Exercises


1. Set both population means to 10. What do you think the percent rejected will be?
Explanation

2. Specify a within-subjects design with n = 16, populations means of 15 and 10, rho = 0.2, and a population standard deviation of 10. Estimate power (the percent rejected)? Now increase rho to 0.6. Do you expect the power to increase or decrease? Test out your prediction.
Explanation

3. First, specify a between-subjects design with n = 16, populations means of 15 and 10, and a population standard deviation of 10. Simulate the experiment eight times recording the numerator and denominator of the t ratio for each simulation. Compute the mean numerator and mean denominator.

Second, specify a within-subjects design with n = 16, populations means of 15 and 10, rho = 0.6, and a population standard deviation of 10. Before simulating the experiment, think about how the values of the numertors and denominators of the t ratios will differ from the previous simulations. Simulate the experiment eight times recording the numerator and denominator of the t ratio for each simulation. Compute the mean numerator and mean denominator. Compare the results with the previous simulations.
Explanation

4. As part of a class assignment, two teams of students each do an experiment on the time it takes people to name the ink color that words are printed in. Specifically they compare the time it takes to name ink colors when the word is a conflicting color name (e.g., blue, yellow) to the time it takes to name the colors of neutral words (e.g., dog, cat). In both cases, the correct responses are "red" and "green." Each subject names 50 colors.

Assume that the population mean time to name the colors of conflicting color words is 55 seconds (sd = 16 sec) , the mean time to name the colors of neutral words is 40 seconds (sd = 16 sec), and that the population correlation between the two tasks is 0.55.

Team A uses a between-subjects design and tests 40 subjects (20 subjects in each group). Team B uses a within-subjects design and tests a total of 12 subjects. Which team is more likely to reject the null hypothesis?
Explanation

5. For the default value of n = 8, what are the degrees of freedom for between-subjects and for within-subjects tests? What is the formula for computing the degrees of freedom? If you don't already know the formula, experiment with different values of n and see if you can figure it out.
Explanation

6. The formulas for the between- and within-subjects designs are the same except that r is always 0 for between-subjects designs. Based on the formula, what is the relationship between r and t?
Explanation

7. How is the size of r related to the degree the lines in the graph are parallel?
Explanation


8. The case study "Weapons and Aggression" tested whether people are faster at naming an aggressive word when it is preceded by a "weapon word" than when it is preceded by a "neutral word." A within-subjects design was used. Use the Analysis Lab to determine the sample size, means, standard deviations, and correlation between these two conditions (The data library is "RVLS_case_studies," the dataset is "aggress_prime," the "weapon word" condition is called "aw" and neutral wordcondition is called "an"). Assume these sample values are the population parameters and estimate the power of the test of the difference between the two conditions. Use the average of the two sample standard deviations for your population standard deviation.) Estimate power if the experiment were done with a separate group for each condition.

 


Explanations
1. The percent rejected should be about 0.05. There is no real difference between the means in the population and therefore significant results are Type I errors. Because the significance level in this applet it set to 0.05, about 5% of the simulations should show significant differences.

2. Power goes up as the correlation between the two groups increases. A higher correlation between the two scores results in more consistent differences between across subjects. Look at the slopes of the lines for a samples using rho=0.2 compared to a samples using rho=0.6. With 0.2, the slopes of the lines vary much more than they do for 0.6, and less variance means more power. Set the correlation to 0.99 and see how it affects the consistency of the slopes.

In terms of the formula for t, notice that one of the terms in the denominator contains r: the higher the value of r, the lower the denominator and therefore the higher the t.

3. Since the numerator of the t ratio is the difference between means, you would expect the averge value of the numerators to be about 5. This does not differ for between- and within-subject designs. As discussed in Exercise 2, denominators are smaller in within-subject designs.

4. Team B has a greater chance. No new concepts here, just a illustration of the importance of choosing a within-subjetcs design.

5. For between-subject designs, the df = k(n-1) where k is the number of groups and n is the number of subjects per group. Since there are two groups and 8 subjects per group, df = 2(8-1) = 14.

For within-subject designs, df = n-1 = 7 in this case.

6. The larger the r the larger the t.

7. With larger r's, the lines are more parallel.