Question 4:
The times that a cashier spends processing each person’stransaction are independent and identically distributed randomvariables with a mean of µ and a variance of σ2 . Thus,if Xi is the processing time for each transaction, E(Xi) = µ and Var(Xi) = σ2 .
Let Y be the total processing time for 100 orders: Y =X1 + X2 + · · · + X100
(a) What is the approximate probability distribution of Y , thetotal processing time of 100 orders? Hint: Y = 100X, where X = 1100 P100 i=1 Xi is the sample mean.
(b) Suppose for Z ∼ N(0, 1), a standard normal randomvariable:
P(a < Z < b) = 100(1 − α)%
Using your distribution from part (a), show that an approximate100(1 − α)% confidence interval for the unknown population mean µis:
(Y − 10bσ)/100 < µ < (Y − 10aσ)/100
(c) Now suppose that the population mean processing time isknown to be µ = 1.5 minutes, and the population standard deviationprocessing time is known to be σ = 1 minute. What is theprobability that it takes less than 120 minutes to process the 100orders? If you use R, please provide the commands used to determinethe probability. Could you show all steps in the hand writtenworking for this question please.
