because its results are easier to calculate by hand and it builds on the standard normal distribution that they probably already understand. Additionally, students don’t need to know about the degrees of freedom .

Z and T test results converge as the sample size approaches infinity. Indeed, for sample sizes greater than 30, the differences between the two analyses become small.

William Sealy Gosset developed the t test specifically to account for the additional uncertainty associated with smaller samples. Conversely, Z tests are too sensitive to mean differences in smaller samples and can produce statistically significant results incorrectly (i.e., false positives).

When to use a T Test vs Z Test

Let’s put a button on it.

When you know the population standard deviation, use a Z test.

When you have a sample estimate of the standard deviation, which will be the vast majority of the time, the best statistical practice is to use a t test regardless of the sample size.

However, the difference between the two analyses becomes trivial when the sample size exceeds 30.

Learn more about a T-Test Overview: How to Use & Examples and How T-Tests Work .

Z Test Hypotheses

This analysis uses sample data to evaluate hypotheses that refer to population means (µ). The hypotheses depend on whether you’re assessing one or two samples.

One-Sample Z Test Hypotheses

  • Null hypothesis (H 0 ): The population mean equals a hypothesized value (µ = µ 0 ).
  • Alternative hypothesis (H A ): The population mean DOES NOT equal a hypothesized value (µ ≠ µ 0 ).

When the p-value is less or equal to your significance level (e.g., 0.05), reject the null hypothesis. The difference between your sample mean and the hypothesized value is statistically significant. Your sample data support the notion that the population mean does not equal the hypothesized value.

Related posts : Null Hypothesis: Definition, Rejecting & Examples and Understanding Significance Levels

Two-Sample Z Test Hypotheses

  • Null hypothesis (H 0 ): Two population means are equal (µ 1 = µ 2 ).
  • Alternative hypothesis (H A ): Two population means are not equal (µ 1 ≠ µ 2 ).

Again, when the p-value is less than or equal to your significance level, reject the null hypothesis. The difference between the two means is statistically significant. Your sample data support the idea that the two population means are different.

These hypotheses are for two-sided analyses. You can use one-sided, directional hypotheses instead. Learn more in my post, One-Tailed and Two-Tailed Hypothesis Tests Explained .

Related posts : How to Interpret P Values and Statistical Significance

Z Test Assumptions

For reliable results, your data should satisfy the following assumptions:

You have a random sample

Drawing a random sample from your target population helps ensure that the sample represents the population. Representative samples are crucial for accurately inferring population properties. The Z test results won’t be valid if your data do not reflect the population.

Related posts : Random Sampling and Representative Samples

Continuous data

Z tests require continuous data . Continuous variables can assume any numeric value, and the scale can be divided meaningfully into smaller increments, such as fractional and decimal values. For example, weight, height, and temperature are continuous.

Other analyses can assess additional data types. For more information, read Comparing Hypothesis Tests for Continuous, Binary, and Count Data .

Your sample data follow a normal distribution, or you have a large sample size

All Z tests assume your data follow a normal distribution . However, due to the central limit theorem, you can ignore this assumption when your sample is large enough.

The following sample size guidelines indicate when normality becomes less of a concern:

  • One-Sample : 20 or more observations.
  • Two-Sample : At least 15 in each group.

Related posts : Central Limit Theorem and Skewed Distributions

Independent samples

For the two-sample analysis, the groups must contain different sets of items. This analysis compares two distinct samples.

Related post : Independent and Dependent Samples

Population standard deviation is known

As I mention in the Z test vs T test section, use a Z test when you know the population standard deviation. However, when n > 30, the difference between the analyses becomes trivial.

Related post : Standard Deviations

Z Test Formula

These Z test formulas allow you to calculate the test statistic. Use the Z statistic to determine statistical significance by comparing it to the appropriate critical values and use it to find p-values.

The correct formula depends on whether you’re performing a one- or two-sample analysis. Both formulas require sample means (x̅) and sample sizes (n) from your sample. Additionally, you specify the population standard deviation (σ) or variance (σ 2 ), which does not come from your sample.

I present a worked example using the Z test formula at the end of this post.

Learn more about Z-Scores and Test Statistics .

One Sample Z Test Formula

One sample Z test formula.

The one sample Z test formula is a ratio.

The numerator is the difference between your sample mean and a hypothesized value for the population mean (µ 0 ). This value is often a strawman argument that you hope to disprove.

The denominator is the standard error of the mean. It represents the uncertainty in how well the sample mean estimates the population mean.

