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Statistics for psychology : Pearson new international edition

Aron, Arthur; Aron, Elaine; Coups, Elliot J

Chapter 9

Introduction to the Analysis of Variance - all with Video Answers

Educators

Chapter Questions

04:14

Problem 1

For each of the following studies, decide whether you can reject the null hypothesis that the groups come from identical populations. Use the .05 level. Note that study (b) provides $S$, not $S^2$.
$$
\begin{array}{lccc}
\hline \text { (a) } & \text { Group 1 } & \text { Group 2 } & \text { Group 3 } \\
\hline n & 10 & 10 & 10 \\
M & 7.4 & 6.8 & 6.8 \\
S^2 & 82 & .90 & .80 \\
\hline
\end{array}
$$
$$
\begin{array}{lcccc}
\hline \text { (b) } & \text { Group 1 } & \text { Groap 2 } & \text { Group 3 } & \text { Group 4 } \\
\hline n & 25 & 25 & 25 & 25 \\
M & 94 & 101 & 124 & 105 \\
S & 24 & 28 & 31 & 25 \\
\hline
\end{array}
$$

Lucas Finney
Lucas Finney
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06:39

Problem 2

For each of the following studies, (a) and (b), decide whether you can reject the null hypothesis that the groups come from identical populations. Use the .01 level. (c) Figure the effect size for each study. (d) ADVANCED TOPIC: For study (a), carry out an analysis of variance using the structural model method.
$$
\begin{array}{|c|c|c|c|}
\hline \text { (a) } & \text { Group } 1 & \text { Group } 2 & \text { Group } 3 \\
\hline & 8 & 6 & 4 \\
\hline & 8 & 6 & 4 \\
\hline & 7 & 5 & 3 \\
\hline & 9 & 7 & 5 \\
\hline
\end{array}
$$
$$
\begin{array}{rrc}
\hline \text { (b) Group 1 } & \text { Group 2 } & \text { Group 3 } \\
\hline 12 & 10 & 8 \\
4 & 2 & 0 \\
12 & 10 & 8 \\
4 & 2 & 0 \\
\hline
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
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03:15

Problem 3

A psychologist at a private psychiatric hospital was asked to determine whether there was any clear difference in the length of stay of patients with different categories of diagnosis. Looking at the last four patients in each of the three major categories, the results (in terms of weeks of stay) were as follows:
(table can't copy)
(a) Using the .05 level and the five steps of hypothesis testing, is there a significant difference in length of stay among diagnosis categories? (b) Sketch the distributions involved. (c) Figure the effect size for the study. (d) Explain your answer to part (a) to someone who understands everything involved in conducting a $t$ test for independent means but is unfamiliar with the analysis of variance. (e) Test the significance of planned contrasts (using the . 05 level without a Bonferroni correction) for affective disorders versus drug-related conditions and (f) cognitive disorders versus drug-related conditions. (g) Explain your answers to parts $(\mathrm{e})$ and $(f)$ to a person who understands analysis of variance but is unfamiliar with planned contrasts.

Maxime Rossetti
Maxime Rossetti
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Problem 4

A study compared the felt intensity of unrequited love (loving someone who doesn't love you) among three groups: 50 individuals who were currently experiencing unrequited love who had a mean experienced intensity $=3.5, S^2=5.2$; 50 who had previously experienced unrequited love and described their experiences retrospectively, $M=3.2, S^2=5.8$; and 50 who had never experienced unrequited love but described how they thought they would feel if they were to experience it, $M=3.8, S^2=4.8$. Determine the significance of the difference among groups, using the $5 \%$ level. (a) Use the steps of hypothesis testing; (b) sketch the distributions involved; (c) figure the effect size for the study; and (d) explain your answer to part (a) to someone who has never had a course in statistics.

