17. 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, ; and for University
Z, . 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) Steps of hypothesis testing:

Step 1: Specify the null hypothesis (H0) and the alternative hypothesis (Ha):
In this study, the null hypothesis (H0) would be that there is no difference in the sociability level among students from the three universities. The alternative hypothesis (Ha) would be that there is a difference in sociability among students from the three universities.

Step 2: Choose the significance level (α):
The significance level, denoted by α, determines the likelihood of rejecting the null hypothesis when it is actually true. In this case, α is given as 0.05.

Step 3: Calculate the test statistic:
To compare the means of the three universities, we can use a one-way analysis of variance (ANOVA) test. This test will generate an F-statistic that will help us determine if there is a significant difference in sociability.

Step 4: Determine the critical value:
The critical value is the threshold beyond which we reject the null hypothesis. For an ANOVA test, we need to compare the calculated F-statistic with the critical value from the F-distribution table.

Step 5: Compare the test statistic with the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. If it is smaller, we fail to reject the null hypothesis.

(b) Effect size for the study:
The effect size measures the magnitude of the difference between groups in a study. In this case, we can calculate the effect size using eta-squared (η²) for the ANOVA. It is the proportion of the total variance that is accounted for by the group differences.

(c) Explanation for (a) and (b):
To determine if there is a difference in sociability among students from the three universities, we conduct a hypothesis test. We set up a null hypothesis and an alternative hypothesis, calculate the test statistic (F-statistic) using ANOVA, and compare it with the critical value from the F-distribution table. If the test statistic exceeds the critical value, we reject the null hypothesis and conclude that there is a significant difference in sociability among the universities.

The effect size helps us understand the practical significance of this difference. It measures how much of the total variability in sociability is attributed to the universities. A larger effect size indicates a stronger impact of the university on sociability.

It is important to note that these conclusions are based on the sample data collected from the universities. The hypothesis test and effect size calculations provide statistical evidence to support the conclusions but do not guarantee the same results for the entire population of students at these universities.