Chapter 9: 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, M=4,S2=1.5 ; and for University
Z,M=6,S2=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

Here are a few hints:

Check out a one-way ANOVA for this kind of problem. Do the calculations to find the F-ratio to compare to the critical or cutoff value from the F-distribution table (used for ANOVA tests). Find the critical or cutoff value at .05 level of significance using the table to reject the null hypothesis (which would state that all population means are equal). If the null is rejected in favor of the alternate hypothesis (which would state that all population means are not equal), then you can conclude a difference.

the cutoff is between 2 and 6

experience with hypothesis testing.

(a) Steps of hypothesis testing:
1. Formulate the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis: There is no difference in sociability among students at the three universities.
- Alternative hypothesis: There is a difference in sociability among students at the three universities.

2. Choose a significance level (alpha) to determine the threshold for accepting or rejecting the null hypothesis. In this case, the significance level is .05.

3. Collect data and calculate the test statistic:
- In this scenario, we have the means (M) and estimated population variances (S2) for each university.

4. Determine the critical value(s):
- With a significance level of .05, we can evaluate the critical value for a two-tailed test using a t-distribution table with degrees of freedom equal to the total sample size minus one.

5. Compare the test statistic to the critical value(s) and make a decision:
- If the test statistic falls within the range of critical values, we fail to reject the null hypothesis.
- If the test statistic falls outside the range of critical values, we reject the null hypothesis in favor of the alternative hypothesis.

(b) Effect size for the study:
Effect size is a measure of the magnitude of the difference between groups or the strength of the relationship. In this case, we can use Cohen's d as the effect size measure:

Cohen's d = (Mean1 - Mean2) / √[(S12 + S22) / 2]

where Mean1 and Mean2 represent the means of two groups, and S12 and S22 represent the estimated population variances for each group.

(c) Explanation for parts (a) and (b):
Hypothesis testing allows us to determine if there is evidence to support a claim about a population based on the sample data. In this case, we are investigating whether there are differences in sociability among students at three universities.

To perform hypothesis testing, we follow a series of steps, including formulating the null and alternative hypotheses, choosing a significance level, collecting data, calculating the test statistic, determining critical values, and making a decision based on the comparison.

For this study, we can calculate Cohen's d as a measure of effect size. Effect size helps us understand the practical significance of the results by quantifying the difference between groups.

By following these steps and considering the calculated test statistic and critical values, we can determine whether there is evidence to support the claim that students at the various universities differ in sociability.