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1. A researcher randomly assigns members of a statistics class to two groups. In one group, each participant is assigned a tutor. The tutor is available to meet with the student 20 minutes before each class. The other group is a control group not assigned a tutor. Suppose the researcher finds that the tutored group scores significantly better on exams.

a. Can the researcher conclude that the experimental group students learned statistical information in tutoring sessions that enabled them to perform better on the exam? Why or why not?

Hint: Is the tutoring the only difference between the groups? See pages 426-427.

b. What changes would you recommend in the study?

Hints: Your control group is an empty control group? What are the problems with such groups? How can you replace the empty control group with a placebo control group? If you can't find one really great control group, could you use still have decent construct validity by using two good control groups? See pages 430-432.

2. Suppose people living in homes for the elderly were randomly assigned to two groups:  a no treatment group and a transcendental mediation (TM) group.  Transcendental mediation involves more than sitting with eyes closed.  The technique involves both a "mantra, or meaningless sound selected for its value in facilitating or settling down process and a specific procedure for using it mentally without effort again to facilitate transcending" (Alexander, Langer, Newman, Chandler, & Davies, 1989).  Thus, the TM group was given instruction in how to perform the technique, then "they met with their instructors 1/2 hour each week to verify that they were mediating correctly and regularly.  They were to practice their program 20 minutes twice daily (morning and afternoon) sitting comfortably in their own room with eyes closed and using a timepiece to ensure correct length of practice."  (Alexander, Langer, Newman, Chandler, & Davies). Suppose that the TM group performed significantly better than other groups on a mental health measure.

a. Could the researcher conclude that it was the transcendental meditation that caused the effect?

No, because the control group was an empty control group.

b. What besides the specific aspects of TM could cause the difference between the two groups?

The extra attention the TM group received, the structure of a routine that was imposed on the TM group, as well as the fact that those who weren't able to learn the TM technique or who didn't continue to apply the technique would be dropped from the study. Thus, people may be dropping out of the experimental group, but not out of the control group.

c. What control groups would you add?

A group that had to undergo some training (e.g., critical thinking) and would have to practice what they had learned twice a day and meet with their instructors once a week.

d. Suppose you added these control groups and then got a significant F for the treatment variable? What could you conclude? Why?

Conclusion: That at least one of the groups differ from the others. In other words, at least one of the treatments had an effect. However, we would not be able to say which groups differed from each other until we did a post hoc test.

3. Assume you want to test the effectiveness of a new kind of  therapy. This therapy involves screaming and hugging people in group sessions followed by individual meetings with a therapist. What control group(s) would you use? Why?


  1. What type of control group would allow you to see whether therapy was more harmful than no therapy?

  2. What control group would allow you to see whether hugging was an essential part of the therapy?

  3. What control group would allow you to see whether screaming was an essential part of the therapy?

  4. What control group would allow you to see whether the group meetings were a useful part of the therapy?


4. Assume a researcher is looking at the relationship between caffeine consumption and sense of humor.

a. How many levels of caffeine should the researcher use? Why?

At least three because the relationship might be nonlinear. For example, people might have little sense of humor with no caffeine (they're not awake) and little with an extreme amount of caffeine (they are too hyped up and irritable), but a good sense of humor under moderate levels of caffeine. Using three or more levels of  caffeine would allow us to detect some nonlinear trends and help us make predictions about the effects of levels of caffeine that we had not directly tested.

b. What levels would you choose? Why?

Three to four levels. A no caffeine group, a low caffeine group, a moderate caffeine group, and a high caffeine group. Make sure that the amounts of caffeine are evenly spaced (e.g., 0 mg., 20, 40, 60, 80) so that trend analyses can be performed.

c. If a graph of the data suggests a curvilinear relationship, can the researcher assume that the functional relationship between the independent and dependent variable is curvilinear? Why or why not?

No—the researcher do a post hoc trend analysis to make sure the observed pattern is reliable.

d. Suppose the researcher used the following four levels of caffeine: 0 mg., 20 mg., 25 mg., 26 mg. Can the researcher do a trend analysis? Why or why not?

No—the levels are not evenly or proportionately spaced.

e. Suppose the researcher ranked participants based on their sense of humor. That is, the person that laughed least got a score of "1", the person who laughed second least got a "2", etc.  Can the researcher use this data to do a trend analysis? Why or why not?

