Table of Contents - Momentum Web view6.3 If a person has a z-score of 2.0 on the psychopathic deviancy score of the MMPI, what is their MMPI scale score ... Combined Word and Numerical

Problem Sets for Elements of Mental Tests 1

Mathematical and Multiple Choice

Problem Setsfor

The Elements of Mental TestsSECOND EDITION

John D. Mayer

January 25th, 2016 Version


Table of ContentsProblem Set for Chapter 1________________________________________________________3

Problem Set For Chapter 2________________________________________________________4



Problem Sets For Chapter 5_______________________________________________________8

Problem Sets For Chapter 6______________________________________________________11




Problem Sets For Chapter 10_____________________________________________________22



Appendix: Hints, Intermediate Values and Answers for Selected Problems_________________30


PROBLEM SET FOR CHAPTER 1Types of multiple choice questions that address the content of Chapter 1

Questions about the History of Testing 1.1. Journalists have criticized psychological tests beginninga. 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 Is1.2. Which feature is NOT a defining aspect of a mental test:a. an integrated mental procedureb. collects data about mental lifec. uses ratio scales d. indicates the presence of magnitude of features

Questions about the Advantages and Drawbacks of Testing1.3. Psychological tests have this relationship with informal interviewing (say, a job interview):a. they have not been comparedb. psychological tests are more reliable and less biasedc. psychological tests are more biased and less validd. psychological tests lose or miss more information than does the interviewer



Questions about the History and Classification of Tests2.1. Mental testing began by trying to distinguish those with higher versus lowera. wealthb. sociabilityc. mental health (versus psychopathology)d. mental abilities

Questions about Specific Widely-Used Tests2.2. The Positive and Negative Affect Test (PANAS) measuresa. moodb. mental ability to affect the correct answersc. good versus bad attituded. 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 reportb. observer report and informant reportc. personal-report data, external source datad. institutional data and self-report data


PROBLEM SET FOR CHAPTER 3Reverse-Scoring a TestTo 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 X2 I enjoy boisterous parties X3 I prefer to study alone. X4 I think it is fun to be the center of attention at a

partyX

5 I like playing practical jokes X6 People often tell me I’m quiet X7 I belong to many social organizations. X8 I prefer to be alone more than I am. X9 I like to read in the evenings. X10 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 71 I enjoy reading books X 2 I enjoy boisterous parties X3 I prefer to study alone. X4 I think it is fun to be the center of attention at a

party X

5 I like playing practical jokes X6 People often tell me I’m quiet X 7 I belong to many social organizations. X8 I prefer to be alone more than I am. X9 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 participants took 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 data in 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 shown in 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.



Questions about Types of Items4.1. Empirically-keyed data begins with responses to _______ that then are evaluated according to ________.a. self-judgment items; how well they distinguish between groupsb. projective stimuli; open-ended respondingc. intelligence test items; open-ended respondingd. creativity tests; divergent responding

4.2. Thematic-report data is often used in: a. self-report tests using likert scalesb. projective tests using open-ended respondingc. intelligence tests using open-ended respondingd. creativity tests


PROBLEM SETS FOR CHAPTER 5Basic 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-Scores5.5. 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:

X73154

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-Scores5.7. 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 AlternativesPeople 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 ScaleParticipant Test X Test Y

7-point scalea Yes/No scaleb

Rebecca 7 2Jonah 6 2Caitlyn 4 1Rylee 2 1Ella 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?




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 6Converting 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 TriangleTable 6.1Rows 1 through 7 of Pascal’s Triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 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 7Calculating the Correlation Coefficient from Z-ScoresTable 7.1Raw Data and Z-Scores for Participants on Tests T and U

T U ZT ZU

1 3 3 1 12 3 3 1 13 3 3 1 14 3 3 1 15 3 3 1 16 1 1 -1 -17 1 1 -1 -18 1 1 -1 -19 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 7.2Raw Data and Z-Scores for Participants on Tests V and W

V W Zv Zw1 8 10 1 -12 8 10 1 -13 8 10 1 -14 8 10 1 -15 8 10 1 -16 10 8 -1 17 10 8 -1 18 10 8 -1 19 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?



Table 7.3Raw Data and Z-Scores for Participants on Tests X and Y

X Y Zx Zy1 2 5 -.214 .02 1 4 -.286 -.0713 5 5 .0 04 4 10 -.071 .3675 5 15 0 .7146 10 7 .367 .1427 15 0 .714 -.3678 7 1 .142 -.2869 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?




