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.