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 a course in statistics.
(a) To answer this question, we need to conduct a one-way analysis of variance (ANOVA) test to determine if there is a significant difference in the mean amount of time spent socializing among the three universities. Here are the steps of hypothesis testing:
Step 1: State the null hypothesis (H0) and alternate hypothesis (Ha):
- Null Hypothesis (H0): There is no significant difference in the mean amount of time spent socializing among the three universities.
- Alternate Hypothesis (Ha): There is a significant difference in the mean amount of time spent socializing among the three universities.
Step 2: Set the significance level:
- In this case, the significance level is .05 (5%).
Step 3: Compute the test statistic:
- The test statistic for ANOVA is F-statistic, which compares the variation between the groups to the variation within the groups.
Step 4: Determine the critical value:
- We need to consult the F-table or use statistical software to find the critical value of F at the .05 level of significance and degrees of freedom for the numerator and denominator.
Step 5: Calculate the F-statistic:
- Using the given mean and estimated population variance, we can calculate the F-statistic for each university.
Step 6: Compare the calculated F-statistic with the critical value:
- If the calculated F-statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in the mean amount of time spent socializing among the universities. Otherwise, we fail to reject the null hypothesis.
(b) Effect size measures the magnitude of the difference between the groups. In this case, we can calculate eta-squared (η^2) as a measure of effect size for the ANOVA test. Eta-squared represents the proportion of variance in the dependent variable explained by the independent variable (universities) in the population.
(c) To someone unfamiliar with statistics, you can explain the answers as follows:
(a) Hypothesis testing involves setting a null hypothesis, which assumes there is no significant difference, and an alternate hypothesis, which suggests there is a difference. We then calculate a test statistic (F-statistic) and compare it to a critical value at a given significance level. If the test statistic is larger than the critical value, we reject the null hypothesis and conclude that there is a significant difference in how sociable students are at the three universities. If not, we fail to find enough evidence to support a significant difference.
(b) Effect size measures the magnitude of the difference between the groups. In this case, we can use eta-squared to represent the proportion of variability in the time spent socializing that can be explained by the universities. By calculating this effect size, we can understand the practical significance or importance of any differences found among the universities. A larger effect size indicates a stronger association between the universities and the time spent socializing.