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04:15

Problem 5

A researcher studying genetic influences on learning compares the maze performance of four genetically different strains of mice, using eight mice per strain. Performance for the four strains were as follows:
$$
\begin{array}{lcc}
\hline \text { Strain } & \text { Mean } & S \\
\hline J & 41 & 3.5 \\
\text { M } & 38 & 4.6 \\
0 & 14 & 3.8 \\
W & 37 & 4.9 \\
\hline
\end{array}
$$
Using the .01 significance level, is there an overall difference in maze performance among the four strains? (a) Use the steps of hypothesis testing; (b) sketch the distributions involved; (c) figure the effect size for the study; and (d) explain your answer to part (a) to someone who is familiar with hypothesis testing with known populations but unfamiliar with the $t$ test or the analysis of variance. (e) Test the significance of planned contrasts using the overall .05 level (with a Bonferroni correction for testing each of the five contrasts) for strain J versus strain M, (f) for strain J versus strain Q, (g) for strain $J$ versus strain $\mathrm{W}$, (h) for strain $\mathrm{Q}$ versus strain $\mathrm{M}$, and (i) for strain $\mathrm{Q}$ versus strain W. (j) Explain your answers to parts (e) through (i) to a person who understands analysis of variance but is unfamiliar with planned contrasts and the Bonferroni correction.

Raymond Matshanda
Raymond Matshanda
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01:39

Problem 6

What is the Bonferroni corrected significance level for each of the following situations?
$$
\begin{array}{lcccc}
\hline \text { Situation } & \text { (a) } & \text { (b) } & \text { (c) } & \text { (d) } \\
\hline \text { Overal significance level } & .05 & .05 & .01 & .01 \\
\text { lammer of planned contrasts } & 2 & 4 & 3 & 5 \\
\hline
\end{array}
$$

Dominador Tan
Dominador Tan
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03:09

Problem 7

For each of the following studies, test whether a comparison in which the researcher figures an $F$ of 17.21 would be significant using the Scheffé method.
$$
\begin{array}{lccc}
\hline & \begin{array}{c}
\text { Number of } \\
\text { Groups }
\end{array} & \begin{array}{c}
\text { Participants in } \\
\text { Each Group }
\end{array} & \begin{array}{c}
\text { Significance } \\
\text { Level }
\end{array} \\
\hline \text { (a) } & 5 & 10 & .05 \\
\text { (b) } & 6 & 10 & .05 \\
\text { (c) } & 5 & 20 & .05 \\
\text { (d) } & 5 & 10 & .01 \\
\hline
\end{array}
$$

Beth Stone
Beth Stone
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Problem 8

What is the power of each of the following planned studies, using the analysis of variance with $p<.05$ ?
$$
\begin{array}{lccc}
\hline & \begin{array}{c}
\text { Predicted Effect } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Number of } \\
\text { Groups }
\end{array} & \begin{array}{c}
\text { Participants in } \\
\text { Each Group }
\end{array} \\
\hline \text { (a) } & \text { Smal } & 3 & 20 \\
\text { (4) } & \text { Smal } & 3 & 30 \\
\text { (c) } & \text { Smal } & 4 & 20 \\
\text { (d) } & \text { Medum } & 3 & 20 \\
\hline
\end{array}
$$

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Problem 9

About how many participants do you need in each group for $80 \%$ power in each of the following planned studies, using the analysis of variance with $p<.05$ ?
$$
\begin{array}{llc}
\hline & \begin{array}{c}
\text { Predicted Elfect } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Number of } \\
\text { Groups }
\end{array} \\
\hline \text { (a) } & \text { Small } & 3 \\
\text { (d) } & \text { Large } & 3 \\
\text { (c) } & \text { Small } & 4 \\
\text { (d) } & \text { Medium } & 3 \\
\hline
\end{array}
$$

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01:00

Problem 10

Grilo and colleagues (1997) are clinical psychologists interested in the relationship of depression and substance use to personality disorders. Personality disorders are persistent, problematic traits and behaviors that exceed the usual range of individual differences. The researchers conducted interviews assessing personality disorders with adolescents who were psychiatric inpatients and had one of three diagnoses: (1) those with major depression, (2) those with substance abuse, and (3) those with both major depression and substance abuse. The mean number of disorders was as follows: major depression $M=1.0$, substance abuse $M=.7$, those with both conditions $M=1.9$. The researchers reported, "The three study groups differed in the average number of diagnosed personality disorders, $F(2,112)=10.18, p<.0001$." Explain this result to someone who is familiar with hypothesis testing with known populations but is unfamiliar with the $t$ test or the analysis of variance.

