Research Methods and Statistics A Critical Thinking Approach, 4th International Edition by Sherri L. Jackson – Test Bank

Research Methods and Statistics A Critical Thinking Approach, 4th International Edition by Sherri L. Jackson – Test Bank

Chapter 10

Inferential Statistics: Two-Group Designs

Chapter Outline

Parametric Statistics
t-Test for Independent Groups (Samples): What It Is and What It Does
Calculations for the Independent-groups t-Test
Interpreting the t-Test
Graphing the Means
Effect Size – Cohen’s d and r²
Assumptions of the Independent-groups t-Test
t-Test for Correlated Groups: What It Is and What It Does
Calculations for the Correlated-groups t-Test
Interpreting the Correlated-groups t-Test and Graphing the Means
Effect Size: Choen’s d and r²

Assumptions of the Correlated-groups t-Test
Nonparametric Statistics

Wilcoxon Rank-Sum Test: What It Is and What It Does
Calculations for the Wilcoxon Rank-Sum Test
Interpreting the Wilcoxon Rank-Sum Test
Assumptions of the Wilcoxon Rank-Sum Test

Wilcoxon Matched-Pairs Signed-Ranks T Test: What It Is and What It Does
Calculations for the Wilcoxon Matched-Pairs Signed-Ranks T Test
Interpreting the Wilcoxon Matched-Pairs Signed-Ranks T Test
Assumptions of the Wilcoxon Matched-Pairs Signed-Ranks T Test

Chi-Square (c²) Test of Independence: What It Is and What It Does
Calculations for the c² Test of Independence
Interpreting the c² Test of Independence
Effect Size: Phi Coefficient
Assumptions of the c² Test of Independence

Summary

Review of Key Terms

Chi-square test for independence—A nonparametric inferential test used when frequency data have been collected to determine how well an observed breakdown of people over various categories fits some expected breakdown.

Cohen’s d—An inferential statistic for measuring effect size with a t test.

Correlated-groups t test—A parametric inferential test used to compare the means of two related (within- or matched-participants) samples.

Difference scores—Scores representing the difference between participants’ performance in one condition and their performance in a second condition.

Effect size—The proportion of variance in the dependent variable that is accounted for by the manipulation of the independent variable.

Independent-groups t test—A parametric inferential test for comparing sample means of two independent groups of scores.

Phi Coefficient—An inferential test used to determine effect size for a chi-square test.

Standard Error of the Difference Between Means—The standard deviation of the sampling distribution of differences between the means of independent samples in a two-sample experiment.

Standard Error of the Difference Scores—The standard deviation of the sampling distribution of mean differences between dependent samples in a two-group experiment.

Wilcoxon matched-pairs signed-ranks T test—A nonparametric inferential test for comparing sample medians of two dependent or related groups of scores.

Wilcoxon’s Rank-sum Test—A nonparametric inferential test for comparing sample medians of two independent groups of scores.

Relevant Articles from Handbook forTeaching Statistics and Research Methods (1st ed.)

Dillbeck, M. C. Teaching statistics in terms of the knower. Pp. 20-23.

Dillon, K. M. Statisticophobia. P. 3.

Forsyth, G. A. A task-first individual-differences approach to designing a statistics and methodology course. Pp. 15-17.

Hastings, M. W. Statistics: Challenge for students and the professor. Pp. 6-7.

Magnello, M. E., & Spies, C. J. Using organizing concepts to facilitate the teaching of statistics. Pp. 12-15.

Ward, E. F. Statistics mastery: A novel approach. Pp. 17-20.

Web Resources

For step-by-step practice and information, have your students check out the Statistics and Research Methods Workshops at www.cengage.com/psychology/workshops. In addition, practice quizzes, vocabulary flashcards, and more are available at www.cengage.com/psychology/jackson.

Answers to Chapter Exercises

1. a. An independent samples t-test should be used.
2. H₀: m_Females = m_Males

H_a: m_Females ¹ m_Males

1. t (12) = -.78, not significant.
2. Fail to reject H₀. There are no significant differences in the amount of study time per week for females versus males.
3. Not necessary
4. Not necessary
5. The 95% confidence interval is –8.64 – 4.07

2. a. An independent samples t-test should be used.
3. H₀: m_Music ³ m_{No music}

H_a: m_Music < m_{No music}

2. t (16) = 2.60, p < .01.
3. Reject H₀. Studying with no music lead to significantly higher test scores.
4. d = 1.22—there is a large effect size.

f.