Learn more about the Standard Error of the Mean .

Two Sample Z Test Formula

Two sample Z test formula.

The two sample Z test formula is also a ratio.

The numerator is the difference between your two sample means.

The denominator calculates the pooled standard error of the mean by combining both samples. In this Z test formula, enter the population variances (σ 2 ) for each sample.

Z Test Critical Values

As I mentioned in the Z vs T test section, a Z test does not use degrees of freedom. It evaluates Z-scores in the context of the standard normal distribution. Unlike the t-distribution , the standard normal distribution doesn’t change shape as the sample size changes. Consequently, the critical values don’t change with the sample size.

To find the critical value for a Z test, you need to know the significance level and whether it is one- or two-tailed.

0.01 Two-Tailed ±2.576
0.01 Left Tail –2.326
0.01 Right Tail +2.326
0.05 Two-Tailed ±1.960
0.05 Left Tail +1.650
0.05 Right Tail –1.650

Learn more about Critical Values: Definition, Finding & Calculator .

Z Test Worked Example

Let’s close this post by calculating the results for a Z test by hand!

Suppose we randomly sampled subjects from an honors program. We want to determine whether their mean IQ score differs from the general population. The general population’s IQ scores are defined as having a mean of 100 and a standard deviation of 15.

We’ll determine whether the difference between our sample mean and the hypothesized population mean of 100 is statistically significant.

Specifically, we’ll use a two-tailed analysis with a significance level of 0.05. Looking at the table above, you’ll see that this Z test has critical values of ± 1.960. Our results are statistically significant if our Z statistic is below –1.960 or above +1.960.

The hypotheses are the following:

  • Null (H 0 ): µ = 100
  • Alternative (H A ): µ ≠ 100

Entering Our Results into the Formula

Here are the values from our study that we need to enter into the Z test formula:

  • IQ score sample mean (x̅): 107
  • Sample size (n): 25
  • Hypothesized population mean (µ 0 ): 100
  • Population standard deviation (σ): 15

Using the formula to calculate the results.

The Z-score is 2.333. This value is greater than the critical value of 1.960, making the results statistically significant. Below is a graphical representation of our Z test results showing how the Z statistic falls within the critical region.

Graph displaying the Z statistic falling in the critical region.

We can reject the null and conclude that the mean IQ score for the population of honors students does not equal 100. Based on the sample mean of 107, we know their mean IQ score is higher.

Now let’s find the p-value. We could use technology to do that, such as an online calculator. However, let’s go old school and use a Z table.

To find the p-value that corresponds to a Z-score from a two-tailed analysis, we need to find the negative value of our Z-score (even when it’s positive) and double it.

In the truncated Z-table below, I highlight the cell corresponding to a Z-score of -2.33.

Using a Z-table to find the p-value.

The cell value of 0.00990 represents the area or probability to the left of the Z-score -2.33. We need to double it to include the area > +2.33 to obtain the p-value for a two-tailed analysis.

P-value = 0.00990 * 2 = 0.0198

That p-value is an approximation because it uses a Z-score of 2.33 rather than 2.333. Using an online calculator, the p-value for our Z test is a more precise 0.0196. This p-value is less than our significance level of 0.05, which reconfirms the statistically significant results.

See my full Z-table , which explains how to use it to solve other types of problems.

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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.

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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.

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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 test statistic calculator hypothesis testing

Z-Test for Statistical Hypothesis Testing Explained

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

Egor Howell

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: Definition & Two Proportion Z-Test

What is a z test.

z test

For example, if someone said they had found a new drug that cures cancer, you would want to be sure it was probably true. A hypothesis test will tell you if it’s probably true, or probably not true. A Z test, is used when your data is approximately normally distributed (i.e. the data has the shape of a bell curve when you graph it).

When you can run a Z Test.

Several different types of tests are used in statistics (i.e. f test , chi square test , t test ). You would use a Z test if:

  • Your sample size is greater than 30 . Otherwise, use a t test .
  • Data points should be independent from each other. In other words, one data point isn’t related or doesn’t affect another data point.
  • Your data should be normally distributed . However, for large sample sizes (over 30) this doesn’t always matter.
  • Your data should be randomly selected from a population, where each item has an equal chance of being selected.
  • Sample sizes should be equal if at all possible.

How do I run a Z Test?