Jessica Reyna
Jessica Reyna
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02:52

Problem 11

A researcher wants to know whether the need for mental health care among prisoners varies according to the different types of prison facilities. The researcher randomly selects 40 prisoners from each of the three main types of prisons in a particular Canadian province and conducts exams to determine their need for mental health care. In the article describing the results, the researcher reported the means for each group and then added: "The need for mental health care among prisoners in the three types of prison systems appeared to be clearly different, $F(2,117)=5.62, p<.01$. A planned comparison [contrast] of System 1 to System 2 was significant, $F(1,117)=4.03, p<.05$." Explain this result to a person who has never had a course in statistics.

Maxime Rossetti
Maxime Rossetti
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Problem 12

Based on Table 11 from the Hazan and Shaver (1987) study, indicate for which variables, if any, (a) the Avoidants are significantly different from the other two groups, (b) the Anxious-Ambivalents are different from the other two groups, (c) the Secures are different from the other two groups, and (d) all three groups are different. (e) Explain, to a person who understands analysis of variance but does not know anything about post hoc comparisons, what is meant in the table note that the results are "according to a Scheffé test."

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Problem 13

Which type of English word is longer: nouns, verbs, or adjectives? Go to a book of at least 400 pages and turn to random pages using the random numbers listed at the end of this paragraph. Go down the page until you come to a noun. Note its length (in number of letters). Do this for 10 different nouns. Do the same for 10 verbs and then for 10 adjectives. Using the .05 significance level, (a) carry out an analysis of variance comparing the three types of words, and (b) figure a planned contrast of nouns versus verbs. (Be sure also to give the full bibliographic information on the book you used: authors, title, year published, publisher, city.)
$73,320,179,323,219,176,167,102,228,352,4,335,118,12,333,123,38,49$, $399,17,188,264,342,89,13,77,378,223,92,77,152,34,214,75,83,198,210$

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01:55

Problem 14

For each of the following studies, decide whether you can reject the null hypothesis that the groups come from identical populations. Use the . 05 level.
$$
\begin{array}{lccc}
\hline \text { (a) } & \text { Group 1 } & \text { Group 2 } & \text { Group 3 } \\
\hline n & 5 & 5 & 5 \\
M & 10 & 12 & 14 \\
s^2 & 4 & 6 & 5 \\
\hline
\end{array}
$$
$$
\begin{array}{lccc}
\hline \text { (b) } & \text { Group 1 } & \text { Group 2 } & \text { Group 3 } \\
\hline n & 10 & 10 & 10 \\
M & 10 & 12 & 14 \\
s^2 & 4 & 6 & 5 \\
\hline
\end{array}
$$

Jameson Kuper
Jameson Kuper
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Problem 15

For each of the following studies, (a) and (b), decide whether you can reject the null hypothesis that the groups come from identical populations. Use the .01 level. (c) Figure the effect size for each study. (d) ADVANCED TOPIC: Carry out an analysis of variance for study (a) using the structural model method.
$$
\begin{array}{|c|c|c|c|}
\hline \text { (a) } & \text { Group } 1 & \text { Group } 2 & \text { Group } 3 \\
\hline & 1 & 1 & 8 \\
\hline & 2 & 2 & 7 \\
\hline & 1 & 1 & 8 \\
\hline & 2 & 2 & 7 \\
\hline
\end{array}
$$
$$
\begin{array}{ccc}
\hline \text { (b) Group 1 } & \text { Group 2 } & \text { Group 3 } \\
\hline 1 & 4 & 8 \\
2 & 5 & 7 \\
1 & 4 & 8 \\
2 & 5 & 7 \\
\hline
\end{array}
$$

