# Hypothesis Testing for a Proportion This lesson explains how to conduct a hypothesis test for a population proportion.

## Checking Conditions

Before we can conduct a hypothesis test for a population proportion, we first need to make sure the following conditions are met to ensure that our hypothesis test will be valid:

• Random: A random sample or random experiment should be used to collect the data.
• Successes & Failures: The sample includes at least 10 successes and at least 10 failures.
• Size: The sample size is no bigger than 5% of the population size.

If these conditions are met, we can then conduct the hypothesis test. The following two examples show how to conduct a hypothesis test for a population proportion.

## Hypothesis Test for Population Proportion (Two-Tailed)

A phone company claims that 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 85% responded yes.

Test the null hypothesis that 90% of customers are satisfied with their service against the alternative hypothesis that not 90% of customers are satisfied with their service. Use a 0.05 level of significance.

Step 1. State the hypotheses.

The null hypothesis (H0): P = 0.90

The alternative hypothesis: (Ha): P ≠ 0.90

Step 2. Determine a significance level to use.

The problem tells us that we are to use a .05 level of significance.

Step 3. Find the test statistic.

z  =  (p-P) / (√P(1-P) / n)

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

z = (.85-.90) / (√.90(1-.90) / 200) = (-.05) / (.0212) = -2.358

Use the Z Score to P Value Calculator with a z score of -2.358 and a two-tailed test to find that the p-value = 0.018.

Step 4. Reject or fail to reject the null hypothesis.

Since the p-value is less than our significance level of .05, we reject the null hypothesis.

Step 5. Interpret the results.

Since we rejected the null hypothesis, we have sufficient evidence to say that it’s not true that 90% of customers are satisfied with their service.

## Hypothesis Test for Population Proportion (One-Tailed)

A phone company claims that at least 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 85% responded yes.

Test the null hypothesis that at least 90% of customers are satisfied with their service against the alternative hypothesis that less than 90% of customers are satisfied with their service. Use a 0.01 level of significance.

Step 1. State the hypotheses.

The null hypothesis (H0): P ≥ 0.90

The alternative hypothesis: (Ha): P < 0.90

Step 2. Determine a significance level to use.

The problem tells us that we are to use a .01 level of significance.

Step 3. Find the test statistic.

z  =  (p-P) / (√P(1-P) / n)

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

z = (.85-.90) / (√.90(1-.90) / 200) = (-.05) / (.0212) = -2.358

Use the Z Score to P Value Calculator with a z score of -2.358 and a one-tailed test to find that the p-value = 0.009.

Step 4. Reject or fail to reject the null hypothesis.

Since the p-value is less than our significance level of .01, we reject the null hypothesis.

Step 5. Interpret the results.

Since we rejected the null hypothesis, we have sufficient evidence to say that the company’s claim that at least 90% of its customers are satisfied with their service is not true.