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HUDM 5123 - Linear Models and Experimental Design

HW 03 -Comparing Means

Instructions.

You are encouraged to discuss problems with classmates, but all work you submit must be your own.

If applicable, any plots should have appropriate axis and overall labels.

In general, do not include computer output (either SPSS or R) in your write-up.Instead, summarize relevant points using text, tables, or plots.

When in doubt about formatting issues (e.g., for references, tables, notes, etc.), use APA sty le.

Complete exercises 1-5 in MDK, chapter 4. For your convenience, and to eliminate ambiguity in case you are using a version of the text other than the third, I have posted a screenshot of them below. Note that solutions to the starred exercises are available at the book website here if you want to check your work for those exercises.

EXERCISES

1 point *1. Write out the coefficients for contrasts to be used for testing each of the following hypotheses in afour-group study.

a. Ho: μ1 = μ2

b. Ho: μ1 = .5(μ2 + μ3)

c. Ho: μ2 = μ4

d. H : μ4 = 1/3(μ1 + μ2 + μ3)

INDIVIDUAL COMPARISONS OF MEANS

1 point 2. Which of the contrasts in Exercise I are pairwise? Which are complex?

*3. A psychologist collected data for three groups. The sample means are as follows: Y = 12, Y, = 10, and Y,=6. The value of MSw is 25, and there are 10 subjects in each group. The psychologist is interested in comparing the average of the Group 1 and 2 means to the Group 3 mean.

1 point a. The psychologist forms a contrast whose coefficients are given by .5, .5, and -1. Test this contrast for statistical significance.

1 point b. A colleague has suggested that it would be simpler to test a contrast with coefficients of I, I, and-2. Does this produce the same result as Part a?

1 point c. What is the relationship between ()? of Part a and ()? of Part b? What is the relationship of Σe in Part a to Σe in Part b? Does this explain why the Σef term is needed in Equation 32? Justify your answer.

1 point 4. Yet another contrast that might be used in Exercise 3 is one with coefficients of-1, -1, and 2. How does the F value for this contrast compare with the Fvalue obtained in Exercise 3? What general rule does this illustrate?

5. Exercises 3 and 4 asked you to test a complex comparison in a three-group study. This exercise asks you to form a confidence interval for the same complex comparison. As before, the sample means are: y =12, Y,= 10, and Y, = 6. The value of MSw is 25, and there are 10 subjects per group. Continue to assume that the psychologist is interested in comparing the average of the Group 1 and 2 means to the Group 3 mean.

1 point a. Form a 95% confidence interval for the contrast of interest. (Notice that with the available information, you must assume homogeneity of variance.)

1 point b. Does the confidence interval you found in Part a agree with the results of the hypothesis test in Exercise 3? Explain your answer.

1 point c. Express the mean difference for this contrast as a standardized difference. How would you interpret this result?

1 pointd. d.We saw in Exercise 3 that we could use coefficients of 1, 1, and-2 without changing the result of the hypothesis test. Can we also use these coefficients for forming a confidence interval without changing the result? Why or why not?