A market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included in their basic cell phone contract. The analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 25 minutes. Consider a random sample of 120 customers who exceed the time allowance included in their basic cell phone contract. Denote by X the excess time used by one individual cell phone customer who exceeds her contracted time allowance. Then X has the exponential distribution of mean = 25 minutes. Denote by X the mean excess time used by a sample of n = 120 customers 37 who exceed their contracted time allowance; that is,

where It is the excess time used by customer 1:. {a} Determine the mean and standard deviation of the mean excess time X. {b} Using the central limit theorem, nd an estimate of the probability' that the mean excess time used by the 101'] customers in the sample is between 24 and 25 minutes; that is, nd 13124 a: \"X -:T. 2E}. [ch Suppose that one customer who exceeds the time limit for his cell phone contract- is randome selected. Find the probabilitg.r that this individual customer's excess time is between 24 and 26 minutes. {d} Explain why the probabilities in {b} and {c} are di'erent. {e} Find the 95th percentile for the sample mean excess time for samples of III} customers who exceed their basic contract time allowances; that is, determine I: such that P{f q 1:} = 0.95