2. The 95% confidence interval is ^–2.83 – ^–-28.

3. a. A correlated groups t-test should be used.
4. H₀: m_Before ³ m_After

H_a: m_Before < m_After

6. t (5) = 6.71 (or 6.82 if calculated by hand and each step is rounded to two decimal places) , p < .005.
7. Reject H₀. Participating in sports lead to significantly higher self-esteem scores.
8. d = 2.73. This is a large effect size.

2. The 95% confidence interval is ^–-2.07 – ^–-.93.

4. a. A correlated groups t-test should be used.
5. H₀: m_Music ³m_{No music}

H_a: m_Music < m_{No music}

2. t (5) = 2.78, p < .025.
3. Reject H₀. Participants had significantly higher scores on the test when they studied without music.
4. d = 1.12. This is a large effect size.

f.

1. The 95% confidence interval is ^–-1.93 – ^–-.07.

5. a. The Wilcoxon’s rank-sum test should be used.
6. H₀: Md_{No service} ≥ Md_Service

H_a: Md_{No service} < Md_Service

23. W (n₁=6, n₂=6) = 23.5, p < .01.
24. Yes, reject H₀. Students who completed community service had significantly higher maturity scores.

6. a. The Wilcoxon’s rank-sum test should be used.
7. H₀: Md_Red ≤ Md_Green

H_a: Md_Red > Md_Green

1. W (n₁=7, n₂=7) = 43, not significant.
2. Fail to reject H₀. Tastiness scores for the two different colored sauces did not differ significantly.

7. a. The Wilcoxon matched-pairs signed-ranks T test should be used.
   b. H₀: Md_Red ≤ Md_Green

H_a: Md_Red > Md_Green
c. T(N=7) = 3, p < .05.
d. Reject H₀. Taste scores for the two sauces differed significantly. The red was preferred.

e

8. a. The chi-square test for independence should be used.
9. H₀: There is no difference in the frequency of seating preferences for males versus females.

H_a: There is a difference in the frequency of seating preferences for males versus females.

6. c² (N=93) = 6.80, p < .01.
7. Yes, reject H₀. There is a significant difference in seating preferences for males versus females. More males sit in the back and more females sit in the front.

9. a. A t-test for independent groups should be used.
10. A Wilcoxon’s rank-sum test should be used.
11. A chi-square test for independence should be used.
12. A t-test for independent groups should be used.

Test Items

Multiple Choice Questions

1. An independent-groups t-test:
   a. compares sample means for two related groups.
   b. compares sample means for two unrelated groups.
   c. compares standard deviations for two unrelated group.
   d. compares sample means for three or more unrelated groups.

Answer: b

2. As sample variance _____, the value of the t-test _____.
   a. increases; increases
   b. decreases; decreases
   c. increases; decreases
   d. increases; stays the same

Answer: c

3. If, after calculating an independent-groups t-test you find that it is equal to zero, then:
   a. the two sample means differ significantly.
   b. the two sample means do not differ significantly.
   c. you have done something wrong.
   d. the two population means differ significantly.

Answer: b

4. Which of the following t-test results has the greatest chance of statistical significance?
   a. t(28) = 1.70
   b. t(14) = 1.70
   c. t(18) = 1.70
   d. t(10) = 1.70

Answer: a

^w5. If the null hypothesis is true, then the t-test should be close to:
a. 0.00.
b. ±1.65.
c. ±1.96.
d. ±3.00.

Answer: a

6. Imagine that you conducted an independent-groups t-test with 12 participants in each group. For a two-tailed test, the t_cv at a = .05 would be:
   a. ±1.717.
   b. ±2.074.
   c. ±1.711.
   d. ±2.064.

Answer: b

7. If a researcher reported for an independent-groups t-test that t(18) = 2.90, p< .01, how many participants were there in the study?
   a. 18
   b. 20
   c. 10
   d. 9

Answer: b

8. The null hypothesis for an independent-groups t-test states that:
   a. H₀: m₁ ¹ m_2.
   b. H₀: m₁ = m_2.
   c. H_a: m₁ ¹ m_2.
   d. H_a: m₁ = m_2.

Answer: b

9. The alternative hypothesis for an independent-groups t-test states that:
   a. H₀: m₁ ¹ m_2.
   b. H₀: m₁ = m_2.
   c. H_a: m₁ ¹ m_2.
   d. H_a: m₁ = m_2.

Answer: c

10. If an independent-groups t-test has 7 participants in each group, then the degrees of freedom are:
    a. 7.
    b. 14.
    c. 6.
    d. 12.