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00:59

Problem 16

An organizational psychologist was interested in whether individuals working in different sectors of a company differed in their attitudes toward the company. The results for the three people surveyed in development were 10,12 , and 11; for the three in the marketing department, 6,6 , and 8 ; for the three in accounting, 7, 4, and 4; and for the three in production, 14, 16, and 13 (higher numbers mean more positive attitudes). Was there a significant difference in attitude toward the company among employees working in different sectors of the company at the .05 level? (a) Use the steps of hypothesis testing; (b) sketch the distributions involved; (c) figure the effect size for the study; (d) explain your answer to part (a) to someone who understands everything involved in conducting a $t$ test for independent means but is unfamiliar with the analysis of variance; (e) test the significance of planned contrasts using the overall .05 level (with a Bonferroni correction for testing each of the five contrasts) for engineering versus production, (f) marketing versus production, (g) accounting versus production, (h) development versus marketing, and (i) development versus accounting. (j) Explain your answers to parts (e) through (i) to a person who understands analysis of variance but is unfamiliar with planned contrasts or Bonferroni corrections. (k) ADVANCED TOPIC: Carry out an analysis of variance for the study using the structural model method.

Robin Corrigan
Robin Corrigan
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Problem 17

Do students at various universities differ in how sociable they are? Twenty-five students were randomly selected from each of three universities in a region and were asked to report on the amount of time they spent socializing each day with other students. The result for University $\mathrm{X}$ was a mean of 5 hours and an estimated population variance of 2 hours; for University Y, $M=4, S^2=1.5$; and for University $\mathrm{Z}, M=6, S^2=2.5$. What should you conclude? Use the .05 level. (a) Use the steps of hypothesis testing, (b) figure the effect size for the study; and (c) explain your answers to parts (a) and (b) to someone who has never had a course in statistics.

Rashmi Sinha
Rashmi Sinha
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04:19

Problem 18

A psychologist studying artistic preference randomly assigns a group of 45 participants to one of three conditions in which they view a series of unfamiliar abstract paintings. The 15 participants in the Famous condition are led to believe that these are each famous paintings; their mean rating for liking the paintings is $6.5(S=3.5)$ The 15 in the Critically Acclaimed condition are led to believe that these are paintings that are not famous but are very highly thought of by a group of professional art critics; their mean rating is $8.5(S=4.2)$. The 15 in the Control condition are given no special information about the paintings; their mean rating is $3.1(S=2.9)$. Does what people are told about paintings make a difference in how well they are liked? Use the . 05 level. (a) Use the steps of hypothesis testing; (b) sketch the distributions involved; (c) figure the effect size for the study; (d) explain your answer to part (a) to someone who is familiar with the $t$ test for independent means but is unfamiliar with analysis of variance; (e) test the significance of planned contrasts (using the . 05 significance level without a Bonferroni correction) for Famous versus Control and (f) Critically Acclaimed versus Control. (g) Explain your answers to parts (e) and (f) to a person who understands analysis of variance but is unfamiliar with planned contrasts.

Prabhakar Kumar
Prabhakar Kumar
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01:39

Problem 19

What is the Bonferroni corrected significance level for each of the following situations?
$$
\begin{array}{lcccc}
\hline \text { Situation } & \text { (a) } & \text { (b) } & \text { (c) } & \text { (d) } \\
\hline \text { Overall significance level } & .01 & .01 & .05 & .05 \\
\text { Mumber of planned contrasts } & 4 & 2 & 4 & 3 \\
\hline
\end{array}
$$

Dominador Tan
Dominador Tan
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Problem 20

