p-value calculator for Z-test

$p$ value of the test

p-value of the test is the probability that the test statistic under null hypothesis will take on values as extreme as or more extreme than the observed value of test statistic.

The smaller p-value of the test indicate strong evidence against the null hypothesis $H_0$.

In this tutorial we discuss about how to find the p-value of the Z-test. The p-value of the Z-test depends on the direction of the alternative hypothesis.

p-Value for Z-test

If the test statistic $Z$ has standard normal distribution, then the $p$-value of the test for testing

a. left-tailed hypothesis is $p$-value = $P(Z\leq z_{obs})$.

b. right-tailed hypothesis is $p$-value = $P(Z\geq z_{obs})$.

c. two-tailed hypothesis is $p$-value = $2P(Z\geq |Z_{obs}|)$.

Interpretation from p-value

If the $p$-value of the test is less than or equal to the level of significance ($\alpha$) (i.e., $p \leq \alpha$), we reject the null hypothesis $H_0$ at $\alpha$ level of significance.

If the $p$-value of the test is greater than the level of significance ($\alpha$) (i.e., $p > \alpha$), we fail to reject the null hypothesis $H_0$ at $\alpha$ level of significance.

$p$ value Calculator for Z test

Use this calculator to compute the p-value for Z-test.

Z-test p Value Calculator
Z-value : ($z$)
Tail : Left tailedRight tailedTwo tailed
Results
p-value:

How to use p-value calculator for z-test?

Step 1 - Enter the value of $Z$-test statistic.

Step 2 - Select the alternative hypothesis (i.e. Left-tailed / Right-tailed / Two-tailed)

Step 3 - Click on "Calculate" button to get the p-value.

Example 1 - Find p-value for right tailed

Find the $p$-value for a right-tailed hypothesis testing problem with a test statistic $Z_{obs}=2.65$. What is your decision if the level of significance is $\alpha = 0.05$?

Solution

The test statistic is $Z_{obs}=2.65$. The alternative hypothesis is right-tailed.

Because the alternative hypothesis is right-tailed, the $p$-value of the test is given by

$$ \begin{aligned} p&=P(Z\geq Z_{obs})\\ &=P(Z\geq 2.65)\\ &= 0.004 \end{aligned} $$

p-value for Z test
p-value for Z test

Because the p-value ($0.004$) is $\text{less than}$ the significance level $\alpha = 0.05$, we $\text{reject}$ the null hypothesis at $0.05$ level of significance.

Example 2 - Find p-value for left tailed

Find the $p$-value for a left-tailed hypothesis testing problem with a test statistic $Z_{obs}=-2.023$. What is your decision if the level of significance is $\alpha = 0.01$?

Solution

The test statistic is $Z_{obs}=-2.023$. The alternative hypothesis is left-tailed.

Because the alternative hypothesis is left-tailed, the $p$-value of the test is given by
$$ \begin{aligned} p&=P(Z\leq Z_{obs})\\ &=P(Z\leq -2.023)\\ &= 0.0215 \end{aligned} $$

p-value for Z test
p-value for Z test

Because the p-value ($0.0215$) is $\text{greater than}$ the significance level $\alpha = 0.01$, we $\text{fail to reject}$ the null hypothesis at $0.01$ level of significance.

Example 3 - Find p-value for two tailed

Find the $p$-value for a two-tailed hypothesis testing problem with a test statistic $Z_{obs}=1.674$. What is your decision if the level of significance is $\alpha = 0.05$.

Solution

The test statistic is $Z_{obs}=1.674$. The alternative hypothesis is two-tailed.

Because the alternative hypothesis is two-tailed, the $p$-value of the test is given by

$$ \begin{aligned} p&=2*P(Z\geq |Z_{obs}|)\\ &=2*P(Z\geq |1.674|)\\ &=2*0.0471\\ &= 0.0942 \end{aligned} $$

p-value for Z test
p-value for Z test

Because the p-value ($0.0942$) is $\text{greater than}$ the significance level $\alpha = 0.05$, we $\text{fail to reject}$ the null hypothesis at $0.05$ level of significance.

Z-tests calculators

Here you will find some easy to use statistics calculators with illustrated examples on Z-tests.

Endnote

In this tutorial, you learned about how to find the p-value for z-test. You also learned about how to solve numerical problems based on p-value for z-test.

To learn more about other parametric hypothesis testing, please refer to the following tutorials:

Hypotesis Testing

Let me know in the comments if you have any questions on p-value calculator for z-test and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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