Answer: d

11. When using an independent-groups t-test, the difference between the means is divided by:
    a. the standard error of the mean.
    b. the standard error of the standard deviation.
    c. the standard error of the difference between means.
    d. the standard error of the difference between standard deviations.

Answer: c

12. Which of the following are aspects of a study that can increase power?
    a. greater differences produced by the independent variable
    b. smaller variability of raw scores in each condition
    c. increased sample size
    d. greater differences produced by the independent variable, smaller variability or raw scores in
    each condition, and increase sample size

Answer: d

13. For an independent-groups t-test, effect size can be measured using:
    a. Cohen’s d.
    b. the standard error of the difference between means.
    c. the phi coefficient.
    d. the c² test for independence.

Answer: a

14. Which of the following is not an assumption of the t-test for independent groups?
    a. The data are interval or ratio.
    b. The underlying distributions are bell-shaped.
    c. The observations are independent.
    d. There is not homogeneity of variance.

Answer: d

15 t_cv = 3.35 and t_obt = -3.55. Based on these results we _____.
a. reject H_o.
b. fail to reject H_o.
c. accept H_o.
d. reject H_a.

Answer: a

^w16. For an independent-groups t-test, the value of Cohen’s d describes:
a. the size of the treatment effect.
b. the chance of a Type I error.
c. the chance of a Type II error.
d. the overall chance level in the experiment.

Answer: a

17. For a correlated-groups t-test, the null hypothesis states that:
    a. H₀: m₁-m₂ = 0.
    b. H₀: m₁-m₂ > 0.
    c. H_a: m₁-m₂ = 0.
    d. H_a: m₁-m₂ > 0.

Answer: a

18. For a correlated-groups t-test, the alternative hypothesis states that:
    a. H₀: m₁-m₂ = 0.
    b. H₀: m₁-m₂ > 0.
    c. H_a: m₁-m₂ = 0.
    d. H_a: m₁-m₂ > 0.

Answer: d

19. If a correlated-groups t-test and an independent-groups t-test both have df=16, which experiment used more participants?
    a. they both used the same number of participants (n=18)
    b. they both used the same number of participants (n=17)
    c. the correlated-groups t-test
    d. the independent-groups t-test

Answer: d

^w20. In a research study comparing two conditions, researchers obtain 24 scores from each condition. If this was a within-participants design, then there were _____ participants in the study.
a. 24
b. 25
c. 26
d. 48

Answer: a

21. If researchers reported that, for a correlated-groups design, t(20) = 3.57, p<.05, you can conclude that:
    a. a total of 21 people participated in the study.
    b. a total of 22 people participated in the study.
    c. a total of 40 people participated in the study.
    d. there is no way to determine how many people participated in the study.

Answer: a

22. If researchers reported that, for a correlated-groups design, t(20) = 3.57, p<.05, you can conclude that:
    a. the null hypothesis should be rejected.
    b. that you should fail to reject the null hypothesis.
    c. the alternative hypothesis was incorrect.
    d. the study needs to be repeated.

Answer: a

23. When using a correlated-groups t-test, df =:
    a. n₁ + n₂ – 2.
    b. n – 1.
    c. n₁ + n₂ – 1.
    d. n – 2.

Answer: b

24. In a correlated-groups design, if n = 20, then df =:
    a. 20.
    b. 19.
    c. 21.
    d. 40.

Answer: b

25. Imagine that you conducted a correlated-groups t-test with 12 participants. For a two-tailed test, the t_cv at a = .05 would be:
    a. ±1.796.
    b. ±2.201.
    c. ±1.782.
    d. ±2.179.

Answer: b

26. One advantage of the correlated-groups t-test over the independent-groups t-test is that is reduces the error variance due to:
    a. the degrees of freedom.
    b. the independent variable.
    c. the dependent variable.
    d. individual differences.

Answer: d

27. Wilcoxon rank-sum test is to _____ as Wilcoxon matched-pairs signed-ranks T test is to _____.
    a. ordinal data; nominal data
    b. interval data, ordinal data
    c. between-participants design; within-participants design
    d. within-participants design; between-participants design

Answer: c

^w28. Chi-square tests are to _____ data as Wilcoxon tests are to _____ data.
a. ordinal; interval
b. nominal; ordinal
c. ordinal; nominal
d. ratio, ordinal

Answer: b

29. The Wilcoxon tests are used with _____ data.
    a. ordinal
    b. interval
    c. ratio
    d. nominal

Answer: a

30. When using the Wilcoxon tests, the obtained value must be ______ the critical value.
    a. less than or equal to
    b. greater than or equal to
    c. greater than
    d. equal to

Answer: a

31. Which of the following is not an assumption of the Wilcoxon rank-sum test?
    a. The data are ratio, interval or ordinal in scale, all of which must be converted to ranked
    (ordinal) data before conducting the test.
    b. The underlying distribution is not normal.
    c. The observations are independent.
    d. There is homogeneity of variance.

