Problem Sets for Elements of Mental Tests 1

Problem Sets for Elements of Mental Tests 1

Mathematicaland

Multiple Choice

Problem Sets

for

The Elements of Mental Tests

Second Edition

John D. Mayer

January 25th, 2016 Version

Table of Contents

Problem Set for Chapter 1

Problem Set For Chapter 2

Problem Set For Chapter 3

Problem Set For Chapter 4

Problem Sets For Chapter 5

Problem Sets For Chapter 6

Problem Sets For Chapter 7

Problem Sets For Chapter 8

Problem Sets For Chapter 9

Problem Sets For Chapter 10

Problem Sets For Chapter 11

Problem Sets For Chapter 12

Appendix: Hints, Intermediate Values and Answers for Selected Problems

Problem Set for Chapter 1

Types of multiple choice questions that address the content of Chapter 1

Questions about the History of Testing

1.1. Journalists have criticized psychological tests beginning

a. in the past decade.

b. with the introduction of intelligence tests in the early 1900s.

c. only mildly at first, but with increasing intensity through the 20th century.

d. only in the last few years.

Questions about What a Test Is

1.2. Which feature is NOT a defining aspect of a mental test:

a. an integrated mental procedure

b. collects data about mental life

c. uses ratio scales

d. indicates the presence of magnitude of features

Questions about the Advantages and Drawbacks of Testing

1.3. Psychological tests have this relationship with informal interviewing (say, a job interview):

a. they have not been compared

b. psychological tests are more reliable and less biased

c. psychological tests are more biased and less valid

d. psychological tests lose or miss more information than does the interviewer

Problem Set For Chapter 2

Types of multiple choice questions that address the content of Chapter 2

Questions about the History and Classification of Tests

2.1. Mental testing began by trying to distinguish those with higher versus lower

a. wealth

b. sociability

c. mental health (versus psychopathology)

d. mental abilities

Questions about Specific Widely-Used Tests

2.2. The Positive and Negative Affect Test (PANAS) measures

a. mood

b. mental ability to affect the correct answers

c. good versus bad attitude

d. none of the above

Questions about Data

2.3. Data about a person can be divided between inner and outer sources called:

a. self-report and personal report

b. observer report and informant report

c. personal-report data, external source data

d. institutional data and self-report data

Problem Set For Chapter 3

Reverse-Scoring a Test

To solve problems 3.1-3.6 use the following information:
Trevor and Adrianne each took an extraversion scale. Their responses, on the actual extraversion scale itself are shown in Tables 3.1 and 3.2. The two test-takers responded to each test item on a scale that went from 1 (disagree) to 7 (agree). One view of extraversion, by Costa and McCrae, includes six defining qualities: (a) warmth, (b) gregariousness, (c) assertiveness, (d) activity, (e) excitement-seeking, and (f) positive emotions. Introversion involves (a) coldness, (b) withdrawal, (c) passivity, (d) low activity level, (e) excitement-avoidance, and (f) negative emotions.
Table 3.1
Trevor’s Data on a Measure of Extraversion
Disagree / Agree
1 / 2 / 3 / 4 / 5 / 6 / 7
1 / I enjoy reading books / X
2 / I enjoy boisterous parties / X
3 / I prefer to study alone. / X
4 / I think it is fun to be the center of attention at a party / X
5 / I like playing practical jokes / X
6 / People often tell me I’m quiet / X
7 / I belong to many social organizations. / X
8 / I prefer to be alone more than I am. / X
9 / I like to read in the evenings. / X
10 / I like to go out with a group of people. / X

Problem Set For Chapter 3 (Continued)

Table 3.2
Adrianne’s Data on a Measure of Extraversion
1 / 2 / 3 / 4 / 5 / 6 / 7
1 / I enjoy reading books / X
2 / I enjoy boisterous parties / X
3 / I prefer to study alone. / X
4 / I think it is fun to be the center of attention at a party / X
5 / I like playing practical jokes / X
6 / People often tell me I’m quiet / X
7 / I belong to many social organizations. / X
8 / I prefer to be alone more than I am. / X
9 / I like to read in the evenings. / X
10 / I like to go out with a group of people. / X

3.1.In Tables 3.1 and 3.2, how many participants’ test data are there? That is, how many participantstook the test? (Hint: this is easy) _____

3.2. Judging from Tables 3.1 and 3.2, how many items were on the test? ______

3.3.Using the test datain Table 3.1, construct a raw data table that includes the item-level responses for Trevor. Please create a 10 (row) by 2 (column) chart in which each row represents an item (the items can be numbered). Column 1 should label the item numbers; column 2 should include the response.

