Exercises 81a. The most productive 10% of the faculty will have a raise equal to or greater than
$.
b. The 5% of the faculty who have done nothing useful in years will receive no more than
$ each.3.10 We have sent out everyone in a large introductory course to check whether people use seat
belts. Each student has been told to look at 100 cars and count the number of people wear-
ing seat belts. The number found by any given student is considered that student’s score.
The mean score for the class is 44, with a standard deviation of 7.
a. Diagram this distribution, assuming that the counts are normally distributed.
b. A student who has done very little work all year has reported finding 62 seat belt users
out of 100. Do we have reason to suspect that the student just made up a number rather
than actually counting?
3.11 A number of years ago a friend of mine produced a diagnostic test of language problems.
A score on her scale is obtained simply by counting the number of language constructions
(e.g., plural, negative, passive) that the child produces correctly in response to specific
prompts from the person administering the test. The test had a mean of 48 and a standard
deviation of 7. Parents had trouble understanding the meaning of a score on this scale, and
my friend wanted to convert the scores to a mean of 80 and a standard deviation of 10 (to
make them more like the kinds of grades parents are used to). How could she have gone
about her task?
3.12 Unfortunately, the whole world is not built on the principle of a normal distribution. In the
preceding example the real distribution is badly skewed because most children do not have
language problems and therefore produce all or most constructions correctly.
a. Diagram how the distribution might look.
b. How would you go about finding the cutoff for the bottom 10% if the distribution is not
normal?
3.13 In October 1981 the mean and the standard deviation on the Graduate Record Exam (GRE)
for all people taking the exam were 489 and 126, respectively. What percentage of students
would you expect to have a score of 600 or less? (This is called the percentile rank of 600.)
3.14 In Exercise 3.13 what score would be equal to or greater than 75% of the scores on the
exam? (This score is the 75th percentile.)
3.15 For all seniors and non-enrolled college graduates taking the GRE in October 1981, the
mean and the standard deviation were 507 and 118, respectively. How does this change the
answers to Exercises 3.13 and 3.14?
3.16 What does the answer to Exercise 3.15 suggest about the importance of reference groups?
3.17 What is the 75th percentile for GPA in Appendix Data Set? (This is the point below which
75% of the observations are expected to fall.)
3.18 Assuming that the Behavior Problem scores discussed in this chapter come from a popula-
tion with a mean of 50 and a standard deviation of 10, what would be a diagnostically mean-
ingful cutoff if you wanted to identify those children who score in the highest 2% of the
population?
3.19 In Section 3.6, I said that Tscores are designed to have a mean of 50 and a standard devia-
tion of 10 and that the Achenbach Youth Self-Report measure produces Tscores. The data
in Figure 3.3 do not have a mean and standard deviation of exactly 50 and 10. Why do you
suppose that this is so?
3.20 Use a standard computer program to generate 5 samples of normally distributed variables
with 20 observations per variable. (For SPSS the syntax for the first sample would be
COMPUTE norm1 5 RV.NORMAL(0,1).)