Answer: d

^w32. Which of the following is not an assumption of the Wilcoxon matched-pairs signed-ranks T test?
a. The data are ratio, interval or ordinal in scale, all of which must be converted to ranked
(ordinal) data before conducting the test.
b. The underlying distribution is not normal.
c. The observations are independent.
d. The other alternatives are all assumptions of the Wilcoxon matched-pairs signed-ranks test.

Answer: c

33. The _____ is a nonparametric inferential test for comparing sample medians of two dependent or related groups of scores.
    a. Wilcoxon matched-pairs signed-ranks T test.
    b. Wilcoxon rank-sum test.
    c. c² test for independence
    d. c² goodness-of-fit test

Answer: a

34. The _____ is a nonparametric inferential test for comparing sample medians of two independent groups of scores.
    a. Wilcoxon matched-pairs signed-ranks T test.
    b. Wilcoxon rank-sum test.
    c. c² test for independence
    d. c² goodness-of-fit test

Answer: b

35. Chi-square is calculated by summing the[(_____ – _____)² divided by _____].
    a. observed; expected; observed
    b. expected; observed; observed
    c. observed; expected; expected
    d. observed; observed; expected

Answer: c

36. Nonparametric is to parametric as _____ is to _____.
    a. t-test; z-test
    b. z-test; t-test
    c. c²-test; t-test
    d. t-test; c²-test

Answer: c

37. Which of the following is not an assumption of the c²-test?
    a. It is a nonparametric test.
    b. It is appropriate only for ordinal data.
    c. The frequency in each cell should be at least 5.
    d. The sample should be randomly selected.

Answer: b

38. Parametric is to nonparametric as _____ is to _____.
    a. c² test for independence; independent-groups t test
    b. correlated-groups t test; independent-groups t test
    c independent-groups t test; c² test for independence
    d. none of the other alternatives is correct

Answer: c

39. Which of the following is true regarding the c² test of independence?
    a. It requires a numerical (quantitative) score for each individual.
    b. The sample must be random.
    c. Population parameters such as m and s must be known.
    d. The observations must not be independent.

Answer: b

^w40. The calculation of the df for the c² test of independence is:
a. r x c – 1.
b. (r-1)(c-1).
c. (r-c)n.
d. r x c.

Answer: b

41. The c² test of independence is used with _____ data.
    a. continuous
    b. categorical
    c. ordinal
    d. quantitative

Answer: b

Short Answer/Essay Questions

1. Imagine I conducted the following experiment: I used gender as a nonmanipulated independent variable and measured performance in my class. I measured performance by ranking everyone in the class, giving the person with the highest grade a 1, the person with the 2^nd highest grade a 2, etc. Which statistic would I use to determine any differences in performance between these two groups?

   Because the data are ordinal (ranked) you would use a Wilcoxon test. Furthermore, because the nonmanipulated independent variable is between-participants (gender), the exact test would be the Wilcoxon Rank-Sum Test.

2. According to some research, males have better spatial skills than do females; and according to other research, females have better reading skills than males. A student is interested in determining which gender performs better on a word-search puzzle (a puzzle in which words are hidden vertically, horizontally, and diagonally within an array of letters) since this type of puzzle involves both spatial and reading skills. A sample of males and females volunteer to participate and are given 10 minutes to work on a 50-word puzzle. The number of words correctly recognized is recorded for each subject, and the resulting data are as follows:

Males Females

12 15

8 12

9 11

11 18

10 13

12 14

7 17

Conduct the appropriate analysis of these data and determine whether there are any significant differences.

An independent samples t test should be performed on these data, with t (12) = -3.64, p<.01. Yes, there are significant differences between the groups. Females performed significantly better on the task than males.

3. A college student is interested in whether there is a difference between male and female students in the amount of time spent working out each week. The student gathers information from a random sample of male and female students on her campus. Amount of time spent working out is normally distributed. The data appear below.

Males Females

7 5

5 9

9 8

10 3

6 10

2 5

4 9

1. a) What statistical test should be used to analyze these data?
   The independent samples t test.

2. b) Identify H₀ and H_a for this study.
   H₀: m₁ = m₂
   H_a: m₁¹m2

3. c) Conduct the appropriate analysis.
   t (12) = -.58.

