The researcher in exercise 2 decides to conduct the same study using a within- participants design to control for differences in cognitive ability. He selects a random sample of subjects and has them study different material of equal difficulty in both the music and no music conditions. The study is completely counterbalanced to control for order effects. The data appear next. As before, they are measured on an interval- ratio scale and are normally distributed; he believes that studying under quiet conditions will lead to better performance.

Music No Music 7 7 6 8 5 7 6 7 8 9 8 8
a. What statistical test should be used to analyze these data?
The statistical test that should be used is the 2- sample t test.
b. Identify H0 and Ha for this study.
c. Conduct the appropriate analysis.
d. Should H0 be rejected? What should the researcher conclude?
e. If significant, compute and interpret the effect size.

b. H0 (null hypothesis): There is no difference in performance between studying with music and studying without music.

Ha (alternative hypothesis): Studying under quiet conditions will lead to better performance.

c. To conduct the appropriate analysis, we will use the 2-sample t-test. This test compares the means of two independent groups to determine if there is a significant difference between them.

Here are the steps to conduct the analysis:

1. Calculate the mean and standard deviation for each condition (music and no music).
2. Calculate the difference between the scores of each pair of subjects (music - no music).
3. Calculate the mean of the differences.
4. Calculate the standard deviation of the differences.
5. Use the formula for the t-test to calculate the t-value.
6. Determine the degrees of freedom (df) for the t-test. In this case, it is the number of pairs of scores minus 1 (n - 1).
7. Look up the critical value of t for the chosen alpha level and df.
8. Compare the calculated t-value with the critical value to determine if the difference is statistically significant.

d. To determine if H0 should be rejected, we compare the calculated t-value with the critical value of t. If the calculated t-value exceeds the critical value, we reject H0. If not, we fail to reject H0.

e. If the difference is statistically significant, we can compute the effect size using Cohen's d. Cohen's d is calculated by dividing the mean difference by the pooled standard deviation. This effect size measure indicates the magnitude of the difference between the two groups.

To interpret the effect size, we can use benchmarks provided by Cohen:
- Small effect size: d = 0.2
- Medium effect size: d = 0.5
- Large effect size: d = 0.8

By examining the effect size, we can determine the practical significance or meaningfulness of the observed difference.