Running a Z test on your data requires five steps:

  • State the null hypothesis and alternate hypothesis .
  • Choose an alpha level .
  • Find the critical value of z in a z table .
  • Calculate the z test statistic (see below).
  • Compare the test statistic to the critical z value and decide if you should support or reject the null hypothesis .

You could perform all these steps by hand. For example, you could find a critical value by hand , or calculate a z value by hand . For a step by step example, watch the following video: Watch the video for an example:

You could also use technology, for example:

  • Two sample z test in Excel .
  • Find a critical z value on the TI 83 .
  • Find a critical value on the TI 89 (left-tail) .

Two Proportion Z-Test

A Two Proportion Z-Test (or Z-interval) allows you to calculate the true difference in proportions of two independent groups to a given confidence interval .

There are a few familiar conditions that need to be met for the Two Proportion Z-Interval to be valid.

  • The groups must be independent. Subjects can be in one group or the other, but not both – like teens and adults.
  • The data must be selected randomly and independently from a homogenous population. A survey is a common example.
  • The population should be at least ten times bigger than the sample size. If the population is teenagers for example, there should be at least ten times as many total teenagers as the number of teenagers being surveyed.
  • The null hypothesis (H 0 ) for the test is that the proportions are the same.
  • The alternate hypothesis (H 1 ) is that the proportions are not the same.

Example question: let’s say you’re testing two flu drugs A and B. Drug A works on 41 people out of a sample of 195. Drug B works on 351 people in a sample of 605. Are the two drugs comparable? Use a 5% alpha level .

Step 1: Find the two proportions:

  • P 1 = 41/195 = 0.21 (that’s 21%)
  • P 2 = 351/605 = 0.58 (that’s 58%).

Set these numbers aside for a moment.

Step 2: Find the overall sample proportion . The numerator will be the total number of “positive” results for the two samples and the denominator is the total number of people in the two samples.

  • p = (41 + 351) / (195 + 605) = 0.49.

Set this number aside for a moment.

two-proprtion-z-test

Solving the formula, we get: Z = 8.99

We need to find out if the z-score falls into the “ rejection region .”

z alpha

Step 5: Compare the calculated z-score from Step 3 with the table z-score from Step 4. If the calculated z-score is larger, you can reject the null hypothesis.

8.99 > 1.96, so we can reject the null hypothesis .

Example 2:  Suppose that in a survey of 700 women and 700 men, 35% of women and 30% of men indicated that they support a particular presidential candidate. Let’s say we wanted to find the true difference in proportions of these two groups to a 95% confidence interval .

At first glance the survey indicates that women support the candidate more than men by about 5% . However, for this statistical inference to be valid we need to construct a range of values to a given confidence interval.

To do this, we use the formula for Two Proportion Z-Interval:

z test statistic calculator hypothesis testing

Plugging in values we find the true difference in proportions to be

z test statistic calculator hypothesis testing

Based on the results of the survey, we are 95% confident that the difference in proportions of women and men that support the presidential candidate is between about 0 % and 10% .

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P Value from Z Score Calculator

This is very easy: just stick your Z score in the box marked Z score, select your significance level and whether you're testing a one or two-tailed hypothesis (if you're not sure, go with the defaults), then press the button!

If you need to derive a Z score from raw data, you can find a Z test calculator here .

Z score:

Enter your z score value, and then press the button.

Additional Z Statistic Calculators

If you're interested in using the z statistic for hypothesis testing and the like, then we have a number of other calculators that might help you.

Z-Test Calculator for a Single Sample Z-Test Calculator for 2 Population Proportions Z Score Calculator for a Single Raw Value (Includes z from p )

z test statistic calculator hypothesis testing

Z Table. Z Score Table. Normal Distribution Table. Standard Normal Table.

What is Z-Test?

Z-Test is a statistical test which let’s us approximate the distribution of the test statistic under the null hypothesis using normal distribution .

Z-Test is a test statistic commonly used in hypothesis test when the sample data is large.For carrying out the Z-Test, population parameters such as mean, variance, and standard deviation should be known.

This test is widely used to determine whether the mean of the two samples are different when the variance is known. We make use of the Z score and the Z table for running the Z-Test.

Z-Test as Hypothesis Test

A test statistic is a random variable that we calculate from the sample data to determine whether to reject the null hypothesis. This random variable is used to calculate the P-value, which indicates how strong the evidence is against the null hypothesis. Z-Test is such a test statistic where we make use of the mean value and z score to determine the P-value. Z-Test compares the mean of two large samples taken from a population when the variance is known.

