1. A sports researcher gave a standard written test of eating habits to 12 randomly selected professionals, four each from baseball, football, and basketball. The results were as follows:
Eating Habits Scores
Baseball Players Football Players Basketball Players
34 27 35
18 28 44
21 67 47
65 42 61
Is there a difference in eating habits among professionals in the three sports? (Use the .05 significance level.)
a. Use the five steps of hypothesis testing.
b. Sketch the distribution involved.
c. Determine effect size.
2. To study the effectiveness of treatments for insomnia, a sleep researcher conducted a study with 12 participants. Four participants were instructed to count sheep (Sheep Condition), four were told to concentrate on their breathing (Breathing Condition), and four were not given any special instructions. Over the next few days, measures were taken of how many minutes it took each participant to fall asleep. The average times for the participants in the Sheep Condition were 14, 28, 27, and 31; for those in the Breathing Condition, 25, 22, 17, and 14; and for those in the control condition, 45, 33, 30, and 41.
Do these results suggest that the different techniques have different effects?
(Use the .05 significance level.)
a. Use the five steps of hypothesis testing.
b. Sketch the distribution involved.
c. Figure the effect size of the study.
d. Explain your findings (including the logic of comparing within-group to between-group population variance estimates, how each of these is figured, and the F distribution).
3. High school juniors planning to attend college were randomly assigned to view one of four videos about a particular college, each differing according to what aspect of college life was emphasized: athletics, social life, scholarship, or artistic/cultural opportunities. After viewing the videos, the students took a test measuring their desire to attend this college. The results were as follows:
Desire to Attend this College
Athletics Social Life Scholarship Art/Cultural
68 89 74 76
56 78 82 71
69 81 79 69
70 77 80 65
Do these results suggest that the type of activity emphasized in a college film affects desire to attend that college? (Use the .01 significance level.)
a. Use the five steps of hypothesis testing.
b. Sketch the distribution involved.
c. Figure the effect size of the study.
d. Explain the logic of what you have done to a person who is unfamiliar with the analysis of variance.
4. A team of psychologists designed a study in which 12 psychiatric patients diagnosed as having generalized anxiety disorder were randomly assigned to one of three new types of therapy, here labeled X, Y, and Z. One year after therapy, the patients’ overall mental health scores were as follows:
Mental Health Assessment
Therapy X Therapy Y Therapy Z
85 78 79
79 81 83
84 68 75
67 75 74
Do these results suggest that the different therapies have different effects on mental health?
(Use the .05 level.)
a. Use the five steps of hypothesis testing.
b. Sketch the distribution involved.
c. Explain your results.
1.
a. The five steps of hypothesis testing are:
1. State the null hypothesis (H0) and the alternative hypothesis (Ha).
2. Select the significance level (alpha).
3. Calculate the test statistic.
4. Determine the critical value(s) or p-value(s).
5. Make a decision and interpret the results.
In this case, the null hypothesis (H0) is that there is no difference in eating habits among professionals in the three sports. The alternative hypothesis (Ha) is that there is a difference in eating habits among professionals in the three sports.
b. To sketch the distribution involved, we need to analyze the data using analysis of variance (ANOVA). ANOVA compares the means of three or more groups to determine if there are any significant differences. In this case, we have three groups (baseball, football, and basketball).
c. To determine the effect size, we can use a measure called eta-squared (η^2). Eta-squared measures the proportion of variance in the dependent variable (eating habits scores) that can be accounted for by the independent variable (sports).
2.
a. The five steps of hypothesis testing are the same as mentioned above.
b. To sketch the distribution involved, we need to analyze the data using ANOVA. In this case, we have three conditions: sheep, breathing, and control.
c. To figure the effect size of the study, we can calculate eta-squared (η^2) as mentioned above.
d. To explain the findings, we can compare the within-group and between-group population variance estimates. Within-group variance estimates the variability of scores within each group, while between-group variance estimates the variability of means between groups. The F distribution is used to compare the variability of means between groups against the natural variability within each group.
3.
a. Apply the five steps of hypothesis testing as mentioned above.
b. Sketch the distribution involved using ANOVA. In this case, we have four types of college videos (athletics, social life, scholarship, and art/cultural).
c. Calculate the effect size using eta-squared (η^2) as mentioned above.
d. To explain the logic to someone unfamiliar with the analysis of variance, we can say that ANOVA helps determine if there are significant differences between groups by comparing the variability of means between groups to the variability within each group. The F distribution is used to make this comparison.
4.
a. Apply the five steps of hypothesis testing as mentioned above.
b. Sketch the distribution involved using ANOVA. In this case, we have three therapy types (X, Y, and Z).
c. Explain the results based on the analysis of variance and significance level chosen. If the p-value is less than the significance level (0.05), we can reject the null hypothesis and conclude that the different therapies have different effects on mental health. If the p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that there is no significant difference in the effects of the therapies on mental health.