Question

According to the National Institute of Occupational Safety and Health, job stress poses a major threat to the health of workers. A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. A sample of 100 employees of a restaurant chain finds that 68 answer “Yes” when asked, “Does work stress have a negative impact on your personal life?” Is this good reason to think that the proportion of all employees of this chain who would say “Yes” differs from the national proportion of 0.75? [Test of Hypothesis for the Proportion]

a) State the null hypothesis and the alternative hypothesis.

b) What is the value of the test statistic in this problem? Explain how you obtain your answer.

c) What is the p-value associated with the test statistic in this problem? Explain how you obtain your answer.

d) Using a 5% level of significance, what would be your decision in the hypothesis test? Justify your answer.

e) What would be a 95% confidence interval for the population proportion?

Answer 1

The following information is provided: The sample size is N = 100 the number of favorable cases is X = 68 and the sample proportion is , and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.75

Ha: p ≠​ 0.75

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is z_c = 1.96

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |z| = 1.617 ≤zc​=1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.106 , and since p = 0.106 ≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pp is different than p0​, at the 0.05α=0.05 significance level.

The corresponding confidence interval is computed as shown below:

CI = (0.589, 0.771)