Please use statistics principles and methods to work with the two cases below. Explanation is needed.

I. The 40-hour work week did not become a U.S. standard until 1940. Today, many

white-collar employees work more than 40 hours per week. A random sample of 14 whitecollar

employees was obtained and the number of hours each worked during the last week

was collected. The sample mean number of hours worked for the week was found to be

44.39 with a standard deviation of 3.38.

(a) Construct and interpret a 95% confidence interval for the true mean number of hours

worked each week by white-collar workers.

(b) What assumption(s) must be satisfied for the interval obtained in part (a) to be

statistically valid?

II. Researchers investigated how the size of a bowl affects how much ice cream people

tend to scoop when serving themselves. At an "ice cream social", people were randomly

given either a 17 fluid oz (503 mL) or a 34 fluid oz (1006 mL) bowl. They were then

invited to scoop as much ice cream as they liked. Did the bowl size change the selected

portion size? Here are the summary statistics:

Small Bowl Large Bowl

Sample Size 36 32

Sample Mean 5.07 oz 6.58 oz

Sample Standard Deviation 1.84 oz 2.91 oz

(a) Construct a 95% confidence interval for the true difference in mean portion size of ice

cream.

(b) What assumptions must be satisfied for the inference in part (a) to be valid?