7.2 I drew 50 samples of 5 scores each from the same population that the data in Exercise 7.1
came from, and calculated the mean of each sample. The means are shown below. Plot the
distribution of these means.
2.8 6.2 4.4 5.0 1.0 4.6 3.8 2.6 4.0 4.8
6.6 4.6 6.2 4.6 5.6 6.4 3.4 5.4 5.2 7.2
5.4 2.6 4.4 4.2 4.4 5.2 4.0 2.6 5.2 4.0
3.6 4.6 4.4 5.0 5.6 3.4 3.2 4.4 4.8 3.8
4.4 2.8 3.8 4.6 5.4 4.6 2.4 5.8 4.6 4.8
7.3 Compare the means and the standard deviations for the distribution of digits in Exercise 7.1
and the sampling distribution of the mean in Exercise 7.2.
a. What would the Central Limit Theorem lead you to expect in this situation?
b. Do the data correspond to what you would predict?
7.4 In what way would the result in Exercise 7.2 differ if you had drawn more samples of size 5?
7.5 In what way would the result in Exercise 7.2 differ if you had drawn 50 samples of size 15?
7.6 Kruger and Dunning (1999) published a paper called “Unskilled and unaware of it,” in
which they examined the hypothesis that people who perform badly on tasks are unaware of
their general logical reasoning skills. Each student estimated at what percentile he or she
scored on a test of logical reasoning. The eleven students who scored in the lowest quartile
reported a mean estimate that placed them in the 68th percentile. Data with nearly the same
mean and standard deviation as they found follow: [40 58 72 73 76 78 52 72 84 70 72.]
Is this an example of “all the children are above average?” In other words is their mean per-
centile ranking greater than an average ranking of 50?
7.7 Although I have argued against one-tailed tests, why might a one-tailed test be appropriate
for the question asked in the previous exercise?
7.8 In the Kruger and Dunning study reported in the previous two exercises, the mean estimated
percentile for the 11 students in the top quartile (their actual mean percentile 5 86) was 70
with a standard deviation of 14.92, so they underestimated their abilities. Is this difference
significant?
7.9 The over- and under-estimation of one’s performance is partly a function of the fact that if
you are near the bottom you have less room to underestimate your performance than to
overestimate it. The reverse holds if you are near the top. Why doesn’t that explanation ac-
count for the huge overestimate for the poor scorers?
7.10 Compute 95% confidence limits on mfor the data in Exercise 7.8.
7.11 Everitt, in Hand et al., 1994, reported on several different therapies as treatments for
anorexia. There were 29 girls in a cognitive-behavior therapy condition, and they were
weighed before and after treatment. The weight gains of the girls, in pounds, are given be-
low. The scores were obtained by subtracting the Before score from the After score, so that
a negative difference represents weight loss, and a positive difference represents a gain.
1.7 0.7 2 0.1 2 0.7 2 3.5 14.9 3.5 17.1 2 7.6 1.6 11.7
6.1 1.1 2 4.0 20.9 2 9.1 2.1 2 1.4 1.4 2 0.3 2 3.7 2 0.8
2.4 12.6 1.9 3.9 0.1 15.4 2 0.7
a. What does the distribution of these values look like?
b. Did the girls in this group gain a statistically significant amount of weight?
7.12 Compute 95% confidence limits on the weight gain in Exercise 7.11.
7.13 Katz, Lautenschlager, Blackburn, and Harris (1990) examined the performance of 28 stu-
dents, who answered multiple choice items on the SAT without having read the passages to
which the items referred. The mean score (out of 100) was 46.6, with a standard deviation
of 6.8. Random guessing would have been expected to result in 20 correct answers.
a. Were these students responding at better-than-chance levels?
b. If performance is statistically significantly better than chance, does it mean that the
SAT test is not a valid predictor of future college performance?218 Chapter 7 Hypothesis Tests Applied to Means