Exercises
3.1 Assume that the following data represent a population with m54 and s51.63: X 5
[1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7]
a. Plot the distribution as given.
b. Convert the distribution in part (a) to a distribution of X2m.
c. Go the next step and convert the distribution in part (b) to a distribution of z.
3.2 Using the distribution in Exercise 3.1, calculate zscores for X 5 2.5, 6.2, and 9. Interpret
these results.
3.3 Suppose we want to study the errors found in the performance of a simple task. We ask a large
number of judges to report the number of people seen entering a major department store in
one morning. Some judges will miss some people, and some will count others twice, so we
don’t expect everyone to agree. Suppose we find that the mean number of shoppers reported is
975 with a standard deviation of 15. Assume that the distribution of counts is normal.
a. What percentage of the counts will lie between 960 and 990?
b. What percentage of the counts will lie below 975?
c. What percentage of the counts will lie below 990?
3.4 Using the example from Exercise 3.3:
a. What two values of X(the count) would encompass the middle 50% of the results?
b. 75% of the counts would be less than.
c. 95% of the counts would be between and.
3.5 The person in charge of the project in Exercise 3.3 counted only 950 shoppers entering the
store. Is this a reasonable answer if he was counting conscientiously? Why or why not?
3.6 A set of reading scores for fourth-grade children has a mean of 25 and a standard deviation
of 5. A set of scores for ninth-grade children has a mean of 30 and a standard deviation
of 10. Assume that the distributions are normal.
a. Draw a rough sketch of these data, putting both groups in the same figure.
b. What percentage of the fourth graders score better than the average ninth grader?
c. What percentage of the ninth graders score worse than the average fourth grader? (We
will come back to the idea behind these calculations when we study power in Chapter 8.)
3.7 Under what conditions would the answers to parts (b) and (c) of Exercise 3.6 be equal?
3.8 A certain diagnostic test is indicative of problems only if a child scores in the lowest 10%
of those taking the test (the 10th percentile). If the mean score is 150 with a standard devia-
tion of 30, what would be the diagnostically meaningful cutoff?
3.9 A dean must distribute salary raises to her faculty for the next year. She has decided that the
mean raise is to be $2000, the standard deviation of raises is to be $400, and the distribution
is to be normal.80 Chapter 3 The Normal Distribution
Key Terms
Normal distribution (Introduction)
Bar chart (Introduction)
Abscissa (3.1)
Ordinate (3.1)
Standard normal distribution (3.2)
Pivotal statistic (3.2)
Deviation score (3.2)
zscore (3.2)
Quantile-quantile
(Q-Q) plots (3.5)Kolmogorov-Smirnov test (3.5)
Standard scores (3.6)
Percentile (3.6)
Tscores (3.6)