PROBLEM 1. Direct Mail

Difficulty: !!; confidence intervals for proportions.

Based on Chapter 15: "Confidence Intervals", problem 47.

Direct mail advertisers send solicitations (junk mail) to thousands of potential customers hoping that some will buy the product. The response rate is usually quite low. Suppose a company wants to test the response to a new flyer and sends it to 1,000 randomly selected people. The company gets orders from 123 of the recipients and decides to do mass mailing to everyone on its mailing list of 200,000 people. Create 95% confidence interval for the percentage of those people who will order something.

A numerical answer appears at the end of the chapter. This can serve as a check on your answers. The deliverable is a detailed calculation that validates this answer.

PROBLEM 2. Smartphones Difficulty: !!; confidence intervals.

Based on problem 50, Chapter 15: "Confidence Intervals".

On October 10, 2011, the Wall Street Journal reported that 44% of smartphone purchases made by people with income between $35,000 to

$50,000 were Android phones.

1. If the story is based on a sample of 200 purchases, then should we conclude that Android phones were getting less than half of the market at that time?
2. How large a sample is needed for 95% confidence interval for the pop- ulation proportion to just exclude 50%? (That is, find the n needed so that the margin of error (ME) is 6%.)
3. Hint: Use the margin of error formula for proportions to determine the n needed to bring the ME to this level.
4. 2

PROBLEM 3. Cadillac Escalade I Difficulty: !!; hypothesis testing.

The average Cadillac buyer is older than 60, past the prime middle years typically associated with high spending. Cadillac's Escalade brand draws younger customers. Suppose we are given a sample of 50 Escalade purchasers with average age of 45 and estimated population standard deviation (s) of 25.

Cadillac's management would like to know whether this is compelling ev- idence that Escalade buyers are younger on average than the typical Cadillac buyer.

1. State the Null and Alternative Hypotheses for Cadillac's management.
2. In about one line each, describe in words the Type I (false positive) and Type II (false negative) errors in the context of this problem.
3. Calculate the t-value of sample mean under your Null Hypothesis.
4. Find the p-value of the test using both the standard normal and t- distributions. In your opinion, does the data supply enough evidence to support rejection of the Null Hypothesis under either of the two distributions?

This question is based on Chapter 16, problem 46. Problem 48 is related and is optional. It illustrates how to take into account skewness in assessing validating a test. We will provide answers as part of the answer key. You have the option to submit answers to this problem (48) but your answers will not affect the grade.

PROBLEM 4. Cadillac Escalade II Difficulty: !!; hypothesis testing.

Cadillac formulates the following null hypothesis about the average age of its customers:

H0 :=60.

Suppose that the population standard deviation is known to be = 28, so

the standard deviation for the mean of samples of size n = 49 is 28 = 4.

Cadillac adopts the following procedure: 3

49

"Reject H0 if the sample mean is more than 2 standard deviations from = 60. Accept H0 otherwise."

What is the probability that this procedure will fail to reject H0 if the true population mean age is 50? In other words, if the true average were 50, how likely is to observe a sample that would cause Cadillac to accept the null hypothesis (when in fact it is false)? (That is, what is the probability of Type II error under this procedure when the true mean is 50.)