CHAPTER 12SOLUTIONS AND MINI-PROJECT NOTES
CHAPTER 12
RELATIONSHIPS BETWEEN CATEGORICAL VARIABLES
EXERCISE SOLUTIONS
12.1No, it is not sufficient. We do not know how many of the men or women were Democrats versus Republicans.
12.21.6 per 100,000 is 0.0016% of the population. It is difficult to interpret such a small percentage. The rate of 1.6 per 100,000 is easier to understand.
12.3The proportion of surgical operations requiring an overnight hospital stay is 0.40; the risk of having to spend the night at the hospital following a surgical procedure is 0.40. The odds of having to spend the night at the hospital versus getting to go home on the same day are 40 to 60, or 4 to 6, or 2 to 3.
12.4To compute the relative risk use 1/5 divided by 1/50, or 50/5 = 10. Patients who have had heart surgery might be doing themselves a favor by having religious beliefs giving them strength and comfort, or by participating in social activities. An observational study found that patients who had neither of these had a 10 times higher risk of dying within 6 months than patients who had both.
12.5a.Relative risk = 1 + (increased risk/100) = 1 + 0.25 = 1.25.
b.The baseline risk is missing.
12.6a.Observational study.
b.Reasons 3, 4 and 5 are all possibilities. Reason 3 says that firing someone or having a high-stakes deadline may be a contributor to the heart attack. Reason 4 says that there may be confounding variables; in this case, overall job responsibility and consequent stress are likely to be involved. Reason 5, that both having to fire someone (or having a deadline) and having a heart attack result from a common cause is possible; they could both indirectly result from having the type of personality that leads one to be driven to succeed, to take high risks, and so on.
c.i.A person who has to fire someone or is subjected to a high-stakes deadline has double the odds of having a heart attack in the following week than someone who has not had to do those things. (Notice that this is not the same thing as saying this person has twice the odds that that same person would have had if they hadn't fired someone or had the deadline.)
ii. Although someone who has to fire someone or who has a high-stakes deadline has twice the risk of having a heart attack during the next week as someone who has not done those things, the increased risk is quite small. For a healthy 50-year-old man or 60-year-old woman, the risk in any given hour without a trigger is only about 1 in a million.
iii. A person who has to fire someone or is subjected to a high-stakes deadline has twice the risk of having a heart attack in the following week than someone who has not had to do those things.
12.7a.The odds of a successful guess compared to an unsuccessful guess, by chance alone, are 25 to 75 or 1 to 3.
b.In the experiment, the odds of guessing successfully to guessing unsuccessfully were 34 to 66 or 17 to 33 (or about 1 to 2).
12.8This does not mean that each, or even any, individual has equal chances of the two events. Someone who drives mainly in cities has a higher chance of being a carjacking victim, while someone who drives mainly on highways has a higher chance of being killed in a traffic accident.
12.9a.i.212/1,525 = 0.1390 = 13.9%
ii.465/4,377 = 0.1062
iii.This is the same as in part i, 0.1390 or 13.9%.
iv.The odds are 465 to 3912, or about 0.12 to 1, 12 to 100, 3 to 25.
b.The relative risk is 0.1390/0.1062 = 1.31. We would say: People aged 18 to 29 are 1.31 times as likely to report seeing a ghost than those over 30 (or some similar wording).
c.People aged 18 to 29 are 31% more likely to report seeing a ghost than those over 30.
12.10a.Increased risk.
b.Relative risk, in both cases.
c.Proportion.
12.11Relative risk = 1 + (increased risk/100) = 1 + 0.50 = 1.50.
12.12The relative risk is 1.30. Nonsmokers who are exposed to their spouses' smoke have a chance of death from heart disease that is 1.3 times the chance for nonsmokers who are not exposed.
12.13a.A student in 1980 was 10 times as likely to use cocaine as one in 1993.
b.They are proportions, although they are expressed differently than proportions are usually expressed. The statement "one in three" means that 1/3 of students smoked in 1980. The odds would be: In 1980, the odds that a student smoked marijuana were 1 to 2. By 1993 they were reduced to 1 to 6. Equivalently, you could say that the odds of not smoking were 2 to 1 in 1980 and 6 to 1 in 1993.
c.No. This is a good example of a relationship due to changes over time. It could be that drug use declined due to changing priorities, etc., and had nothing to do with the "efforts to discourage them from using drugs."
12.14a.Here is the combined table:
Admit / Deny / Total / Percent AdmittedMen / 450 / 550 / 1,000 / 450/1,000 = 45%
Women / 175 / 325 / 500 / 175/500 = 35%
Total / 625 / 875 / 1,500
A higher percentage of men than women were admitted, so it appears that the women could have been discriminated against.
b.Program A admitted 400/650 = 61.5% of the men, and 50/75 = 67% of the women. Program B admitted 50/350 = 14.3% of the men and 125/425 = 29.4% of the women. Therefore, it appears that both programs had a slight discrimination against men!
c.Simpson's Paradox occurs when combining groups reverses the direction of the relationship from what it was when the groups were separate. In this case, each program admitted a higher percentage of women, yet overall a lower percentage of women were admitted. What happened was that Program B was harder to get into, and was the one for which the majority of women applied. Of the applicants to Program A, which was relatively easy to get into, only about 10% were women. However, over half of the applicants to Program B, which was hard to get into, were women. These may have been something like Mathematics (Program A) and Psychology (Program B).
12.15a.Here is the Table:
Lung Cancer / No Lung Cancer / TotalBird / 98 / 101 / 199
No Bird / 141 / 328 / 469
Total / 239 / 429 / 668
b.For bird owners the risk is 98/199 = 0.4925 = 49.25%. For nonbird owners it is 141/469 =0.301 =30.1%.
c.No. This was a case-control study, not based on a random sample from the population. Clearly we would not find the risk of lung cancer among bird owners in general to be 0.49.
d.The relative risk is 0.49/0.30 = 1.63 (or 1.64 if you don't round off as soon) so the increased risk is 63%.
e.You would want to know the baseline risk of lung cancer without owning a bird. If it is very small, then an increased risk of 63% may not mean much to someone who really likes birds.
12.16a.The odds are 1,382 to 130, or about 10.6 to 1 for African Americans and 2,813 to 87 or about 32.3 to 1 for Caucasians.
b.Odds ratio is (2,813/87) (1,382/130) = 3.04.
12.17a.Here is the contingency table:
Application Approved?Yes / No / Total
Black / 3,117 / 979 / 4,096
White / 71,950 / 12,997 / 84,947
Total / 75,067 / 13,976 / 89,043
b.76.1% of blacks were approved and 84.7% of whites.
c.The ratio is 76.1/84.7 = 0.898, a selection ratio.
d.Yes, it would pass because it is over 80% or 4/5.
12.18a.Teens who are frequently bored are 50% more likely to do those things. The increased risk is 50%.
b.The relative risk is 1.50, computed from the increased risk.
c.56% is a proportion of 0.56. (See page 535 of the Appendix.)
d.68% have no friends who use marijuana so 32% do have friends who use it. (See page 3 of Original Source 13 or Question 65 on page 49.)
12.19a.Relative risk.
b.Proportion or risk.
c.Relative risk.
d.Increased risk.
12.20a.The baseline risk is indeed missing, so we can’t say what the death rates are for the two groups. Of course it depends on age, so it is not possible to give a baseline risk. Everyone eventually dies!
b.The time period of the risk is not identified. We don’t know if it’s in a given year, or in a decade, or since the beginning of the study in 1965.
c.The reported risk is not necessarily your risk at all. There are many possible confounding variables in this study, such as social support. Your risk may in fact increase if you attend church, or may not change at all, or may decrease. The reported comparison is an average for everyone.
12.21a.The increased risk of dying of circulatory disease is reported to be 21%.
b.Relative risk is 1.21.
c.The increased risk of dying of digestive disease is reported to be 99%.
d.Relative risk is 1.99.
12.22a.Women under the age of 70 who attended religious services less than once a week or never were 1.22 times more likely to die during the course of the study than those who attended religious services at least weekly, after adjusting for factors such as education and income.
b.Men under the age of 70 who attended religious services less than once a week or never had a 27 percent greater overall risk of dying than men who attended religious services at least weekly, after adjusting for factors such as education and income.
c.Men age 70 and older who attended religious services less than once a week or never were no more or less likely to die of cancer than men who attended religious services at least once a week.
d.Men under age 70 who attended religious services less than once a week or never were only .74 times (about three-fourths) as likely to die of cancer than men who attended religious services at least weekly, after adjusting for factors such as education and income.
NOTES ABOUT MINI-PROJECTS FOR CHAPTER 12
Mini-Project 12.1
This project is relatively straightforward in that it involves the same kinds of computations as the exercises. Be careful about collecting the data, to be sure the individuals are independent of each other and the same variables are measured on each one. Some guidelines to check are as follows. For part a, make sure the convention of putting the explanatory variable in the rows and the response in the columns is followed, if that distinction can be made. For part b make sure all parts are answered and that relative risk is computed rather than increased risk. For part c remember that a cause-and-effect conclusion can be made only if the data were collected as a randomized experiment.
Mini-Project 12.2
The project is designed to illustrate that relative risk is a common feature in news articles about studies. It should be easy to find such an example. Make sure the report discusses all three of the features listed in Section 12.3.
Mini-Project 12.3
This is a fairly long assignment and you will need to check what students write against the original articles. One of the major short-comings is that these are all observational studies, so a cause-and-effect conclusion cannot be made.
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