We are interested in whether heavy marketing of a new product increases units of sales. A random sample of 50 regions was chosen in which to heavily market the product. From that sample, the average increase in sales, in thousands of units, was 6.1697 and the standard deviation was 13.7765. Let be the mean increased sales for all heavily marketed regions.

a. Produce a 95% confidence interval for the true average increase in sales from heavy marketing.

b. Suppose the company makes a net profit of $10 per product sold (excluding marketing costs) and that heavy marketing of a given region costs $40,000. Use the confidence interval in part a) to decide whether there is clear evidence that the company should use heavy marketing or not.

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 1000 randomly selected people. The company gets orders from 128 of the recipients and decides to do a mass mailing to everyone on its mailing list. Create a 95% confidence interval for the percentage of people who will order something.

A study is done in San Antonio, TX to see whether the average weekly grocery bill per five-person family there is significantly different from the national average. A random sample of 50 five-person families in San Antonio showed a mean of $133.474 and a standard deviation of $11.193.

a. Assume the national average weekly grocery bill for a five-person family is $131. Can we conclude that San Antonio grocery bills are significantly higher than the national average? What decision would you make for a = 0.10?

b. What decision would you make for a = 0.05?

c. For which values of the sample mean (i.e., average weekly grocery bill) would you be able to reject the null hypothesis at the 1% significance level?