A survey is designed to investigate the average weekly exercise hours of adults in California. It is known that the population mean weekly exercise hours is 7, and the standard deviation is 2.5. It is also known that the interquartile range of the weekly exercise hours is 5.
(a) Based on the information provided, are the weekly exercise hours normally distributed? Explain why or why not.
(b) A random sample of 100 adults is drawn from the population. With the information available to you, is it possible to find the probability that the average weekly exercise hours of these 100 adults is less than 6.5? If so, compute the probability and state all the assumptions that you need for your calculation (if any). Also state whether the probability obtained is an approximate or exact probability. If not, briefly explain why not.
(c) A random sample of 5 adults is drawn from the population. With the information available to you, is it possible to find the probability that at most 2 adults exercise more than 8 hours per week? If so, compute the probability and state all the assumptions that you need for your calculation (if any). Also state whether the probability obtained is an approximate or exact probability. If not, briefly explain why not.
(d) Regardless of your answer to part (a), now assume the distribution of exercise hours is Normal with the mean and standard deviation values provided above. Find the probability that in a random sample of 200 adults, more than 15% of the adults exercise more than 10 hours per week. State all the assumptions you need for your calculation.