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.
You might try a one-way ANOVA on this data. Use the appropriate table to compare your test statistic to the critical value from the table at .05 level of significance. If the test statistic exceeds the critical value from the table, reject the null and conclude a difference. If the test statistic does not exceed the critical value from the table, do not reject the null (you cannot conclude a difference in this case).
(a) To conduct the hypothesis test, there are several steps to follow:
1. Establish the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): There is no significant difference in sociability among students at the three universities.
- Alternative hypothesis (Ha): There is a significant difference in sociability among students at the three universities.
2. Determine the appropriate test statistic:
Since we are comparing means from three independent groups, we can use a one-way analysis of variance (ANOVA) test.
3. Set the significance level (α):
In this case, the significance level is given as .05.
4. Compute the test statistic:
Using the ANOVA test, we calculate the F-statistic.
5. Determine the critical value(s):
By comparing the test statistic to the critical value(s) from the F-distribution table, we can determine if the results are statistically significant.
6. Make a decision and interpret the results:
If the test statistic exceeds the critical value(s), we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, if the test statistic falls within the critical region, we fail to reject the null hypothesis.
(b) The effect size for the study can be measured using eta-squared (η²), which represents the proportion of total variation in the dependent variable (sociability) explained by the independent variable (university).
(c) Explanation:
(a) Hypothesis testing involves setting up two competing hypotheses (null and alternative), selecting an appropriate test statistic, determining a significance level, calculating the test statistic, comparing it to critical values, and making a decision based on the results. In this case, the hypothesis testing would determine if there is a significant difference in sociability among students at the three universities.
(b) Effect size measures the magnitude of the relationship between variables. In this study, eta-squared quantifies the proportion of variation in sociability that can be attributed to the university attended by students.
To summarize, part (a) involves conducting a hypothesis test to assess if there are differences in sociability among students at the three universities. Part (b) calculates the effect size, which describes the strength of the relationship between university attended and sociability.