Witte & Witte, 9e Page 1 of 4 Pages
Chapter 5 Exercises
Chapter 5 Standard (z) Scores and Normal Distributions
Exercise 1
- The average distance in kilometers from a school in Chicago to the closest fast-food restaurant is 0.60 with a standard deviation of 0.45.
Source: Austin, S. B., Melly, S. J., Sanchez, B. N., Patel, A., Buka, S., & Gortmaker, S. L. (2005). Clustering of fast-food restaurants around schools: A novel application of spatial statistics to the study of food environments. American Journal of Public Health, 95, 1575-1581.
Express each of the following distances as a z score:
a. A distance of 1.00 kilometer.
b. A distance of 0.20 kilometers.
c. A distance of 1.25 kilometers.
d. A distance of 0.60 kilometers.
Answers:
a. z = 0.89
b. z = -0.89
c. z = 1.44
d. z = 0
- The addiction scale developed by Gossop and Eysenck was used by mental health researchers to assess liability to development of a drug dependency among secondary-school students in West Germany. Higher scores on this scale indicate a greater liability for drug dependency development. For students who never participated in endurance sports, the addiction mean and standard deviation were 12.36 and 4.83, respectively. For students who often participated in endurance sports, the addiction mean and standard deviation were 9.69 and 4.58, respectively.
Source: Kirkcaldy, B. D., Shephard, R. J., & Siefen, R. G. 2002. The relationship between physical activity and self-image and problem behaviour among adolescents. Social Psychiatry and Psychiatric Epidemiology, 37, 544-550.
Express each of the following addiction scores as a z score:
a. An addiction score of 10 for a student who never participated in endurance sports.
b. An addiction score of 10 for a student who often participated in endurance sports.
c. An addiction score of 15 for a student who never participated in endurance sports.
d. An addiction score of 15 for a student who often participated in endurance sports.
Answers:
a. -0.49
b. 0.07
c. 0.55
d. 1.16
Exercise 2
Using Table A in Appendix C, find the proportion of the total area identified with the following statements:
1. above a z score of 1.50
2. below a z score of 1.50
3. between the mean and a z score of 0.56
4. below a z score of -2.00
5. above a z score of -2.00
6. between the mean and a score of 2.58
7. between z scores of 0 and -1.65
Answers:
1. .0668
2. .9332
3. .2123
4. .0228
5. .9772
6. .4951
7. .4505
Exercise 3
The Detroit Tests of Learning Aptitude include 10 subtests (e.g., Story Construction, Word Sequences, and Story Sequences). The subtest standardized scores have a mean of 10 and a standard deviation of 3. Find the proportion that corresponds to each of the target areas by converting the standardized score to a z score and then referring to Table A in Appendix C.
1. greater than 13
2. less than 13
3. less than 6
4. greater than 6
5. greater than 15
6. less than 2
7. between 7 and 13
8. between 6 and 14
9. less than 7 and greater than 13
10. less than 6 and greater than 14
Answers:
1. z = 1, proportion = .1587
2. z = 1, proportion = .8413
3. z = -1.33, proportion = .0918
4. z = -1.33, proportion = .9082
5. z = 1.67, proportion = .0475
6. z = -2.67, proportion = .0038
7. z = -1 and z = 1; proportion = (2)( .3413) = .6826
8. z = -1.33 and z = 1.33; proportion = (2)(.4082) = .8164
9. z = -1 and z = 1; proportion = (2)(.1587) = .3174
10. z = -1.33 and z = 1.33; proportion = (2)(.0918) = .1836
Exercise 4
Assume that GRE Verbal scores approximate a normal curve with a mean of 500 and a standard deviation of 100.
1. A university admissions office has decided to admit only those applicants who score in the top 10 percent of the GRE Verbal distribution. In order to be admitted, a student must have received a score of ____ or higher.
2. A university admissions office has decided to admit only those applicants who score in the top 25 percent of the GRE verbal distribution. In order to be admitted, a student must have received a score of ____ or higher.
3. A GRE training program will only admit individuals who scored in the bottom 25 percent of the distribution. In order to be admitted, a student must have received a score of ____ or less.
Answers:
1. z = 1.28; GRE Verbal = 628
2. z = 0.67; GRE Verbal = 567
3. z = -0.67; GRE Verbal = 433
Exercise 5
Megan is working with a distribution of reaction times that is positively skewed. Megan transforms the reaction times to z scores.
1. Will the distribution of these z scores have an approximately normal curve shape?
2. The mean of these z scores is equal to ___.
3. The standard deviation of these z scores is equal to ___.
4. The variance of these z scores is equal to ___.
Answers:
1. No; the distribution of the z scores will be positively skewed.
2. 0
3. 1
4. 1
Exercise 6
To calculate grades in her statistics course, Professor Clark converted the raw scores on all four exams to z-scores. Each student’s final course grade was then based on his or her highest z score. Sara’s raw scores are shown below along with the mean and standard deviation for each of the four exams. On which exam did Sara have the highest z score?
Exam / Raw Score / Mean / Standard DeviationA / 50 / 45 / 10
B / 30 / 30 / 10
C / 25 / 20 / 2
D / 25 / 20 / 3
Answer:
Exam A z = 0.50; Exam B z = 0; Exam C z = 2.5; Exam D z = 1.67
The highest z score was obtained on Exam C.
Exercise 7
After transforming each of the raw scores below into a z score, transform each z score into a standard score with a mean of 50 and a standard deviation of 10.
Exam / Raw Score / Mean / Standard DeviationA / 50 / 45 / 10
B / 30 / 30 / 10
C / 25 / 20 / 2
D / 25 / 20 / 3
Answers:
A. 55
B. 50
C. 75
D. 66.7