Do students at various colleges differ in how sociable they are? Twenty-five students were randomly selected from each of three colleges in a particular region and were asked to report on the amount of time they spent socializing each day with other students. The results for College X was a mean of 5 hours and an estimated population variance of 2 hours; for College Y, M = 4, S2 = 1.5; and for College 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 (a) and (b) to someone who has never had a course in statistics.
|(1) We use ANOVA to examine if there is a significant difference in the mean values of X, Y, Z.
(a) H0: There is no significant difference in the mean scores of the three colleges.
Ha: There is a significant difference in the mean scores of the three colleges.
(b) Decision rule : Reject H0 if the test F- score > the critical value of F at a = 0.05
The General format for the one-way ANOVA table is shown below:
Correction Factor CF = 375^2/75 = 1875
Total Sum of squares = 2075 – 1875 = 200
Treatment sum of squares = (48125/25) – 1875 = 50
Error = 200 – 50 = 150
(c) ANOVA table
(d) Conclusion: Since 12 > 3.124, we reject H0 and accept Ha. It appears that there is a significant difference in the mean scores of the three colleges
(2) Effect Size is a statistical measure of the magnitude of a treatment effect. One popular effect size measure is h^2. Eta squared is the proportion of the total variance that is attributed to an effect.
= 50/200 = 0.25
(3) The present problem examined if the students at the three colleges were significantly different in the mean amount of time they spent in socializing each day with other students. The mean times for the 3 colleges were 5, 4, 6 hours. ANOVA results indicate that these are significantly different and students from College Z spent significantly more time than the those of the other two colleges.
The effect size measure suggests that 25% of total variation can to attributed to the variations among colleges.