Z-Test is usually used to conduct a hypothesis test when the sample size is greater than 30. This is because of the central limit theorem where when the sample gets larger, the distributed data graph starts resembling a bell curve and is considered to be distributed normally. Since the Z-Test follows normal distribution under the null hypothesis, it is the most suitable test statistic for large sample data.

Why do we use a large sample for conducting a hypothesis test?

In a hypothesis test, we are trying to reject a null hypothesis with the evidence that we should collect from sample data which represents only a portion of the population. When the population has a large size, and the sample data is small, we will not be able to draw an accurate conclusion from the test to prove our null hypothesis is false. As sample data provide us a door to the entire population, it should be large enough for us to arrive at a significant inference. Hence a sufficiently large data should be considered for a hypothesis test especially if the population is huge.

How to Run a Z-Test

Z-Test can be considered as a test statistic for a hypothesis test to calculate the P-value. However, there are certain conditions that should be satisfied by the sample to run the Z-Test.

The conditions are as follows:

  • The sample size should be greater than 30.

This is already mentioned above. The size of the sample is an important factor in Z-Testing as the Z-Test follows a normal distribution and so should the data. If the same size is less than 30, it is recommended to use a t-test instead

  • All the data point should be independent and doesn’t affect each other.

Each element in the sample, when considered single should be independent and shouldn’t have a relationship with another element.

  • The data must be distributed normally.

This is ensured if the sample data is large.

  • The sample should be selected randomly from a population.

Each data in the population should have an equal chance to be selected as one of the sample data.

  • The sizes of the selected samples should be equal if at all possible.

When considering multiple sample data, ensuring that the size of each sample is the same for an accurate calculation of population parameters.

  • The standard deviation of the population is known.

The population parameter, standard deviation must be given to run a Z-Test as we cannot perform the calculation without knowing it. If it is not directly given, then it assumed that the variance of the sample data is equal to the variance of the entire population.

If the conditions are satisfied, the Z-Test can be successfully implemented.

Following are steps to run the Z-Test:

  • State the null hypothesis

The null hypothesis is a statement of no effect and it supports the data which is already given. It is generally represented as :

  • State the alternate hypothesis

The statement that we are trying to prove is the alternate hypothesis. It is represented as:

This is the representation of a bidirectional alternate hypothesis.

  • H 1 :µ > k

This is the representation of a one-directional alternate hypothesis that is represented in the right region of the graph.

  • H 1 :µ < k

This is the representation of a one-directional alternate hypothesis that is represented in the left region of the graph.

z test statistic calculator hypothesis testing

  • Choose an alpha level for the test.

Alpha level or significant level is the probability of rejecting the null hypothesis when it is true. It is represented by ( α ). An alpha level must be chosen wisely so as to avoid the Type I and Type II errors.

If we choose a large alpha value such as 10%, it is likely to reject a null hypothesis when it is true. There is a probability of 10% for us to reject the null hypothesis. This is an error known as the Type I error.

On the other hand, if we choose an alpha level as low as 1%, there is a chance to accept the null hypothesis even if it is false. That is we reject the alternate hypothesis to favor the null hypothesis. This is the Type II error.

Hence the alpha level should be chosen in such a way that the chance of making Type I or Type II error is minimal. For this reason, the alpha level is commonly selected as 5% which is proven best to avoid errors.

  • Determining the critical value of Z from the Z table.

The critical value is the point in the normal distribution graph that splits the graph into two regions: the acceptance region and the rejection regions. It can be also described as the extreme value for which a null hypothesis can be accepted. This critical value of Z can be found from the Z table .

  • Calculate the test statistic.

The sample data that we choose to test is converted into a single value. This is known as the test statistic. This value is compared to the null value. If the test statistic significantly differs from the null value, the null value is rejected.

  • Comparing the test statistic with the critical value.

Now, we have to determine whether the test statistic we have calculated comes under the acceptance region or the rejection region. For this, the test statistic is compared with the critical value to know whether we should accept or reject a null hypothesis.

Types of Z-Test

Z-Test can be used to run a hypothesis test for a single sample or to compare the mean of two samples. There are two common types of Z-Test

One-Sample Z-Test

This is the most basic type of hypothesis test that is widely used. For running an one-sample Z-Test, all we need to know is the mean and standard deviation of the population. We consider only a single sample for a one-sample Z-Test. One-sample Z-Test is used to test whether the population parameter is different from the hypothesized value i.e whether the mean of the population is equal to, less than or greater than the hypothesized value.

