Question

Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on 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 22 minutes. Consider a random sample of 144 customers who exceed the time allowance included in their basic cell phone contract.

a. Find P(20 < x¯ < 30).

b. Find P(Σx is at least 3,000).

c. Find the 75th percentile for the sample mean excess time of 144 customers.

d. Find the 85th percentile for the sum of 144 excess times used by customers.

Answer 1

Since X has exponential distribution so we have

(a)

Sample size: n=144

The sampling distribution of sample mean will be approximately normal distribution with mean

and standard deviation is

The z-score for is

The z-score for is

So required probability is

(b)

Let

The sampling distribution of sample sum will be approximately normal distribution with mean

and standard deviation is

The z-score for S = 3000 is

The required probability is

(c)

The z-score 0.67 has 0.75 area to its left.So required sample mean excess time is

(d)

The z-score 1.03 has 0.85 area to its left.So required sample mean excess time is