3.4. Using the test data shownin Table 3.2 (and the rules listed in 3.3), construct a raw data table that includes the item-level responses for Adrianne.

3.5. Add a further column for Trevor’s scores that include the scored data (reversed where needed)

3.6. Add a further column for Adrianne’s scores that include the scored data (reversed where needed)

3.7. Calculate Trevor’s total score on the test.

3.8. Calculate Adrianne’s total score on the test.

Problem Set For Chapter 4

Types of multiple choice questions that address the content of Chapter 4

Questions about Types of Items

4.1. Empirically-keyed data begins with responses to ______that then are evaluated according to ______.

a. self-judgment items; how well they distinguish between groups

b. projective stimuli; open-ended responding

c. intelligence test items; open-ended responding

d. creativity tests; divergent responding

4.2. Thematic-report data is often used in:

a. self-report tests using likert scales

b. projective tests using open-ended responding

c. intelligence tests using open-ended responding

d. creativity tests

Problem Sets For Chapter 5

Basic Symbols

5.1 Write out what i means:______

5.2.Write out what X refers to: ______

5.3.Write out what x refers to: ______

5.4. Write out what σ2X refers to:______

Learn the Column-by-Column Method for Calculating Z-Scores

5.5. Calculate the mean, deviation, squared deviations, standard deviation, and z-scores using the step-by-step method shown in the text for the following data:

X
7
3
1
5
4

5.6.Six people obtained the following scores: 1, 9, 12, 3, 7, and 4. Use the step-by-step method you used above to calculate their z-scores.

Check Z-Scores

5.7. Use the step-by-step method to check that the z-scores you obtained in problem 5.5 are z-scores.

5.8. Are these five numbers z-scores? -1.41, .71, 0, -.71, 1.41

(Problems continue on the next page.)

Problem Sets For Chapter 5 (Continued)

Comparing Scales with Different Response Alternatives

People took the same test items twice; their results are in Table 5.1. The first time, they used a 7-point Likert scale, and the second time, a dichotomous scale.

Table 5.1
Scores for the Same Questions Using a 7-Point Scale and a 2-Point Scale
Participant / Test X / Test Y
7-point scalea / Yes/No scaleb
Rebecca / 7 / 2
Jonah / 6 / 2
Caitlyn / 4 / 1
Rylee / 2 / 1
Ella / 1 / 1

a7=Agree; 1=Disagree b2=Yes; 1=No

Create a table and use the column-by-column method to answer questions 5.9 to 5.14:

5.9. What is the mean of Test X?

5.10. What are the deviations of Test X?

5.11. What are the squared deviations of Test X?

5.12. What is the variance of Test X?

5.13. What is the standard deviation of Test X?

5.14. What are the z-scores of Test X?

(Problems continue on the next page.)

Problem Sets For Chapter 5 (Continued)

Create a table and use the column-by-column method to answer questions 5.15 to 5.20:

5.15. What is the mean of Test Y?

5.16. What are the deviations of Test Y?

5.17. What are the squared deviations of Test Y?

5.18. What is the variance of Test Y?

5.19. What is the standard deviation of Test Y?

5.20. What are the z-scores of Test Y?

5.21. (Write out) How do the means of tests X and Y compare?

5.22. (Write out) How do the standard deviations of tests X and Y compare?

5.23. (Write out) What similarities and differences do you notice between the z-scores on Tests X and Y?

Problem Sets For Chapter 6

Converting from Z-Scores to Other Scales

6.1 If a person has a z-score of -1.5 on an IQ test, what is their IQ score (M = 100, S = 15)?

6.2 If a person has a z-score of .5 on the SAT, what is their SAT score (M = 500, S = 100)?

6.3 If a person has a z-score of 2.0 on the psychopathic deviancy score of the MMPI, what is their MMPI scale score (M = 50; S = 10)?

