Z-Test Calculator for Proportion with Examples

$Z$-Test Calculator for single proportion

In this tutorial we will discuss $z$-test calculator for testing single proportion with some numerical examples on $Z$-test for testing single population proportion.

$Z$-test Calculator for proportion

The $Z$-test calculator for testing population proportion makes it easy to calculate the test statistic, $Z$ critical value and the $p$-value given the sample information, level of significance and the type of alternative hypothesis (i.e. left-tailed, right-tailed or two-tailed.)

Z test Calculator for proportion
Population proportion ($p$)
Sample size ($n$)
No.Successes ($X$)
Level of Significance ($\alpha$)
Tail Left tailed
Right tailed
Two tailed
Results
Sample Proportion :
Standard Error of $p$:
Test Statistics Z:
Z-critical value:
p-value:

How to use $z$-test calculator for testing single proportion?

Step 1 - Enter the population proportion $p$ under $H_0$.

Step 2 - Enter the sample size $n$

Step 3 - Enter the observed number of successes $X$

Step 4 - Enter the level of significance $\alpha$

Step 5 - Select the alternative hypothesis (left-tailed / right-tailed / two-tailed)

Step 6 - Click on "Calculate" button to get the result

Z-test for testing proportion Example 1

An insurance company states that 90% of its claims are settled within 30 days. A consumer group selected a random sample of 75 of the company's claims to test this statement. If the consumer group found that 55 of the claims were settled within 30 days, do they have sufficient reason to support their contention that fewer than 90% of the claims are settled within 30 days? Use 5% level of significance.

Solution

Given that $n = 75$, $X= 55$.

The sample proportion is

$$\hat{p}=\frac{X}{n}=\frac{55}{75}=0.733.$$

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : p = 0.9$ against $H_1 : p < 0.9$ ($\text{left-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

$$ \begin{aligned} Z & = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \end{aligned} $$
which follows $N(0,1)$ distribution.

Step 3 Significance Level

The significance level is $\alpha = 0.05$.

Step 4 Critical values

As the alternative hypothesis is $\textit{left-tailed}$, the critical value of $Z$ $\text{ is }$ $\text{-1.64}$.

Z-critical value for left-tailed test
Z-critical value for left-tailed test

The rejection region (i.e. critical region) for the hypothesis testing problem is $\text{Z < -1.64}$.

Step 5 Computation

The test statistic is

$$ \begin{aligned} Z & = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &= \frac{0.733-0.9}{\sqrt{\frac{0.9* (1-0.9)}{75}}}\\ & =-4.811 \end{aligned} $$

Step 6 Decision (Traditional approach)

The test statistic is $Z =-4.811$ which falls $inside$ the critical region, we $\text{reject}$ the null hypothesis.

OR

Step 6 Decision ($p$-value approach)

This is a $\text{left-tailed}$ test, so the p-value is the area to the left of the test statistic ($Z=-4.811$). Thus the $p$-value = $P(Z<-4.811) =0$.

The p-value is $0$ which is $\text{less than}$ the significance level of $\alpha = 0.05$, we $\text{reject}$ the null hypothesis.

Interpretation

There is sufficient reason to support their contention that fewer than 90% of the claims are settled within 30 days.

Z-test for testing proportion Example 2

An auditor for the U.S. Postal Service wants to examine its special two-day priority mail handling to determine the proportion of parcels that actually arrive within the promised two-day period. A randomly selected sample of 1600 such parcels is found to contain 1250 that were delivered on time. Does the sample data provide evidence to conclude that the percentage of on-time parcels is more than 75% (using $\alpha=0.01$)?

Solution

Given that $n = 1600$, $X= 1250$.

The sample proportion is

$$\hat{p}=\frac{X}{n}=\frac{1250}{1600}=0.781.$$

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : p = 0.75$ against $H_1 : p > 0.75$ ($\text{right-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

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

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

Step 3 Significance Level

The significance level is $\alpha = 0.01$.

Step 4 Critical values

As the alternative hypothesis is $\textit{right-tailed}$, the critical value of $Z$ $\text{ is }$ $\text{2.33}$.

Z-critical value for right-tailed test 2
Z-critical value for right-tailed test

The rejection region (i.e. critical region) for the hypothesis testing problem is $\text{Z > 2.33}$.

Step 5 Computation

The test statistic is

$$ \begin{aligned} Z & = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &= \frac{0.781-0.75}{\sqrt{\frac{0.75* (1-0.75)}{1600}}}\\ & =2.887 \end{aligned} $$

Step 6 Decision (Traditional approach)

The test statistic is $Z =2.887$ which falls $inside$ the critical region, we $\text{reject}$ the null hypothesis.

OR

Step 6 Decision ($p$-value approach)

This is a $\text{right-tailed}$ test, so the p-value is the area to the left of the test statistic ($Z=2.887$). Thus the $p$-value = $P(Z<2.887) =0.0019$.

The p-value is $0.0019$ which is $\text{less than}$ the significance level of $\alpha=0.01$, we $\text{reject}$ the null hypothesis.

Interpretation

The sample data provide enough evidence at $\alpha=0.01$ level of significance to conclude that the percentage of on-time parcels is more than 75%.

Z-test for testing proportion Example 3

A survey by Men''s Health magazine stated that 14% of men said they used exercise to reduce stress. A random sample of 100 men was selected, and 10 said that they used exercise to relieve stress. Test the claim at $\alpha= 0.10$.

Solution

Given that $n = 100$, $X= 10$.

The sample proportion is

$$\hat{p}=\frac{X}{n}=\frac{10}{100}=0.1.$$

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : p = 0.14$ against $H_1 : p \neq 0.14$ ($\text{two-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

$$ \begin{aligned} Z & = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \end{aligned} $$
which follows $N(0,1)$ distribution.

Step 3 Significance Level

The significance level is $\alpha = 0.1$.

Step 4 Critical values

As the alternative hypothesis is $\textit{two-tailed}$, the critical value of $Z$ $\text{ are }$ $\text{-1.64 and 1.64}$.

Z-critical value for two-tailed test 2
Z-critical value for two-tailed test

The rejection region (i.e. critical region) for the hypothesis testing problem is $\text{Z < -1.64 or Z > 1.64}$.

Step 5 Computation

The test statistic is

$$ \begin{aligned} Z & = \frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}\\ &= \frac{0.1-0.14}{\sqrt{\frac{0.14* (1-0.14)}{100}}}\\ & =-1.153 \end{aligned} $$

Step 6 Decision (Traditional approach)

The test statistic is $Z =-1.153$ which falls $outside$ the critical region, we $\text{fail to reject}$ the null hypothesis.

OR

Step 6 Decision ($p$-value approach)

This is a $\text{two-tailed}$ test, so the p-value is the area to the left of the test statistic ($Z=-1.153$). Thus the $p$-value = $P(Z<-1.153) =0.249$.

The p-value is $0.249$ which is $\text{greater than}$ the significance level of $\alpha = 0.1$, we $\text{fail to reject}$ the null hypothesis.

Interpretation

There is no sufficient evidence to say that the percentage of men who use exercise to reduce stress is not $14$%.

Endnote

In this tutorial, you learned the about how to solve numerical examples on $Z$-test for testing single proportion. You also learned about the step by step procedure to apply $Z$-test for testing single proportion and how to use Z-test calculator for testing population proportion to get p-value, z-critical value.

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 calculator for proportion with examples 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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