Question

in a random sample of 8 people, the mean commute time to work was 35.5 minutes and the standard deviation was 7.3 minutes. a 98% confidence interval using the t - distribution was calculated to be (27.8,43.2). after researching commute times to work, it was found that the population standard deviation is 8.6 minutes. find the margin of error and construct a 98% confidence interval using the standard normal distribution with the appropriate calculations for a standard deviation that is known. compare the results. a 98% confidence interval using the standard normal distribution is (28.4,42.6). (round to one decimal place as needed.) compare the results. choose the correct answer below. a. the confidence interval found using the standard normal distribution is wider than the confidence interval found using the students t - distribution. b. the confidence interval found using the standard normal distribution is the same as the confidence interval found using the students t - distribution. c. the confidence interval found using the standard normal distribution has smaller lower and upper confidence interval limits. d. the confidence interval found using the standard normal distribution is narrower than the confidence interval found using the students t - distribution.

Explanation:

Step1: Compare the intervals

The t - distribution interval is \((27.8,43.2)\) and the standard normal distribution interval is \((28.4,42.6)\).
Calculate the width of the t - distribution interval: \(43.2 - 27.8=15.4\).
Calculate the width of the standard normal distribution interval: \(42.6 - 28.4 = 14.2\).

Answer:

D. The confidence interval found using the standard normal distribution is narrower than the confidence interval found using the student’s t - distribution.