Learning Pascal’s Triangle

Table 6.1
Rows 1 through 7 of Pascal’s Triangle
1
1 / 1
1 / 2 / 1
1 / 3 / 3 / 1
1 / 4 / 6 / 4 / 1
1 / 5 / 10 / 10 / 5 / 1
1 / 6 / 15 / 20 / 15 / 6 / 1

6.4. Given the rows of of Pascal’s triangle in Table 6.1, what is row 8 of Pascal’s Triangle?

6.5. Given the rows of of Pascal’s triangle in Table 6.1, what is row 9 of Pascal’s Triangle?

6.6. Find the sum of the numbers in each of the rows of Pascal’s triangle shown in Table 6.1

6.7. What happens to the sum of the rows as you move down the rows of Pascal’s Triangle—what is the pattern of the sums?

Problem Sets For Chapter 7

Calculating the Correlation Coefficient from Z-Scores

Table 7.1
Raw Data and Z-Scores for Participants on Tests T and U
T / U / ZT / ZU
1 / 3 / 3 / 1 / 1
2 / 3 / 3 / 1 / 1
3 / 3 / 3 / 1 / 1
4 / 3 / 3 / 1 / 1
5 / 3 / 3 / 1 / 1
6 / 1 / 1 / -1 / -1
7 / 1 / 1 / -1 / -1
8 / 1 / 1 / -1 / -1
9 / 1 / 1 / -1 / -1
10 / 1 / 1 / -1 / -1

7.1. Using the data in Table 7.1, what is the correlation between tests T and U?

Table 7.2
Raw Data and Z-Scores for Participants on Tests V and W
V / W / Zv / Zw
1 / 8 / 10 / 1 / -1
2 / 8 / 10 / 1 / -1
3 / 8 / 10 / 1 / -1
4 / 8 / 10 / 1 / -1
5 / 8 / 10 / 1 / -1
6 / 10 / 8 / -1 / 1
7 / 10 / 8 / -1 / 1
8 / 10 / 8 / -1 / 1
9 / 10 / 8 / -1 / 1
10 / 10 / 8 / -1 / 1

7.2. Using the data in Table 7.2, what is the correlation between tests V and W?

Problem Sets For Chapter 7 (Continued)

Table 7.3
Raw Data and Z-Scores for Participants on Tests X and Y
X / Y / Zx / Zy
1 / 2 / 5 / -.214 / .0
2 / 1 / 4 / -.286 / -.071
3 / 5 / 5 / .0 / 0
4 / 4 / 10 / -.071 / .367
5 / 5 / 15 / 0 / .714
6 / 10 / 7 / .367 / .142
7 / 15 / 0 / .714 / -.367
8 / 7 / 1 / .142 / -.286
9 / 0 / 2 / -.367 / -.214
10 / 1 / 1 / -.286 / -.286

7.3. Using the data in Table 7.3, what is the correlation between tests X and Y?

(Problems continue on the next page.)

Problem Sets For Chapter 7 (Continued)

Calculating the Correlation Coefficient from Observed Scores

Table 7.4
Scores for Four Participants on Test X
Index
Number / Score on Test X
1 / 8
2 / 0
3 / 2
4 / 2

7.4. Using the data in Table 7.4, calculate the z-scores for Test X using the column-by-column method. (There may be more columns than you need).

Table 7.5
Scores for Four Participants on Test Y
Index
Number / Score on Test Y
1 / 11
2 / 1
3 / 11
4 / 1

7.5. Using the data in Table 7.5, calculate the z-scores for Test Y using the column-by-column method. (There may be more columns than you need).

7.6. Calculate the correlation coefficient between Tests X and Y using the data from Tables 7.4 and 7.5.

(Problems continue on the next page.)

Problem Sets For Chapter 7 (Continued)

Table 7.6
Scores for Six Participants on Test L
Index Number / Scores on Test L
1 / 10
2 / 10
3 / 10
4 / 0
5 / 0
6 / 0

7.7. Using the data in Table 7.6, calculate the z-scores for Test L using the column-by-column method.

Table 7.7
Scores for Six Participants on Test M
Index Number / Scores on Text M
1 / 11
2 / 7
3 / 7
4 / 5
5 / 5
6 / 1

7.8. Using the data in Table 7.7, calculate the z-scores for Test M using the column-by-column method. (There may be more columns than you need).

