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.

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:

Source of  Variation

SS

DOF

MS = SS/DOF

F = MST/MSE

F (critical)

Between Colleges (Treatment)

3 – 1 = 2

Within Colleges (Error)

Total SS – Treatment SS

75 – 3 = 72

Total

75 – 1 = 74

 

 

 

 

College

n

Xi.

S2

S

X

25

5

125

2

1.414214

675

15625

Y

25

4

100

1.5

1.224745

437.5

10000

Z

25

6

150

2.5

1.581139

962.5

22500

Total

75

 

375

6

4.220097

2075

48125

 

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

Source of  Variation

SS

DOF

MS = SS/DOF

F = MST/MSE

F (Critical)

Between Colleges (Treatment)

50

2

25

12

3.124

Within Colleges (Error)

150

72

2.083333

Total

200

74

2.702703

 

 

 

(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.

 

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