Question: In the book Essentials of Marketing Research, William R. Dillon, Thomas J. Madden, and Neil H. Firtle discuss an advertising study for a new suntan lotion. In this study, each respondent is assigned to a group whose members will evaluate an ad for the new lotion. Each respondent is asked to rate the ad on six items:

High quality/low quality

Persuasive/nonpersuasive

Informative/uninformative Artful/artless

The rating for each item is made using a seven-point scale, where, for example, a rating of 1 on the informative/uninformative dimension indicates that the ad is extremely uninformative, and a rating of 7 says that the ad is extremely informative.

Suppose experience shows that a "very informative" ad is typically rated by a large group of respondents according to the probability distribution given in the right page margin.

1. Calculate the mean, variance, and standard deviation of the ratings for a typical "very informative" ad.
2. Suppose that a group of 36 randomly selected respondents rates a typical "very informative" ad, and consider the sample mean \(\bar{x}\) of the 36 ratings. Find the mean and standard deviation of the population of all possible sample means. What is the shape of the population of all possible sample means? How do you know?
3. Draw a sketch of the sampling distribution of the sample mean \(\bar{x}\) and compare it to a sketch of the distribution of individual ratings.
4. Suppose that a randomly selected group of 36 respondents rates a typical "very informative" ad. Find the probability that the respondents give the ad a sample mean rating less than 5 .
5. Suppose that 36 randomly selected respondents are exposed to a new ad in order to determine whether the ad is "very informative," and suppose that the sample mean rating is less than 5 . In light of the probability you computed in part d, what would you conclude about whether the new ad is "very informative"? Explain.

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