In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a $95 \%$ confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations for decreasing the margin of error of the interval. Select the two recommendations that would decrease the margin of error of the interval. A. Decrease the confidence level. B. Use fewer degrees of freedom. C. Increase the confidence level. D. Decrease the standard deviation of hours worked. E. Decrease the sample size. F. Increase the sample size. (1) Time Remaining: $00: 23: 47$ Next
Solution
Step 1 ：In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a $95 \%$ confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. We are asked to provide two recommendations for decreasing the margin of error of the interval.
Step 2 ：The margin of error in a confidence interval is influenced by three factors: the confidence level, the standard deviation of the population, and the sample size.
Step 3 ：The higher the confidence level, the wider the confidence interval, and thus the larger the margin of error. Therefore, decreasing the confidence level would decrease the margin of error.
Step 4 ：The larger the standard deviation, the wider the confidence interval, and thus the larger the margin of error. Therefore, decreasing the standard deviation would decrease the margin of error.
Step 5 ：The smaller the sample size, the wider the confidence interval, and thus the larger the margin of error. Therefore, increasing the sample size would decrease the margin of error.
Step 6 ：Based on these observations, the two recommendations that would decrease the margin of error of the interval are: Decrease the confidence level, Decrease the standard deviation of hours worked, Increase the sample size.
Step 7 ：Final Answer: \(\boxed{\text{A, D, F}}\)
