Do students at various colleges differ in how sociable they are? Twenty-Five students were randomly selected from each of three colleges in a particular region and were asked to report on the amount of the time they spent socializing each day with other students. The results for college X was a mean of 5 hours and an estimated population variance of 2 hours: for college Y, M=4, S^2=1.5: and for college z, M =6, S^2=2.5. What should you conclude? Use the .05 level.

(a) Use the steps of hypothesis testing.

(b) figure the effect size for the study;

(c) explain your answers to (a) and (b) to someone who has never had a course in statistics.

You can probably use a One-Way Anova test for this data. The null hypothesis would be that all means are equal; the alternate hypothesis would be that the means are not equal. If the means are not equal, then there are differences between the groups. You can also calculate the effect size as well.

(a) Steps of hypothesis testing:

Step 1: State the hypotheses
- Null hypothesis (H0): There is no significant difference in the sociability of students across the three colleges.
- Alternative hypothesis (Ha): There is a significant difference in the sociability of students across the three colleges.

Step 2: Set the significance level
- In this case, the significance level is given as .05.

Step 3: Compute the test statistic
- We will use the one-way analysis of variance (ANOVA) test to compare the means of the three groups. The ANOVA test compares the variances between groups with the variances within groups.
- Compute the F-test statistic using the formula: F = (MSB / MSW), where MSB is the Mean Square Between and MSW is the Mean Square Within.

Step 4: Determine the critical value
- Look up the critical value for F in the F-distribution table based on the significance level (.05) and degrees of freedom (numerator = number of groups - 1, denominator = total sample size - number of groups).

Step 5: Make a decision
- If the computed F-test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

(b) Effect size for the study:

- One way to measure the effect size in ANOVA is by calculating eta squared (η²).
- Eta squared is calculated by dividing the sum of squares between groups (SSB) by the total sum of squares (SST).
- Eta squared varies from 0 to 1, where 0 indicates no effect and 1 indicates a strong effect.

(c) Explanation to someone without a background in statistics:

(a) Hypothesis testing helps us determine if there is a significant difference in sociability among students across the three colleges. We compare the means of the groups to see if there is enough evidence to reject the null hypothesis and conclude that there is indeed a difference.

(b) Effect size is another way to measure the difference in sociability among the colleges. It helps us understand the magnitude of the difference, beyond just determining if there is a significant difference. In this case, we will calculate eta squared to quantify the effect size.

To summarize, hypothesis testing allows us to draw conclusions about the sociability of students at different colleges, while effect size helps us understand the size of the differences observed.