# Question

A phone company launched an advertising program designed to increase the number of minutes of long-distance calls made by customers. To get a sense of the benefits of the program, it ran a small test of the promotion. It first selected a sample of 100 customers of the type being targeted by the promotion. This sample of 100 customers used an average of 185 minutes per month of long-distance service. The company then included a special flyer in its monthly statement to those customers for the next two billing cycles. After receiving the promotion, these same customers were using 215 minutes per month. Did the promotion work?

Motivation

(a) Explain why it makes sense for the company to experiment with a sample of customers before rolling this program out to all of its subscribers.

(b) What is the advantage of measuring the response using the same customers? What is the weakness?

Method

Let X1 denote the number of minutes used by a customer before the promotion, and let X2 denote the number of minutes after the promotion. Use μ1 for the mean of X1 (before the promotion) and μ2 for the mean of X2 (after).

(c) The data on the number of minutes used by a customer during a given month are rather skewed. How does this affect the use of confidence intervals?

(d) Form a new variable, say Y = X2 - X1, that measures the change in use. How can you use the 95% confidence interval for the mean of Y?

Mechanics

(e) Form 95% confidence intervals for the mean m1 of X1 and the mean m2 of X2.

(f) Form the 95% confidence interval for the mean of the difference Y.

(g) Which of the intervals in parts (e) and (f) is shortest? Is this a good thing? Explain why this interval is far shorter than the others.

Message

(h) Interpret, after appropriate rounding, the confidence interval for Y in the context of this problem. Are there any caveats worth mentioning?

(i) If the advertising program is rolled out to 10,000 subscribers in this region, what sort of increase in phone usage would you anticipate?

Motivation

(a) Explain why it makes sense for the company to experiment with a sample of customers before rolling this program out to all of its subscribers.

(b) What is the advantage of measuring the response using the same customers? What is the weakness?

Method

Let X1 denote the number of minutes used by a customer before the promotion, and let X2 denote the number of minutes after the promotion. Use μ1 for the mean of X1 (before the promotion) and μ2 for the mean of X2 (after).

(c) The data on the number of minutes used by a customer during a given month are rather skewed. How does this affect the use of confidence intervals?

(d) Form a new variable, say Y = X2 - X1, that measures the change in use. How can you use the 95% confidence interval for the mean of Y?

Mechanics

(e) Form 95% confidence intervals for the mean m1 of X1 and the mean m2 of X2.

(f) Form the 95% confidence interval for the mean of the difference Y.

(g) Which of the intervals in parts (e) and (f) is shortest? Is this a good thing? Explain why this interval is far shorter than the others.

Message

(h) Interpret, after appropriate rounding, the confidence interval for Y in the context of this problem. Are there any caveats worth mentioning?

(i) If the advertising program is rolled out to 10,000 subscribers in this region, what sort of increase in phone usage would you anticipate?

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