Confidence Intervals and Hypothesis Testing

Topic: Type Ⅰ and Type Ⅱ errors and power

Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The mean shopping time, μ, spent by customers at the supermarkets has been reported to be 38 minutes, but executives have good reason to believe that μ is greater than 38 minutes. The executives hire a statistical consultant and ask her to perform a statistical test. She collects a random sample of 75 shopping times at the supermarkets. Suppose that the population of shopping times at the supermarkets has a standard deviation of 16 minutes and that the consultant performs her hypothesis test using the 0.05 level of significance.

Based on this information, answer the questions below. Carry your intermediate computations to four or more decimal places, and round your responses as indicated.

(a) What are the null hypothesis, H0, and the alternative hypothesis, H1, that the consultant should use for the test?

(b) Assuming that the actual value of μ is 43 minutes, what is the probability that the consultant accepts the null hypothesis? Round your response to two or more decimal places.

(c) What is the probability that the consultant commits a Type Ⅰ error? Round your response to two or more decimal places.

(d) Suppose that the consultant decides to perform another statistical test using the same population, the same null and alternative hypotheses, and the same sample size, but for this second test the consultant uses a significance level of 0.1 instead of a significance level of 0.05. Assuming that the actual value of μ is 43 minutes,how does the power of this second test compare to the power of the original test?

Select:

- The power of the second test is greater than the power of the original test

- The power of the second test is less than the power of the original test

- The powers of the two tests are equal

Question

Confidence Intervals and Hypothesis Testing

Topic: Type Ⅰ and Type Ⅱ errors and power

Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The mean shopping time, μ, spent by customers at the supermarkets has been reported to be 38 minutes, but executives have good reason to believe that μ is greater than 38 minutes. The executives hire a statistical consultant and ask her to perform a statistical test. She collects a random sample of 75 shopping times at the supermarkets. Suppose that the population of shopping times at the supermarkets has a standard deviation of 16 minutes and that the consultant performs her hypothesis test using the 0.05 level of significance.

Based on this information, answer the questions below. Carry your intermediate computations to four or more decimal places, and round your responses as indicated.

(a) What are the null hypothesis, H0, and the alternative hypothesis, H1, that the consultant should use for the test?

(b) Assuming that the actual value of μ is 43 minutes, what is the probability that the consultant accepts the null hypothesis? Round your response to two or more decimal places.

(c) What is the probability that the consultant commits a Type Ⅰ error? Round your response to two or more decimal places.

(d) Suppose that the consultant decides to perform another statistical test using the same population, the same null and alternative hypotheses, and the same sample size, but for this second test the consultant uses a significance level of 0.1 instead of a significance level of 0.05. Assuming that the actual value of μ is 43 minutes,how does the power of this second test compare to the power of the original test?

Select:

- The power of the second test is greater than the power of the original test

- The power of the second test is less than the power of the original test

- The powers of the two tests are equal

Accepted Answer

Answered: Image /qna-images/answer/37d8a6d9-83e8-4202-ae02-ade12015cf56.jpg