Assume the Gauss model


1. Loosely based on Ch 26 F4: A scale is calibrated using a weight that is known to be exactly 1 kg. Each data set below represents repeated measurements of the weight. Assume the Gauss model (measurement = true weight of 1 kg + measurement error + bias), with measurement errors following the normal curve. Since the weight actually weighs 1 kg, the null hypothesis says that the expected value should be 1 kg (no bias). For each dataset below, make a t-test to see whether the scale is properly calibrated or not. In one case, this is impossible. Which one, and why? The data is listed as the amount above or below 1kg, and the units are micrograms.

a) 1, -2, 9 b) 1, -2, 9, 14, 8, 15, -1

c) 1 d) 1, 14

2. Ch 26 Rev 8: Bookstores like education, one reason being that educated people are more likely to spend money on books. National data show the nationwide average educational level to be 13 years of schooling completed, with an SD of about 3 years, for persons age 18 and over. A bookstore is doing a market survey in a certain county, and takes a simple random sample of 1,000 people age 18 and over. They find the average educational level to be 14 years, and the SD is 5 years. Can the difference in average educational level between the sample and the nation be explained by chance variation? If not, what other explanation can you give?

3. Ch 26 Rev 11: According to the census, the median household income in Atlanta (1.5 million households) was $52,000 in 1999. In June 2003, a market research organization takes a simple random sample of 750 households in Atlanta; 56% of the sample households had incomes over $52,000. Did median household income in Atlanta increase over the period 1999 to 2003?

a) Formulate null and alternative hypotheses in terms of a box model. (It’s a little tricky, think about what percent you should expect the 56% to be if the median household income didn’t change.)

b) Calculate the appropriate test statistic and P.

c) Did median family income go up?

4. Ch 27 A6: One hundred draws are made at random with replacement from box F: the average of these draws is 51 and their SD is 3. Independently, 400 draws are made at random with replacement from box G: the average of these draws is 48 and their SD is 8. Someone claims that both boxes have the same average. What do you think? You don’t need to do a formal hypothesis test, but think about it in that way.

5. Ch 27 B3: In 1970, 59% of college freshmen thought that capital punishment should be abolished; by 2005, the percentage had dropped to 35%. Is the difference real, or can it be explained by chance? You may assume that the percentages are based on two independent simple random samples, each of size 1,000. For this one, do a formal hypothesis test.

6. Ch 27 C2:(Hypothetical.) Is Wheaties a power breakfast? A study is done in an elementary statistics class; 499 students agree to participate. After the midterm, 250 are randomized to the treatment group, and 249 to the control group. The treatment group is fed Wheaties for breakfast 7 days a week. The control group gets Sugar Pops.

a) Final scores averaged 66 for the treatment group; the SD was 21. For the control group, the figures were 59 and 20. What do you conclude?

b) What aspects of the study could have been done “blind?”

7. Ch 27 D4: Many observational studies conclude that low-fat diets protect against cancer and cardiovascular “events” (heart attacks, stroke, and so forth). Experimental results, however, are generally negative. In 2006, the Women’s Health Initiative (WHI) published its results. This was a large-scale randomized trial on women who had reached menopause. As one part of the study, 48,835 women were randomized: 19,541 were assigned to the treatment group and put on a low-fat diet. The other 29,294 women were assigned to the control group and ate as they normally would. Subjects were followed for 8 years. Among other things, the investigators found that 1,357 women on the low-fat diet experienced at least one cardiovascular event, compared to 2,088 in the control group. Can the difference between the two groups be explained by chance? What do you conclude about the effect of the low-fat diet?

8. Ch 27 Rev 3: The Gallup poll asks respondents how they would rate the honesty and ethical standards of people in different fields-very high, high, average, low, or very low. The percentage who rated clergy “very high or high” dropped from 60% in 2000 to 54% in 2005. This may have been due to scandals involving sex abuse; or it may have been chance variation. (You may assume that in each year, the results are based on independent simple random samples of 1,000 persons in each year.)

a) Should you make a one-sample z-test or a two-sample z-test? Why?

b) Formulate the null and alternative hypotheses in terms of a box model. Do you need one box or two? Why? How many tickets go into each box? How many draws? What do the tickets show? What do the null and alternative hypotheses say about the box

c) Can the difference between 60% and 54% be explained as a chance variation? Or was it the scandals? Or something else?

9. Ch 27 Rev 4: This continues the previous exercise. In 2005, 65% of the respondents gave medical doctors a rating of “very high or high,” compared to a 67% rating for druggists. Is the difference real, or chance variation? Or do you need more information to decide? If the difference is real, how would you explain it? Discuss briefly. You may assume that the results are based on a simple random sample of 1,000 persons taken in 2005; each respondent rated clergy, medical doctors, druggists, and many other professions.

10. Ch 27 Rev 7: When convicts are released from prison, they often return to crime and are arrested again (recidivism). The Department of Labor ran a randomized controlled experiment to find out if providing income support to ex-convicts during the first months after their release reduces recidivism. The experiment was done on a group of convicts being released from prisons in Georgia. Income support was provided for the treatment group, like unemployment insurance, and the control group received no payment.

a) 592 prisoners were assigned to the treatment group, and of them 48.3%were rearrested within a year of release. 154 were assigned to the control group, and of them 49.4% were rearrested within a year of release. Did income support reduce recidivism? Answer yes or no, and explain briefly.

b) In the first year after their release from prison, those assigned to the treatment group averaged 16.8 weeks of paid work; the SD was 15.9 weeks. For those assigned to the control group, the average was 24.3 weeks; the SD was 17.3 weeks. Did income support reduce the amount that the ex-convicts worked? Answer yes or no, and explain briefly.

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