Z-Test for Proportion

Z-Test for Proportion

In this tutorial we will discuss about the step by step procedure of one sample $Z$-test for testing population proportion.

Let $X$ be the observed number of individuals possessing certain attributes (say, number of successes) in a random sample of size $n$ from a large population, then $\hat{p}=\frac{X}{n}$ be the observed proportion of successes.

Let $p$ be the population proportion of successes and $q = 1- p$ be the population proportion of failures.


Assumptions for testing a proportion are as follows:

a. The sample is a random sample.

b. The conditions for binomial experiments are satisfied.

c. $n$ is sufficiently large ($n>20$), $np\geq 5$ and $nq\geq 5$.

Step by step procedure

We wish to test the null hypothesis $H_0 : p = p_0$, where $p_0$ is the specified value of the population proportion.

The standard error of $p$ is
$$ \begin{aligned} SE(\hat{p}) &= \sqrt{\frac{p(1-p)}{n}} \end{aligned} $$
The test statistic for testing $H_0$ is

$$ \begin{aligned} Z & = \frac{\hat{p}-p}{SE(\hat{p})} \end{aligned} $$

which follows standard normal distribution $N(0,1)$.

Step 1 State the hypothesis testing problem

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation Hypothesis Testing Problem
Situation A : $H_0: p=p_0$ against $H_a : p < p_0$ (Left-tailed)
Situation B : $H_0: p=p_0$ against $H_a : p > p_0$ (Right-tailed)
Situation C : $H_0: p=p_0$ against $H_a : p \neq p_0$ (Two-tailed)

Step 2 Define the test statistic

The test statistic for testing above hypothesis is

$$ \begin{eqnarray*} Z & = & \frac{\hat{p}-p}{SE(\hat{p})}\\ & = &\frac{\hat{p}-p}{\sqrt{\frac{p*(1-p)}{n}}} \end{eqnarray*} $$

The test statistic $Z$ follows standard normal distribution $N(0,1)$.

Step 3 Specify the level of significance $\alpha$

Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.

  • For left-tailed alternative hypothesis: Find the $Z$-critical value using

$$ \begin{aligned} P(Z<-Z_\alpha) &= \alpha. \end{aligned} $$

  • For right-tailed alternative hypothesis: $Z_\alpha$.

$$ \begin{aligned} P(Z>Z_\alpha) &= \alpha. \end{aligned} $$

  • For two-tailed alternative hypothesis: $Z_{\alpha/2}$.

$$ \begin{aligned} P(|Z|> Z_{\alpha/2}) &= \alpha. \end{aligned} $$

Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using equation

$$ \begin{aligned} Z_{obs} &= \frac{\hat{p}-p_0}{\sqrt{\frac{p_0*(1-p_0)}{n}}} \end{aligned} $$

Step 6 Decision (Traditional Approach)

Traditional approach is based on the critical value(s).

  • For left-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\leq -Z_\alpha$.
  • For right-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\geq Z_\alpha$.
  • For two-tailed alternative hypothesis: Reject $H_0$ if $|Z_{obs}|\geq Z_{\alpha/2}$.


Step 6 Decision ($p$-value Approach)

$p$-value approach is based on the $p$-value of the test.

Alternative Hypothesis Type of Hypothesis $p$-value
$H_a: p < p_0$ Left-tailed $p$-value $= P(Z\leq Z_{obs})$
$H_a: p > p_0$ Right-tailed $p$-value $= P(Z\geq Z_{obs})$
$H_a: p \neq p_0$ Two-tailed $p$-value $= 2P(Z\geq abs(Z_{obs}))$

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.


In this tutorial, you learned the $Z$-test for testing single proportion and the assumptions for $Z$-test for testing population proportion. You also learned about the step by step procedure to apply $Z$-test for testing single proportion.

To learn more about other hypothesis testing problems, hypothesis testing calculators and step by step procedure, please refer to the following tutorials:

Let me know in the comments if you have any questions on $Z$-test for population proportion 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.

Leave a Comment