4. d) Should H₀ be rejected? What should the researcher conclude?
   No, H₀ should not be rejected.

5. e) If significant, compute the effect size and interpret this.
   Not necessary.

6. f) If significant, draw a graph representing the data.
   Not necessary.

4. A student is interested in whether students who study with others devote at much attention to their studies as do students who study alone. He believes those who study alone devote more attention to their studies. He randomly assigns participants to either group or individual study conditions and has them read and study the same passage of information for the same amount of time. Participants are then all given the same 10-item test on the material. Their scores appear below. Scores on the test represent interval/ratio data and are normally distributed.

Group Alone

6 10

5 9

6 7

5 7

6 6

6 6

7 8

8 6

5 9

1. a) What statistical test should be used to analyze these data?
   The independent-groups t test.

2. b) Identify H₀ and H_a for this study.
   H₀: m₁<m₂
   H_a: m₁ > m2

3. c) Conduct the appropriate analysis.
   t (16) = -2.60, p<.01.

4. d) Should H₀ be rejected? What should the researcher conclude?
   Yes, H₀ should be rejected. The researcher should conclude that when students studied alone,
   they did significantly better on the test.

5. e) If significant, compute the effect size and interpret this.
   Cohen’s d is 1.22—a large effect size. We could also use r² which is .30, indicating
   a large effect size.

6. If significant, draw a graph representing the data.
7. Calculate the 95% CI.
   The 95% CI is -2.832 to -.288

5. A researcher believes exercise reduces anxiety in women. She identifies a group of women who had not exercised before, but are now planning to begin exercising. She gives them a 50-item anxiety inventory before they begin exercising and administers it again after 6 months of exercising. The anxiety inventory is measured on an interval scale and higher numbers indicate higher anxiety. In addition, scores on the inventory are normally distributed. The scores appear below.

Before After

46 44

41 40

42 39

47 46

43 42

45 43

1. a) What statistical test should be used to analyze these data?
   A correlated-groups t test should be used.

2. b) Identify H₀ and H_a for this study.
   H₀: m₁>m₂
   H_a: m₁ < m2

5. c) Conduct the appropriate analysis.
   t (5) = 5.0, p = .002.

6. d) Should H₀ be rejected? What should the researcher conclude?
   Yes, H₀  should be rejected. The researcher should conclude that exercise reduces stress

7. e) If significant, compute the effect size and interpret this.
   Cohen’s d is 2.03—a very large effect size. We could also use r² which is .83, also indicating
   a very large effect size.

   f) If significant, draw a graph representing the data.

8. g) Calculate the 95% CI.
   The 95% CI is -.82 to -.249

6. A researcher is interested in comparing the self-esteem of students who volunteer for community service versus those who do not. The researcher assumes that those who complete community service will have higher self-esteem scores. Self-esteem scores tend to be skewed (not normally distributed). The self-esteem scores appear below. Higher scores indicate higher self-esteem levels.

No Community Service Community Service

33 41

41 48

54 61

13 72

22 83

26 55

1. a) What statistical test should be used to analyze these data?
   The Wilcoxon rank-sum test should be used.

2. b) Identify H₀ and H_a for this study.
   H₀: Md₁ < Md₂
   H_a: Md₁ > Md2

3. c) Conduct the appropriate analysis.
   W_s (n₁ =6, n₂ = 6) = 23.5, p , .01).

4. d) Should H₀ be rejected? What should the researcher conclude?
   Yes, H₀ should be rejected. Those who do not complete community service have significantly lower self-esteem scores.

7. You notice that in your introductory psychology class that it appears that more men tend to sit near the door and more women opposite the door. In order to determine whether this difference is significant, you collect data on the seating preferences for the students in your class. The data appear below.

Males Females

Near the Door 25 14

Opposite the Door 12 20

1. a) What statistical test should be used to analyze these data?
   The chi-square test of independence should be used.

2. b) Identify H₀ and H_a for this study.
   H₀: There are no differences in seating preference.
   H_a: There are differences in seating preference.

3. c) Conduct the appropriate analysis.
   c²(1, N = 71) = 4.99, p < .05.

4. d) Should H₀ be rejected? What should the researcher conclude?
   Yes, H₀ should be rejected. The researcher should conclude that significantly more males sit
   near the door and significantly more females sit opposite the door.

8. What are the assumptions of the Wilcoxon rank-sum test?
   The data are ratio, interval, or ordinal in scale and are converted to an ordinal (ranking) scale before conducting the test, the underlying distribution is not normal, and the observations are independent.