No—you need at least interval scale measurement to do a trend analysis. Ranked data is only ordinal. 

f. If a researcher used 4 levels of caffeine, how many trends can the researcher look for?                  

3 (one less than the number of levels)

What is the treatment's degrees of freedom?

3 (one less than the number of levels)

g. If the researcher used 3 levels of caffeine and 30 participants, what are the degrees of freedom for the treatment?

2 (3-1)

the degrees of freedom for the error term?

27 (30-3)

h. Suppose the F is 3.34  Referring to the degrees of freedom you obtained in your answer to "g" (above) and  to the table F-3, are the results statistically significant?

No—if the significance rule is that p < .05

Can the researcher look for linear and quadratic trends?

No—if the results are not statistically significant, then the researcher cannot look for trends.

5. A computer analysis reports that F (6,23)= 2.54. The analysis is telling you that the F ratio was 2.54, and the degrees of freedom for the top part of the F ratio is 6 and the degrees of freedom for the bottom part is 23.

a. How many groups did the researcher use?

Hint: See the "Treatment (between groups") row of Table 11.1 on page 443.

b. How many participants were in the experiment?

Hint: See the last row of Table 11.1 on page 443.

c. Is this result statistically significant at the .05 level (refer to table F-3)?

Hint: If 2.54 is larger than the critical F for (6,23) in Table F-3, the result is significant. 

6. A friend gives you the following Fs and significance levels. On what basis, would you want these Fs (or significance levels) re-checked?

a. F (2, 63)=.10, not significant

Even when the treatment has no effect, F's rarely tend to be zero. Instead, they are usually closer to 1.00. After all, if there is no treatment effect, then, at a conceptual level, you are dividing an estimate of error variance by another, estimate of the same error variance. Dividing anything by itself should result in a number close to 1.

b. F (3, 85) = -1.70, not significant

Fs can’t be negative. You are dividing a square term by another squared term.

c. F (1, 120)= 52.8, not significant

Such a large F with so many degrees of freedom would have to be significant. Indeed, according to the F table in Appendix F, the critical value of F(1,120) is 3.92.

d. F (5, 70) = 1.00, significant

Fs close to one are rarely significant. An F of one is expected even when there is absolutely no effect. Indeed, the lowest critical value of F on the entire F table in Appendix F is 1.46—and that's for an F(30, and an infinite number of degrees of freedom).

7. Complete the following table.

Hints: See the table accompanying summary point 13 on page 451


Source of Variance (SV)

Sum of Squares


degrees of freedom



Mean Square





3 levels of treatment



Hint: There are 3 levels of  the treatment factor, so, according to summary point 10 on p. 450, df is

Hint: MS is SST/dfT

Hint: F is MST/MSE

Error (E)





Hint: A mean square is always the sum of squares divided by its df.



8. Complete the following table.













Error (E)








SS Total=







9. A study compares the effect of having a snack, taking a 10 minute walk, or getting no treatment on energy levels. Sixty participants are randomly assigned to condition and then asked to rate their energy level of a 0 (not at all energetic) to 10 (very energetic) scale. The mean for the "do nothing" group is 6.0, for having a snack 7.0, and for walking 7.8.  The F-ratio is 6.27.

a. Graph the means

    Hints: You can use either a line graph or a bar chart. Remember to put your three conditions on the horizontal axis (the x-axis) and put energy level on the vertical axis (the y-axis).

b. Are the results statistically significant? Hints: What are the degrees of freedom? If you need help with the degrees of freedom, see main point 10 on p.450. If you need help using the F table on page 682, see main point 9 on p. 450--as well as the directions on the top of that table.

c. If so, what conclusions can you draw?  Why?

Hints: Can you conclude that at least one of the groups differs from the others? From an ANOVA alone, can  you say which groups differ from each other?

d. What additional analyses would you do? Why? Hint: What analyses would allow you to know which groups differed from each other? If you are still not sure, re-read pages 446-447.

e. How would you extend this study?

Hints: How could you change the study so that you could

map the functional relationship between length of walk and energy level,

compare different types of exercise?

comparing different types of snacks?

comparing other activities besides exercise?



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