Calculating the Correlation Coefficient from Observed ScoresTable 7.4 Scores for Four Participants on Test XIndexNumber

Score on Test X

1 82 03 24 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 7.5Scores for Four Participants on Test YIndexNumber

Score on Test Y

1 112 13 114 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.




Table 7.6 Scores for Six Participants on Test L Index Number

Scores on Test L

1 102 103 104 05 06 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 MIndex Number

Scores on Text M

1 112 73 74 55 56 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).




Working With the Binomial Effect Size Display (BESD) TableTable 7.8 A Binomial Effect Size Display for r = .00

Degree to Which Liked by Co-Workers

Lo Liking Hi Liking TotalScores on a test of Conscientiousness

HighLowTotal

7.10. Fill in the values for a BESD where r = .00 in Table 7.8Table 7.9 A Binomial Effect Size Display for r = .25

Degree to Which Liked by Co-Workers

Lo Liking Hi Liking TotalScores on the

MSCEIT HighLowTotal

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

Table 7.10A Binomial Effect Size Display for r = .15

LongevityLifespan < 70

YearsLifespan > 70

YearsTotal

Scores on a test of Conscientiousness

HighLowTotal

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


PROBLEM SETS FOR CHAPTER 8Learn 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, σ2

X = 15. Solve for σ2E

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

8.15. Given σT = 3, σ2X = 20. Solve for ρ2

XT

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



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, σ2

X = 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 is the 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.



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 9Convert Probabilities to Odds Ratios

9.1. Convert 0 to an odds ratio

9.2. Convert .20 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




Graph a Logistic Curve9.13. 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 9.1Correlations Among the Four Items of Test X

A B C DAB .62C .64 .58D .54 .54 .46

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

Table 9.2Correlations Among the Four Items of Test Y

A B C DAB .15C .64 -.14D .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 10Review 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 of ρXT.

10.7. What is ρXE?

10.8. What is σ2T?

10.9. Find the value of σ2E?




Mixed Word and Symbol Problems in Classical Test Theory and ReliabilityTo solve problems 10.10-10.15 refer to the following information where needed:

Given: ρ2XT = .60, σ2

X = 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: ρ2

XT = .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?



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?



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 ReliabilityTo 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?

10.36. What is the Coefficient Alpha estimate of the test’s reliability?

10.37. You have a 100-item test with an estimated reliability of r = .9. What would be the reliability of the test if you cut it in half? (Please note, students often find this problem more difficult than the others).

(Problems continues on the next page.)



To solve problems 10.38-10.41 refer to and use the following information where needed:Given: Five people took a test. Table 10.1 displays information about their performance on the odd-numbered items, half-test 1 (H1); Table 10.2 displays information about their performance on the even-numbered items, half-test 2 (H2), and Table 10.3 indicates the calculations that led to the correlation between them. Table 10.1Scores for a 5-person Sample on the Odd Items of a Test (Half-test 1 or H1)

i H1 M x x2 S Z1 1 2 -1 1 1.67 -.602 4 2 2 4 1.67 1.203 0 2 -2 4 1.67 -1.204 1 2 -1 1 1.67 -.605 4 2 2 4 1.67 1.20

14/5=2.8

Table 10.2Scores for a 5-person Sample on the Even Items of a Test (Half-test 2 or H2)

i H2 M x x2 S Z1 0 1 -1 1 1.1 -.912 1 1 0 0 1.1 03 1 1 0 0 1.1 04 0 1 -1 1 1.1 -.915 3 1 2 4 1.1 1.82

6/5=1.2

Table 10.3Calculation of the Correlation Between H1 and H2

I Z1 Z2 Z1Z21 -.60 -.91 .542 1.20 0 03 -1.20 0 04 -.60 -.91 .545 1.20 1.82 2.18

3.26/5 = .652 = r

(Problem continues on the next page.)



10.38. What is the correlation between the two tests?

10.39. What is the variance of the total test?

10.40. What is the coefficient alpha reliability of the test?

10.41. What is the split half reliability of the test?


PROBLEM SETS FOR CHAPTER 11Types of multiple choice questions that address the content of Chapter 11

Questions about Validity in General 11.1. Which statement is most true?a. The name of a test is a good index of what it measures.b. A single test can NOT have high validity if it also has high reliability. c. validity concerns how well a test measures what it is supposed to measure.d. test validity is best reported in general terms.