For each of the following studies, test whether a comparison in which the researcher figures an $F$ of 8.12 would be significant using the Scheffé method.
$$
\begin{array}{lccc}
\hline & \begin{array}{c}
\text { Number of } \\
\text { Groups }
\end{array} & \begin{array}{c}
\text { Participants in } \\
\text { Each Group }
\end{array} & \begin{array}{c}
\text { Significance } \\
\text { Level }
\end{array} \\
\hline \text { (a) } & 4 & 30 & .05 \\
\text { (b) } & 5 & 80 & .05 \\
\text { (c) } & 4 & 5 & .05 \\
\text { (d) } & 8 & 30 & .01 \\
\hline
\end{array}
$$

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Problem 21

What is the power of each of the following planned studies, using the analysis of variance with $p<.05$ ?
$$
\begin{array}{lccc}
\hline & \begin{array}{c}
\text { Predicted Effect } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Namber of } \\
\text { Groups }
\end{array} & \begin{array}{c}
\text { Participants in } \\
\text { Each Group }
\end{array} \\
\hline \text { (a) } & \text { Small } & 4 & 50 \\
\text { (d) } & \text { Medum } & 4 & 50 \\
\text { (c) } & \text { Large } & 4 & 50 \\
\text { (d) } & \text { Medum } & 5 & 50 \\
\hline
\end{array}
$$

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Problem 22

About how many participants do you need in each group for $80 \%$ power in each of the following planned studies, using the analysis of variance with $p<.05$ ?
$$
\begin{array}{lcc}
\hline & \begin{array}{c}
\text { Predicted Effect } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Number of } \\
\text { Groups }
\end{array} \\
\hline \text { (ai) } & \text { Small } & 5 \\
\text { (b) } & \text { Medium } & 5 \\
\text { ic) } & \text { Large } & 5 \\
\text { (d) } & \text { Medium } & 3 \\
\hline
\end{array}
$$

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02:36

Problem 23

An experiment is conducted in which 60 participants each fill out a personality test, but not according to the way the participants see themselves. Instead, 15 are randomly assigned to fill it out according to the way they think their mothers see them (that is, the way they think their mothers would fill it out to describe the participants); 15 as their fathers would fill it out for them; 15 as their best friends would fill it out for them; and 15 as the professors they know best would fill it out for them. The main results appear in Table 17. Explain these results to a person who has never had a course in statistics.

Nick Johnson
Nick Johnson
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01:31

Problem 24

Rosalie Friend (2001), an educational psychologist, compared three methods of teaching writing. Students were randomly assigned to three different experimental conditions involving different methods of writing a summary. At the end
(table can't copy)
of the two days of instructions, participants wrote a summary. One of the ways it was scored was the percentage of specific details of information it included from the original material. Here is a selection from her article describing one of the findings:

The effect of summarization method on inclusion of important information was significant: $F(2,144)=4.1032, p<.019$. The mean scores (with standard deviations in parentheses) were as follows: Argument Repetition, $59.6 \%$ (17.9); Gencralization, $59.8 \%$ (15.2); and Self-Reflection, $50.2 \%$ (18.0). (p. 14)
(a) Explain these results to a person who has never had a course in statistics. (b) Using the information in the preceding description, figure the effect size for the study.

Lynn Larson
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Problem 25

Miller (1997) asked 147 female students to view slides of magazine ads that included, among other things, pictures of attractive men. The participants were measured for physiological arousal (skin conductance) while viewing the ads and also after viewing them; they were asked to rate the attractiveness and how much they would like to meet each person in the ads. As part of the analysis, Miller compared results for women dating no one, women in casual dating relationships, and women in exclusive dating relationships. Table 18 shows Miller's results. (a), (b), and (c) Describe the pattern of results on each variable. (d) Explain, to a person who understands analysis of variance but is unfamiliar with post hoc comparisons, what is meant in a general way by the table note that the results are based on "Duncan's multiple range test." (That is, you don't need to explain this specific test, but you do need to explain why a test like this was used and what it attempts to accomplish.)
(table can't copy)

Rashmi Sinha
Rashmi Sinha
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