(a) No. The comparison was with patients treated by other methods in the past.

64
Chapter 15
(b) No. Ullyot was using historical controls, which is not such a good idea—
such studies are often biased in favor of treatment. See section 1.3. (Ran-
domized controlled trials were much less positive about the value of coro-
nary bypass surgery.)
3.
False. The selection ratio for the pooled data is 68.5%. This is like Simpson’s
Paradox (pp.17ff). A majority of the women took the test in 1977, when the
pass rates were higher.
Technical comment. In the pooled data, the women’s pass rate is a weighted
average of their pass rates for 1975 and 1977, and must be intermediate between
those two values. Likewise for the men. The selection ratio, however, is a more
complicated statistic. The selection ratio in the pooled data can be written as
79
×
10
79
+
102
×
18
102
79
+
102
1312
×
250
1312
+
1259
×
331
1259
1312
+
1259
which equals
79
×
0
.
1266
+
102
×
0
.
1765
79
+
102
1312
×
0
.
1905
+
1259
×
0
.
2629
1312
+
1259
.
The women are in the numerator; the men, in the denominator. The weights for
the women and for the men differ. The women get more weight on the higher
pass rate (102 on 0.1765 vs 79 on 0.1266), while the men get more weight on
the lower pass rate (1312 on 0.1905 vs 1259 on 0.2629). If the weights are
not constrained, the lower bound on the selection ratio in the pooled data is
0.1266/0.2629
≈
48.16%, the numerator being the smaller of the women’s two
pass rates and the denominator being the larger of the men’s two rates. Likewise,
the upper bound is 0.1765/0.1905
≈
92.65%. The observed selection ratio of
68.5% is well inside the permissible range.
4.
Only (a) is right; (b) has area about 200%, and (c) has the wrong units.
5.
About 1 year. One month is way too small (for instance, 68% of the students
can hardly be within one month of average); 5 years is too large.
6.
False. These data are cross sectional, not longitudinal. The 50-year-old men
were born in 1955, and went to school when educational levels were lower than
they are now.
Furthermore, the material taught was less relevant to today’s
economy. In these respects, the 60-year-olds are in worse shape. Education
is one reason why the curve of age-specific average incomes flattens out. The
effect of education is confounded with the effect of age.
Comment.
Other data suggests that for each birth cohort, average income
increases fairly steadily with age.
7.
False. From left to right, the areas of the blocks are 10%, 10%, 10%, 30%,
40%. The 30th percentile is 60, and the 60th percentile is 80. Generally, the
60th percentile will not be twice the 30th percentile. (Remember, a percentile
is a number on the horizontal axis of the histogram.)

Answers to Sprecial Review Exercises
65
8.
(a) True. List (ii) is obtained by doubling each entry on list (i), then adding 1.
This change of scale cancels on conversion to standard units (section 5.6).
(b) False. List (ii) is obtained by doubling each entry on list (i), changing the
sign, then adding 1. This changes the signs in standard units.