7.9. Calculate the correlation coefficient between Tests L and M using the data from Tables 7.6 and 7.7. (There may be more columns than you need).

(Problems continue on the next page.)

Problem Sets For Chapter 7 (Continued)

Working With the Binomial Effect Size Display (BESD) Table

Table 7.8
A Binomial Effect Size Display for r = .00
Degree to Which Liked by Co-Workers
Lo Liking / Hi Liking / Total
Scores on a test of Conscientiousness / High
Low
Total

7.10. Fill in the values for a BESD where r = .00 in Table 7.8

Table 7.9
A Binomial Effect Size Display for r = .25
Degree to Which Liked by Co-Workers
Lo Liking / Hi Liking / Total
Scores on the MSCEIT / High
Low
Total

7.11. Fill in the values for a BESD where r = .25 in Table 7.9

Table 7.10
A Binomial Effect Size Display for r = .15
Longevity
Lifespan < 70 Years / Lifespan > 70 Years / Total
Scores on a test of Conscientiousness / High
Low
Total

7.12. Fill in the values for a BESD where r = .15 in Table 7.10

Problem Sets For Chapter 8

Learn Psychometric Symbols

8.1. Define X

8.2. Define T

8.3. Define E.

8.4. Define σT.

8.5. Define σX.

8.6. Define σ2E.

8.7. Define ρ2XT.

8.8. Define ρXX.

8.9 Define ρXX'.

Work With Psychometric Assumptions and Derivations in Equation Form

8.10. Given: X = 10, E = 2. Solve for T.

8.11. Given: T = 12, X = 10. Solve for E.

8.12. Given: T = 8, E = -3. Solve for X.

8.13. Given σ2T = 10, σ2X = 15. Solve for σ2E

8.14. Given σT = 3, σX = 4. Solve for σ2E

8.15. Given σT = 3, σ2X = 20. Solve for ρ2XT

8.16. Given: ρ2XT = .89. Solve for ρxx

Problem Set For Chapter 8 (Continued)

Work With Psychometric Assumptions and Derivations: Combined Word and Numerical Problems

To solve problems 8.17-8.23 use the following information:
Given ρ2XT = .56, σ2X = 4.

8.17. What is the variance of the true scores?

8.18. What is the variance of the error scores?

8.19. Solve for σT.

8.20. What is the standard deviation of the error scores?

8.21. What is ρ2XX'equal to?

8.22. What is σ2X' equal to?

8.23. For a given participant, X = 10 and E = 3, Solve for T

To solve problems 8.24-8.31 refer to the following information where needed:
Given: A test, X, has a reliability of .75 and an observed standard deviation of 4.0. In addition, there exists a test, X', that is parallel to X, and another test, Y, that intercorrelates both with X and X' = .65.

8.24. Solve for the true-score variance of test X.

8.25. Solve for the obtained-score variance of test X.

8.26. What is the error variance of test X?

8.27. What isthe value of ρET?

8.28. Calculate the correlation between error and obtained scores on test X.

8.29. Find the correlation between the test and its parallel form.

Problem Set For Chapter 8 (Continued)

8.30. Solve for the reliability of the test that is parallel to test X.

8.31. For a given participant, X = 12 and T = 2, Solve for E.

Problem Sets For Chapter 9

Convert Probabilities to Odds Ratios

9.1. Convert 0 to an odds ratio

9.2. Convert .20 to an odds ratio

9.3. Convert .40 to an odds ratio

9.4. Convert .60 to an odds ratio

9.5. Convert .80 to an odds ratio

9.6. Convert 1.00 to an odds ratio

Convert Odds Ratios to Log Odds Units (Logits)

9.7 Convert 0 to logits

9.8 Convert .25 to logits

9.9. Convert .66 to logits

9.10. Convert 1.33 to logits

9.11. Convert 4.0 to logits

9.12. Convert ∞ to logits

(Problems continue on the next page.)

Problem Set For Chapter 9 (Continued)

Graph a Logistic Curve

9.13. A test item has a difficulty level of β = 0. A group of people of varied abilities (θ) took the item as part of a longer test. Graph their likelihood of passing the item on the Y axis as a function of their ability level θ (theta), measured in log odds units, on the X axis.

