I would like to know how to solve (e) and (f) with specific process.

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I would like to know how to solve (e) and (f) with specific
process.

I Would Like To Know How To Solve E And F With Specific Process 1 (57.58 KiB) Viewed 21 times

2. Jeff runs a fast-food restaurant with serving time of an order following a normal distribution with mean a seconds and standard deviation o seconds. Serving time of each order is independent from others. Past experience reveals that 6.68% of the orders were served beyond 90 seconds and 0.62% of the orders were served within 50 seconds. (a) Show that a = 75 and o= 10. [4 marks]
A customer will enjoy a 50% discount on his/her order if the serving time exceeds 90 seconds. (b) Let X be the cost a customer has to pay for a $100 meal. (i) Find the probability distribution of X. [2 marks] [4 marks] (ii) Find E(X) and Var (X). (c) Given that a customer enjoys the 50% discount on his/her order. Find the probability that the serving time actually exceeds 95 seconds. [3 marks] Customers arrive at the fast-food restaurant according to a Poisson distribution with a rate of 48 customers per hour. (d) Find the probability that no more than 4 customers arrive at the fast-food restaurant in 5 minutes. [3 marks] (e) Given that customers' arrivals and their serving times are independent. Find the probability that exactly 4 customers arrive at the fast-food restaurant in 5 minutes, and less than half of them enjoy the 50% discount. [3 marks] Jeff decides to streamline the food serving procedure so as to shorten the serving time of an order. Assume that the improved serving time of an order can be modelled by a normal distribution with mean b seconds and standard deviation 10 seconds. Jeff selects a random sample of n orders and records their serving times. He then uses the sample mean as an estimate of b. (f) The number n is chosen such that the probability of the absolute difference between the estimated value and the true value being within 0.5 standard deviation is at least 0.9. (i) State the probability distribution followed by the sample mean. Express your answer in terms of b and n. [2 marks] (ii) Find the least value of n. [4 marks] (g) Suppose that n = 10 and the sampled serving times (in seconds) are recorded as follows: 69 60 74 59 65 61 58 71 57 66 Construct a 95% confidence interval for b. [5 marks] [Total: 30 marks]