Question

1. One of the most important measures of the quality of service provided by any firm is the speed with which it responds to customer complaints. Comcast, a U.S. global telecommunications conglomerate, wants to greatly improve its customer satisfaction. Comcast states the desired mean call time involving customer complaints is 12 minutes (including wait time). Assume the standard deviation is known to be 0.15 minutes. A sample of 70 customer calls yields a mean time of 12.14 minutes. This sample will be used to obtain a 99% confidence interval for the mean time of a customer complaint call. Round final answers to two decimal places. Solutions only.

(A) The critical value to use in obtaining the confidence interval is.

(B) The confidence interval goes from to.

(C) True, False, or Uncertain: The confidence interval indicates that Comcast is not meeting its goal.

(D) True, False, or Uncertain: The confidence interval is valid only if the length of calls are normally distributed

(E) Suppose the manager had decided to estimate the mean call time to within 0.03 minutes with 99% confidence. Then the sample size would be?

Answer 1

Let X be the call time involving customer complaint of a given customer call

The following are the sample information

n=70 is the sample size of customer calls

minutes is the sample mean call time

minutes is the population standard deviation of call times

is the standard error of mean

a) Significance level for 99% confidence interval is

Since the sample size n=70 is greater than 30 (or we know the population standard deviation) we can use normal distribution to describe the sampling distribution of mean

The critical value is obtained using

This can be written as

From standard normal tables we get for z=2.58 P(Z<2.58) = 0.5+0.4951=0.995

Ans: the critical value to use in obtaining the confidence interval is 2.58

b) Next we calculate the confidence interval

ans: The confidence interval goes from 12.09 to 12.19

C) Comcast states the desired mean call time involving customer complaints is 12 minutes.

12 minutes lies outside of the confidence interval.

ans: True: The confidence interval indicates that Comcast is not meeting its goal.

D) From central limit theorem we know that the sampling distribution of mean has a normal distribution for sample sizes larger than 30. Irrespective of the distribution of the population, we can use normal distribution as an approximation for sampling distribution of mean if either the sample size is greater than 30 or the population standard deviation is known.

In this case the sample size is 70 and also we know the population standard deviation of call times.

Hence we do not need the call times to be normally distributed.

ans: False: The confidence interval is valid only if the length of calls are normally distributed

E) The manager wants to estimate the mean call time with in 0.03 minues, that is if is the sample mean of a sample of size n, we want to estimate the mean call time within

We know that for 99% confidence the critical value z=2.58

The sample size needs to be 167 to estimate the mean call time to within 0.03 minutes with 99% confidence.