Do students at various universities differ in how sociable they are? Twenty-five

students were randomly selected from each of three universities in a region and
were asked to report on the amount of time they spent socializing each day with
other students. The result for University X was a mean of 5 hours and an estimated
population variance of 2 hours; for University Y,m = 4, s^2=1.5 ; and for University
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;
and (c) explain your answers to parts (a) and (b) to someone who has never had
a course in statistics.

(a) To draw a conclusion about the sociability of students at various universities, we can use the steps of hypothesis testing. Here's how you can proceed:

Step 1: State the hypotheses.
- Null Hypothesis (H0): The mean sociability (μ) is the same for all universities.
- Alternative Hypothesis (H1): The mean sociability (μ) differs for at least one university.

Step 2: Set the significance level.
In this case, the significance level is given as α = 0.05, which means we want to be 95% confident in our conclusion.

Step 3: Compute the test statistic.
Since we are comparing means for three groups, we can use ANOVA (analysis of variance) to test for any significant differences. ANOVA compares the between-group variance with the within-group variance. The test statistic is an F-ratio.

Step 4: Determine the critical value.
Using the significance level α = 0.05 and degrees of freedom for the numerator (k-1) and denominator (N-k), where k is the number of groups and N is the total sample size.

Step 5: Calculate the F-statistic and compare with the critical value.
Calculate the F-statistic using the given means (4, 5, 6) and variances (1.5, 2, 2.5) for the three universities. Compare the F-statistic with the critical value. If the calculated F-statistic is greater than the critical value, reject the null hypothesis.

Step 6: Make a decision and interpret the results.
If the calculated F-statistic is greater than the critical value, that means there is a significant difference in sociability among the universities. If it is not greater, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a difference in sociability.

(b) Effect size quantifies the magnitude of the difference between groups. In this case, effect size can be calculated using eta-squared (η²). It represents the proportion of variance in sociability that can be attributed to the differences between universities.

To calculate eta-squared, divide the between-group sum of squares by the total sum of squares:
η² = SSB / SST

Effect sizes can range from 0 to 1, with higher values indicating a larger effect.

(c) When explaining the above to someone unfamiliar with statistics, you can say:
(a) We are testing whether the mean sociability is the same for all universities or if there are differences. We'll compare the means and see if these differences are statistically significant.
(b) Effect size gives an indication of how different the means are between universities. It helps us understand the magnitude of these differences. We'll calculate eta-squared as our effect size, which tells us the proportion of sociability variance that can be attributed to the differences between universities. Higher values mean larger differences.
In summary, we'll use hypothesis testing to determine if there are significant differences in sociability among the universities and calculate effect size to understand the magnitude of these differences.