Confidence interval and testing of hypothesis problems

1. A manufacturer of small appliances employs a market research firm to estimate retail sales of its products by gathering information from a sample of retail stores. This month an SRS of 75 stores in the Midwest sales region finds that these stores sold an average of 24 of the manufacturer's hand mixers, with standard deviation 11.

(a) Give a 95% confidence interval for the mean number of mixers sold by all stores in the region.

(b) The distribution of sales is strongly right-skewed because there are many smaller stores and a few very large stores. The use of t in (a) is reasonably safe despite this violation of the Normality assumption. Why?

2. The data given below show that many people paid $20 per month for Internet Access, presumably because major providers such as AOL charged this amount. Do the data give good reason to think that they mean cost for all Internet users differs from $20 per month?

DATA
50 users

20 40 22 22 21 21 20 10 20 20
20 13 18 50 20 18 15 8 22 25
22 10 20 22 22 21 15 23 30 12
9 20 40 22 29 19 15 20 20 20
20 15 19 21 14 22 21 35 20 22

3. The table given below gives data on the annual returns (percent) for the Vanguard International Growth Fund and its benchmark index, the Morgan Stanley EAFE Index. Does the fund significantly outperform its benchmark?

(a) Explain clearly why the matched pairs t test is the proper choice to answer this question.
(b) Make a stem plot of the differences (fund - EAFE) for the 20 years. There is no reason to doubt the approximate Normality of the differences. (More detailed study shows that the differences follow a Normal distribution quite closely).
(c) Carry out the test and state your conclusion about the fund's performance.

DATA

Year Fund EAFE
1982 5.27 -1.86
1982 43.08 23.69
1984 - 1.02 7.38
1985 56.94 56.16
1986 56.71 69.44
1987 12.48 24.63
1988 11.61 28.27
1989 24.76 10.54
1990 -12.05 -23.45
1991 4.74 12.13
1992 -5.79 -12.17
1993 44.74 32.56
1994 0.76 7.78
1995 14.89 11.21
1997 14.65 6.05
1997 4.12 1.78
1998 16.93 20.00
1999 26.34 26.96
2000 -8.60 -14.17
2001 -18.92 -21.44

4. Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 1019 adults. A newspaper article about the poll said, "Results have a margin of sampling error of plus or minus 3 percentage points.

(a) Over, 54% of the sample (550 of 1019 people) answered "yes". Find a 95 confidence interval for the proportion in the adult population who would say "yes" if asked. Is the report's claim about the margin of error roughly right? (Assume the sample is an SRS).
(b) The news article said that 65% of men, but only 43% of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately.
(c) Would a 95% confidence interval for women alone have a margin of error less than 0.03, about equal to 0.03, or greater than 0.03? Why? You see that the news article's statement about the margin of error for poll results is a bit misleading.