The equation for finding the value of Z is:

z test statistic calculator hypothesis testing

The following are the assumptions that are generally taken for a one-sampled Z-Test:

  • The sample size is equal to or greater than 30.
  • One normally distributed sample is considered with the standard deviation known.
  • The null hypothesis is that the population mean that is calculated from the sample is equal to the hypothetically determined population mean.

Two-Sample Z-Test

A two-sample Z-Test is used whenever there is a comparison between two independent samples. It is used to check whether the difference between the means is equal to zero or not. Suppose if we want to know whether men or women prefer to drive more in a city, we use a two-sample Z-Test as it is the comparison of two independent samples of men and women.

z test statistic calculator hypothesis testing

  • x 1 and x 2 represent the mean of the two samples.
  • µ 1 and µ 2 are the hypothesized mean values.
  • σ 1 and σ 2 are the standard deviations.
  • n 1 and n 2 are the sizes of the samples.

The following are the assumptions that are generally taken for a two-sample Z-Test:

  • Two independent, normally distributed samples are considered for the Z-Test with the standard deviation known.
  • Each sample is equal to or greater than 30.
  • The null hypothesis is stated that the population mean of the two samples taken does not differ.

Critical value

A critical value is a line that splits a normally distributed graph into two different sections. Namely the ‘Rejection region’ and ‘Acceptance region’. If your test value falls in the ‘Rejection region’, then the null hypothesis is rejected and if your test value falls in the ‘Accepted region’, then the null hypothesis is accepted.

z test statistic calculator hypothesis testing

Critical Value Vs Significant Value

Significant level, alpha is the probability of rejecting a null hypothesis when it is actually true. While the critical value is the extreme value up to which a null hypothesis is true. There migh come a confusion regarding both of these parameters.

Critical value is a value that lies in critical region. It is in fact the boundary value of the rejection region. Also, it is the value up to which the null hypothesis is true. Hence the critical value is considered to be the point at which the null hypothesis is true or is rejected.

Critical value gives a point of extremity whose probability is indicated by the significant level. Significant level is pre-selected for a hypothesis test and critical value is calculated from this Alpha value. Critical value is a point represented as Z score and Significant level is a probability.

Z-Test Vs T-Test

Z-Test are used when the sample size exceeds 30. As Z-Test follows normal distribution, large sample size can be taken for the Z-Test. Z-Test indicates the distance of a data point from the mean of the data set in terms of standard deviation. Also. this test can only be used if the standard deviation of the data set is known.

T-Test is based on T distribution in which the mean value is known and the variance could be calculated from the sample. T-Test is most preferred to know the difference between the statistical parameters of two samples as the standard deviation of the samples are not usually given in a two-sample test for running the Z-Test. Also, if the sample size is less than 30, T-Test is preferred.

z test statistic calculator hypothesis testing

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.

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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 statistic calculator hypothesis testing

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.

Related Calculators

Descriptive Statistics Calculator of Grouped Data

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  1. Z Test Statistics Formula

    z test statistic calculator hypothesis testing

  2. Hypothesis Testing using Z-test Statistics

    z test statistic calculator hypothesis testing

  3. One sample Z-test for proportion: Formula & Examples

    z test statistic calculator hypothesis testing

  4. Z-test- definition, formula, examples, uses, z-test vs t-test

    z test statistic calculator hypothesis testing

  5. Two Sample Z Hypothesis Test

    z test statistic calculator hypothesis testing

  6. Z-Test (Z0, Ze & H0) Calculator, Formulas & Examples

    z test statistic calculator hypothesis testing

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  1. Z-test

  2. Using Z-Test in Hypothesis Testing []The Test Statistic

  3. 8 Hypothesis testing| Z-test |Two Independent Samples with MS Excel

  4. Hypothesis Testing

  5. Hypothesis Testing using one-sample T-test and Z-test

  6. Two Sample Z-Test

COMMENTS

  1. Z-test Calculator

    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 ...

  2. Hypothesis Testing Calculator with Steps

    Hypothesis Testing Calculator. 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 ...

  3. 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. ... statistical hypothesis tests are indispensable tools used to determine if evidence exists to reject a prevailing assumption or theory, known as the null ...

  4. Z Test: Uses, Formula & Examples

    Use a Z test when you need to compare group means. Use the 1-sample analysis to determine whether a population mean is different from a hypothesized value. Or use the 2-sample version to determine whether two population means differ. A Z test is a form of inferential statistics. It uses samples to draw conclusions about populations.