11.2. The current validity standards define validity as centered on the interpretation and understanding ofa. tests and test manuals.b. test manuals only.c. test scores. d. administrators of tests and the way they work.

11.3. The standards for educational and psychological testing are actually developed by a joint committee of the…..? (short answer)

Questions about Specific Types of Validity11.4. A test-maker of an arithmetic test claims the test measures whether students have learned to “carry” numbers when they add such figures as 14 + 19 or 23 + 28 (where, in the ones columns, the sum exceeds 10). To establish process evidence for the measure’s validity, the test-maker might a. correlate the test with a pre-existing test known to measure the ability to carry numbers.b. record students who have been instructed to speak aloud as they answer the questions to see if carrying numbers comes up as they are solving problems.c. correlate the test with tests in which carrying is not necessary.d. factor analyze the test.


PROBLEM SETS FOR CHAPTER 12Types of multiple choice questions that address the content of Chapter 12

Questions about How Tests Cause Discomfort12.1. What is not a source of discomfort about tests described in Chapter 12?a. They are used for selection and therefore influence people’s lives.b. They can provide painful feedback.c. They are less valid that other people’s observations.d. They are sometimes misused.

Questions about Improving the Test Experience12.2. Several recommendations in Chapter 12 to improve testing experience include attempts toa. raise test literacy.b. discourage informed skepticism about tests.c. tell people they should usually score well on tests.d. improve test reliability.

Questions about Collaborative Testing12.3. The collaborative testing movement involves collaboration betweena. test makers and test administrators.b. professionals who provide test feedback and test takers.c. test makers and test takers.d. the government and test makers.


APPENDIX: HINTS, INTERMEDIATE VALUES AND ANSWERS FOR SELECTED

PROBLEMSChapter 11.1. B1.2. C1.3. B

Chapter 22.1 D2.2. A2.3. C

Chapter 33.3.Item Trevor’s

Response1 62 23 64 15 16 67 48 69 6

10 1

Chapter 44.1. A4.2. B


Chapter 55.1. index number5.3. a deviation on Test X5.10.

X M x7 4 36 4 24 4 02 4 -21 4 -3

Chapter 66.1. 100-22.5 = 78.56.6. (partial answer): 1, 2, 4, 8,…

Chapter 77.4. Partial check: Note that standard deviation for Test X is 3.7.5. Partial check: Note that the variance for Test Y is 25.7.7. Partial check: Note that the standard deviation for Test L is 5.7.8. Partial check: Note that the variance for Test M is 9.

Chapter 8Hint for Chapter 8 Problem Sets: Matters will go more easily—especially for the later problems—if you first create a reference sheet on which you copy down the following from Chapter 8:

The key assumptions of psychometric theory The definition of a parallel test The remaining equations of the chapter

8.7. The squared population correlation between obtained and true scores on a test8.13. 5


Chapter 9

Convert Probabilities to Odds Ratios9.1. 0:19.3. 2:39.5. 4:1

Convert Odds Ratios to Log Odds Units (Logits)9.7. -∞9.9. -.4159.11. 1.39

Chapter 10

Review of Symbols10.1. The reliability of a test, and the correlation between a test and its parallel form10.4. The standard deviation of the true score10.5. The variance of the obtained scores on a parallel test

Mixed Word and Symbol Problems in Classical Test Theory and Reliability 10.10. Answer: 5.410.11. Answer: 3.610.12. Answer: 2.3210.13. Answer: 1.9010.14. Answer: .6010.15. Answer: 910.18. Answer: 3.4310.19. Answer: 3.5710.20. Answer: .4910.21. Answer: 2.6510.22. Answer: 0

Word Problems: Basic Computation of Test Reliability10.28. Answer: .5710.29. Answer: .8310.30 Answer: 66 10.31. Answer: .6710.33. Answer: .60


10.34. Answer: .7310.35. Answer: .67 Calculation: [3(.4)] / [1+(3-1).4] = 1.2 / 1.8 = .6710.36. Answer: .75 Calculation: [3/(3-1)] * [(60-(10+10+10))/60] = 3/2 * [30/60] = 3/2* .5 = .75

Chapter 11Multiple Choice/Short Answer11.1. C11.2. C11.3. AERA, APA, NCRE11.4. B

Chapter 1212.1. C12.2. B12.3. B

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Table of Contents - Momentum Web view6.3 If a person has a z-score of 2.0 on the psychopathic deviancy score of the MMPI, what is their MMPI scale score ... Combined Word and Numerical