Question

Before COVID, households in the U.S. watched an average of 57 hours of television per week. At the peak of the COVID epidemic, this number rose to 66 hours per week. Researchers now wonder whether the mean weekly viewing time has begun to drop back toward pre-COVID levels. To test this hypothesis, they installed equipment that would record television viewing time for one week in 25 households. Their sample had a mean viewing time of 63.5 hours and a standard deviation of 6.2 hours. You are going to conduct a hypothesis test to determine whether the data support the theory that the mean weekly television-viewing time is now less than the 66 hours per week. Question E1: Formulate the hypotheses for the test. H 0 ​ : x ˉ =66 H A: ​ : x ˉ <66 H 0 ​ :μ=66 H A: ​ :μ<66 H 0 ​ : x ˉ <66 H A ​ : x ˉ =66 H 0 ​ :μ<66 H A ​ :μ=66 ​ Question E2: Choose the appropriate formula for the test statistic and find its value. −2.683 −2.218 −2.439 −2.016 2.952 Question E3: If the researchers used the .05 level of significance, describe the rejection region for this test. It includes only the values below −1.645. It includes only the values below −1.960. It includes only the values below −2.064. t includes only the values below −2.060. It includes only the values below - 1.711. Question E4: What should the researchers conclude, based on the results? They should not reject H 0 ​ . The results support the conclusion that the mean television-viewing time has decreased since the peak of the COVID epidemic. They should reject H 0 ​ . The results do not support the conclusion that the mean Whelevision-viewing time has decreased since the peak of the COVID epidemic. They should reject H 0 ​ . The results support the conclusion that the mean television-viewing time has decreased since the peak of the COVID epidemic. They should not reject H 0 ​ . The results do not support the conclusion that the mean television-viewing time has decreased since the peak of the COVID epidemic.

Answer 1

The correct hypotheses for the test are: H0: μ=66 and Ha: μ<66. The appropriate formula for the test statistic is t-distribution with a value of -2.439. The rejection region for this test includes only the values below -1.711. Based on the results, the researchers should reject the null hypothesis.

Step-by-step explanation:

To conduct a hypothesis test to determine whether the mean weekly television-viewing time has decreased since the peak of the COVID epidemic, we need to formulate the hypotheses for the test. The null hypothesis (H0) is that the mean weekly television-viewing time is still 66 hours, while the alternative hypothesis (Ha) is that the mean weekly television-viewing time has decreased and is less than 66 hours. Therefore, the correct answer for Question E1 is:

H0: μ = 66

Ha: μ < 66

To choose the appropriate formula for the test statistic, we need to consider the sample size and the information provided. Since the sample size is 25 and the population standard deviation is unknown, we should use the t-distribution formula for the test statistic. Therefore, the correct answer for Question E2 is:

-2.439

The rejection region for this test can be determined based on the level of significance. Since the researchers used the 0.05 level of significance, we need to find the critical value from the t-distribution table. For a one-tailed test with 24 degrees of freedom and a 0.05 level of significance, the critical value is -1.711 (approximately). Therefore, the correct answer for Question E3 is:

It includes only the values below -1.711.

Based on the results of the hypothesis test, the researchers should decide whether to reject or not reject the null hypothesis. If the test statistic falls within the rejection region, they should reject the null hypothesis; otherwise, they should not reject the null hypothesis. Since the test statistic (-2.439) is less than the critical value (-1.711), the researchers should reject the null hypothesis. Therefore, the correct answer for Question E4 is: