PART VIII. TESTS OF SIGNIFICANCE
6. SPECIAL REVIEW EXERCISES These exercises cover all of parts I and II
1. In one course, a histogram for the scores on the final looked like the sketch below. True or false: because this isn’t like the normal curve, there must have been something wrong with the test. Explain.
2. Fill in the blanks, using the options below, and give examples to show that you picked the right answers.
(a) The SD of a list is 0. This means . (b) The r.m.s. size of a list is 0. This means . Options:
(i) there are no numbers on the list (ii) all the numbers on the list are the same (iii) all the numbers on the list are 0 (iv) the average of the list is 0
3. A personality test is administered to a large group of subjects. Five scores are shown below, in original units and in standard units. Fill in the blanks.
79 64 52 72
1.8 0.8 −1.4
4. Among first-year students at a certain university, scores on the Verbal SAT follow the normal curve; the average is around 550 and the SD is about 100.
(a) What percentage of these students have scores in the range 400 to 700?
(b) There were about 1,000 students with scores in the range 450–650 on the Verbal SAT. About of them had scores in the range 500 to 600. Fill in the blank; explain briefly.
5. In Cycle III of the Health Examination Survey (like HANES, but done in 1966–70), there were 6,672 subjects. The sex of each subject was recorded at two different stages of the survey. In 17 cases, there was a discrepancy:
the subject was recorded as male at one interview, female at the other. How would you account for this?
6. Among entering students at a certain college, the men averaged 650 on the Math SAT, and their SD was 125. The women averaged 600, but had the same SD of 125. There were 500 men in the class, and 500 women.
(a) For the men and the women together, the average Math SAT score was .
(b) For the men and the women together, was the SD of Math SAT scores less than 125, just about 125, or more than 125?
7. Repeat exercise 6, when there are 600 men in the class, and 400 women. (The separate averages and SDs for the men and women stay the same.)
8. Table 1 on p. 99 reported 100 measurements on the weight of NB 10; the top panel in figure 2 on p. 102 shows the histogram. The average was 405 micro- grams, and the SD was 6 micrograms. If you used the normal approximation to estimate how many of these measurements were in the range 400 to 406 micrograms, would your answer be too low, too high, or about right? Why?
9. A teaching assistant gives a quiz to his section. There are 10 questions on the quiz and no part credit is given. After grading the papers, the TA writes down for each student the number of questions the student got right and the number wrong. The average number of right answers is 6.4 with an SD of 2.0. The average number of wrong answers is with an SD of . Fill in the blanks—or do you need the data? Explain briefly.
10. A large, representative sample of Americans was studied by the Public Health Service, in the Health and Nutrition Examination Survey (HANES2).5The percentage of respondents who were left-handed decreased steadily with age, from 10% at 20 years to 4% at 70. “The data show that many people change from left-handed to right-handed as they get older.” True or false? Why? If false, how do you explain the pattern in the data?
11. For a certain group of women, the 25th percentile of height is 62.2 inches and the 75th percentile is 65.8 inches. The histogram follows the normal curve.
Find the 90th percentile of the height distribution.
12. In March, the Current Population Survey asks a large, representative sample of Americans to say what their incomes were during the previous year.6 A histogram for family income in 2004 is shown at the top of the next page.
(Class intervals include the left endpoint but not the right.) From $15,000 and on to the right, the blocks alternate regularly from high to low. Why is that?
REVIEW EXERCISES 107
0 25 50 75 100 125 150 175
0 1.5
INCOME (THOUSANDS OF DOLLARS)
THOUSAND DOLLARSPERCENT PER
13. To measure the effect of exercise on the risk of heart disease, investigators compared the incidence of this disease for two large groups of London Trans- port Authority busmen—drivers and conductors. The conductors got a lot more exercise as they walked around all day collecting fares.
The age distributions for the two groups were very similar, and all the sub- jects had been on the same job for 10 years or more. The incidence of heart disease was substantially lower among the conductors, and the investigators concluded that exercise prevents heart disease.
Other investigators were skeptical. They went back and found that London Transport Authority had issued uniforms to drivers and conductors at the time of hire; a record had been kept of the sizes.7
(a) Why does it matter that the age distributions of the two groups were similar?
(b) Why does it matter that all the subjects had been on the job for 10 years or more?
(c) Why did the first group of investigators compare the conductors to drivers, not to London Transport Authority executive staff?
(d) Why might the second group of investigators have been skeptical?
(e) What would you do with the sizes of the uniforms?
14. Breast cancer is one of the most common malignancies among women in Canada and the U.S. If it is detected early enough—before the cancer spreads—chances of successful treatment are much better. Do screening pro- grams speed up detection by enough to matter? Many studies have examined this question.
The Canadian National Breast Cancer Study was a randomized controlled experiment on mammography, that is, x-ray screening for breast cancer. The study found no benefit from screening. (The benefit was measured by com- paring death rates from breast cancer in the treatment and control groups.) Dr. Daniel Kopans argued that the randomization was not done properly: in- stead of following instructions, nurses assigned high risk women to the treat- ment group.8Would this bias the study? If so, would the bias make the benefit from screening look bigger or smaller than it really is? Explain your answer.
15. In some jurisdictions, there are “pretrial conferences,” where the judge con- fers with the opposing lawyers to settle the case or at least to define the issues before trial. Observational data suggest that pretrial conferences promote set- tlements and speed up trials, but there were doubts.
In New Jersey courts, pretrial conferences were mandatory. However, an ex- periment was done in 7 counties. During a six-month period, 2,954 personal injury cases (mainly automobile accidents) were assigned at random to treat- ment or control. For the 1,495 control cases (group A), pretrial conferences remained mandatory. For the 1,459 treatment cases, the conferences were made optional—either lawyer could request one. Among the treatment cases, 701 opted for a pretrial conference (group C), and 758 did not (group B).
The investigator who analyzed the data looked to see whether pretrial confer- ences encouraged cases to settle before reaching trial; or, if they went to trial, whether the conferences shortened the amount of trial time. (This matters, because trial time is very expensive.)
The investigator reported the main results as follows; tabular material is quoted from his report.9
(i) Pretrial conferences had no impact on settlement; the same percentage go to trial in group B as in group A+C.
Percentage of cases reaching trial Group B Group A+C
Reached trial 22% 23%
Number of cases 701 2,079
(ii) Pretrial conferences do not shorten trial time; the percentage of short trials is highest in cases that refused pretrial conferences.
Distribution of trial time among cases that go to trial Group B Group A Group C Trial time (in hours)
1. 5 or less 43% 34% 28%
2. Over 5 to 10 35% 41% 39%
3. Over 10 22% 26% 33%
Number of cases 63 176 70
Comment briefly on the analysis.
7. SUMMARY AND OVERVIEW
1. No matter how carefully it was made, a measurement could have turned out a bit differently. This reflects chance error. Before investigators rely on a measurement, they should estimate the likely size of the chance error. The best way to do that:replicatethe measurement.
SUMMARY AND OVERVIEW 109
2. The likely size of the chance error in a single measurement can be esti- mated by the SD of a sequence of repeated measurements made under the same conditions.
3. Bias, orsystematic error, causes measurements to be systematically too high or systematically too low. The equation is
individual measurement=exact value+bias+chance error.
The chance error changes from measurement to measurement, but the bias stays the same. Bias cannot be estimated just by repeating the measurements.
4. Even in careful measurement work, a small percentage ofoutlierscan be expected.
5. The average and SD can be strongly influenced by outliers. Then the his- togram will not follow the normal curve at all well.
6. This part of the book introduced two basic descriptive statistics, the av- erage and the standard deviation; histograms were used to summarize data. For many data sets, the histogram follows the normal curve. Chapter 6 illustrates these ideas on measurement data. Later in the book, histograms will be used for prob- ability distributions, and statistical inference will be based on the normal curve.
This is legitimate when the probability histograms follow the curve—the topic of chapter 18.
7
Plotting Points and Lines
Q. What did the dot say to the line?
A. Come to the point.