An automobile club comes to the aid of stranded motorists who are members. Over the long run, about 5% of the members utilize this service in any 12-month period. A small town has 400 members of the club. Consider the 400 members to be representative of the larger club membership. Let P^ be the proportion of them who will utilize the service in the coming 12-month period.
a. Explain how Conditions 1 to 3 (p. 325) for the approximate normality of the sampling distribution for a sample proportion are met in this scenario. State any assumptions that you need to make to do so.
b. What are the values of n and p in this situation?
c. Describe the sampling distribution of P^, including values for the mean and standard deviation.
d. The towing service in town contracts with the club to come to the aid of up to 28 members in the next 12-month period. What proportion is that of the 400 members in town? What is the approximate probability that the actual proportion requiring aid will exceed that value?