Likelihood of passing the item / θ (theta), on a logit scale
Near zero / -3.00
.20 / -1.39
.40 / -.41
.60 / .41
.80 / 1.39
Near 1.0 / +3.00
Table 9.1
Correlations Among the Four Items of Test X
A / B / C / D
A
B / .62
C / .64 / .58
D / .54 / .54 / .46

9.14. How many factors might there be in Test X. What items make up the factor or factors?

Table 9.2
Correlations Among the Four Items of Test Y
A / B / C / D
A
B / .15
C / .64 / -.14
D / .06 / .54 / .20

9.15. How many factors might there be in Test Y. What items make up the factor or factors?

Problem Sets For Chapter 10

Review of Symbols

10.1. What does ρxx' stand for?

10.2. What does ρXT stand for?

10.3. What does ρXE stand for?

10.4. What does σT stand for?

10.5. What does σ2X’ stand for?

General Problems in Classical Test Theory and Reliability

To solve problems 10.6-10.9 refer to and use the following information where needed:
Given: ρxx' = .8, and σ2X = 25.

10.6. Find the value ofρXT.

10.7. What is ρXE?

10.8. What is σ2T?

10.9. Find the value ofσ2E?

(Problems continue on the next page.)

Problem Sets For Chapter 10 (Continued)

Mixed Word and Symbol Problems in Classical Test Theory and Reliability

To solve problems 10.10-10.15 refer to the following information where needed:
Given: ρ2XT = .60, σ2X = 9

10.10. Solve for the variance of the true scores.

10.11. Solve for the variance of the error scores.

10.12. What is σT?

10.13. What is the standard deviation of the error scores?

10.14. Solve for ρXX'.

10.15. Solve for σ2X'.

10.16. When does X = T?

10.17. When does X = E?

To solve problems 10.18-10.22 refer to the following information where needed:
Given: ρ2XT = .49, σ2X = 7

10.18. What is the variance of the true scores?

10.19. Solve for the variance of the error scores.

10.20. Solve for the parallel test’s reliability.

10.21. What is the standard deviation of the observed scores?

10.22. What is the expected value of the error scores?

Problem Sets For Chapter 10 (Continued)

Word Problems: Basic Computation of Test Reliability

10.23. You split a test in half and observe a correlation of r = .8 between the halves. What is the test reliability of the first half of the test?

10.24. You split a test in half and observe a correlation of r = .65. What is the test reliability?

10.25. You calculate a split-half reliability estimate and observe a correlation of .3 between the two halves. What is the estimated reliability of the test?

10.26. What is the estimated reliability of a test if the variance of the scores on the first half of the test is 20, the variance of scores on the second half of the test is 25, and the variance of total test scores is 60?

10.27. What is the estimated reliability of a test if the variance of scores on the first half of the test is 20, the variance of scores on the second half of the test is 30, and the variance of total test scores is 70?

10.28. You calculate a split-half reliability estimate and observe a correlation of .4 between the two halves. What is the estimated reliability of the test?

10.29. What is the estimated reliability of a test if the variance of the scores on the first half of the test is 10, the variance of scores on the second half of the test is 25, and the variance of total test scores is 60?

To solve problems 10.30-10.32 refer to the following information where needed:
Given: A test is split in half. The scores on the first half of the test have a variance of 10, and the scores on the second half of the test have a variance of 15. The correlation between the scores on the two halves is .50. The variance of the total test score is 37.2.

10.30. Calculate the reliability of the entire test using coefficient alpha.

10.31. Calculate the reliability using the Spearman-Brown formula.

10.32. How do the coefficient alpha and Spearman-Brown estimates of reliability compare?

Problem Sets For Chapter 10 (Continued)

To solve problems 10.33-10.34 refer to and the following information where needed:
Given: A test is split in half. The scores on the first half of the test have a variance of 10, and scores on the second half of the test have a variance of 25. The correlation between the scores on the two halves is .57. The variance of the total test score is 50.

10.33. Calculate the reliability of the entire test using coefficient alpha.

10.34. Calculate the reliability using the Spearman-Brown formula.

Further Problems in the Computation of Test Reliability

To solve problems 10.35-10.36 refer to the following information where needed:
Given: A test is divided into three parts. The parts correlate with one another at an average of r = .4. The variances of the three parts are 10, 10, and 10, and the variance of the total test is 60.

10.35. What is the Spearman-Brown estimate of the test’s reliability?