  5. Efficient Z-test Calculator: Your Statistical Analysis Simplified

    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

  6. Z-test for One Population Mean

    The main properties of a one sample z-test for one population mean 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 ...

  7. z-test calculator

    z-test calculator. Natural Language; Math Input; Extended Keyboard Examples Upload Random. ... Input values. Null hypothesis. Alternative hypothesis. Test statistic. p ‐value. Sampling distribution of test statistic under the null hypothesis. Power function. Significance level 5%. Test conclusions. Assumptions.

  8. Single Sample Z Score Calculator

    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 ...

  9. Hypothesis Test Calculator

    This section answers some common questions about . Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important.

  10. Z Score Calculator

    Please enter the value of p above, and then press "Calculate Z from P". Additional Z Statistic Calculators. If you're interested in using the z statistic for hypothesis testing, then we have a couple of other calculators that might help you. Z-Test Calculator for a Single Sample Z-Test Calculator for 2 Population Proportions

  11. Z-Test (Z0, Ze & H0) Calculator, Formulas & Examples

    The below is the solved examples for Z-statistic calculation by using standard deviation & without using standard deviation. Z-test calculator, formulas & example work with steps to estimate z-statistic (Z0), critical value of normal distribution (Ze) & test of hypothesis (H0) for large sample mean, proportion & two means or proportions ...

  12. Z-Hypothesis Testing (stats)

    Z-Hypothesis Testing (stats) | Desmos. Enter the size of the sample n, sample mean m, population standard deviation s. n = 1. m = 0. s = 1. Enter M_0, the value of the null hypothesis and click on the tab below corresponding to the proper form of the alternative hypothesis. Or click on confidence interval to obtain that (with CL=1-alpha) M0 = 0.

  13. One Sample Z-Test

    Z test online. Target: To check if the assumed μ 0 is statistically correct, based on a sample average. You know the standard deviation from previous researches. Example1: A farmer calculated last year the average of the apples' weight in his apple orchard μ 0 equals 17 kg, based on the entire population. The current year he checked a small sample of apples and the sample average x equals 18 kg

  14. One Sample Z-Test Calculator

    A one sample z-test is used to test whether or not the mean of a population is equal to some value when the population standard deviation is known. To perform a one sample z-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data.

  15. One Sample Z-Test Calculator

    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)) Where: Z is the Z-statistic.

  16. Z-Test for Statistical Hypothesis Testing Explained

    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 ...

  17. Z-Test Calculators

    Z-tests are crucial statistical procedures to test for claims about population parameters using the normal distribution. We can use for one population or two population means provided that the population standard deviations are known. Also, we can use a z-test to test for claims about a population proportion. Also, via the Central Limit Theorem, the...

  18. Z-test for two Means, with Known Population Standard Deviations

    Instructions: This calculator conducts a Z-test for two population means ( \mu_1 μ1 and \mu_2 μ2 ), with known population standard deviations ( \sigma_1 σ1 and \sigma_2 σ2 ). Please select the null and alternative hypotheses, type the significance level, the sample means, the population standard deviations, the sample sizes, and the results ...

  19. Z Test: Definition & Two Proportion Z-Test

    The z-score associated with a 5% alpha level / 2 is 1.96.. Step 5: Compare the calculated z-score from Step 3 with the table z-score from Step 4. If the calculated z-score is larger, you can reject the null hypothesis. 8.99 > 1.96, so we can reject the null hypothesis.. Example 2: Suppose that in a survey of 700 women and 700 men, 35% of women and 30% of men indicated that they support a ...

  20. P Value from Z Score Calculator

    Additional Z Statistic Calculators. If you're interested in using the z statistic for hypothesis testing and the like, then we have a number of other calculators that might help you. Z-Test Calculator for a Single Sample Z-Test Calculator for 2 Population Proportions Z Score Calculator for a Single Raw Value (Includes z from p)

  21. Z Test

    What is Z-Test?. Z-Test is a statistical test which let's us approximate the distribution of the test statistic under the null hypothesis using normal distribution.. Z-Test is a test statistic commonly used in hypothesis test when the sample data is large.For carrying out the Z-Test, population parameters such as mean, variance, and standard deviation should be known.

  22. Hypothesis Testing using the Z-Test on the TI-84+ and TI-89

    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 ...

  23. Z-test for 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 p0, the significance level \alpha α, the sample proportion or number o favorable cases, and the sample size, and the results of the z-test for one proportion will ...