Question

1. In an effort to compare student grades in 4 different high schools, random samples of students from 4 local high schools were selected. Their first year average college grades were computed and recorded. Can we conclude that there are differences in the mean grades of these students from those 4 high schools? (20 points)

1. Apply either the Fischer-LSD or Tukey-Omega method to determine specifically whether schools A and C have a significant difference in population means.

2. Perform the appropriate F-max test, to determine if the ANOVA assumption of equal population variances is satisfied in this problem.

School A	School B	School C	School D
81.5	64.6	56.5	53.1
61.8	67.0	61.7	64.8
61.0	61.1	53.3	65.3
62.4	61.1	68.0	72.1
58.1	77.6	65.4	55.1
77.0	76.4	57.5	74.6
71.4	61.5	51.2	65.2
75.8	62.5	79.4	69.0
65.9	67.5	59.3	74.6
78.8	70.4	57.4	67.9
72.8	59.2	60.6	57.9
64.6	65.1	52.9	56.2
72.9	54.3	68.5	68.5
73.4	64.4	74.8	70.4
64.0	71.4	67.2	68.2
60.5	77.8	58.5	47.2
65.8	62.2		70.0
67.6	56.4		65.5
79.4	67.5		62.6
61.8	65.2
	64.7
	69.1
	63.6
	63.1

Answer 1

One-way ANOVA: Student grade versus School

Source DF SS MS F P
School 3 429.6 143.2 2.82 0.044
Error 75 3802.3 50.7
Total 78 4231.9
Since p-value<0.05 so we conclude that that there the mean grades of these students from those 4 high schools are not all same.

(a)

Grouping Information Using Tukey Method

School N Mean Grouping
A 20 68.825 A
B 24 65.571 A B
D 19 64.642 A B
C 16 62.012 B

Means that do not share a letter are significantly different.

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of School

Individual confidence level = 98.97%

School = A subtracted from:

School Lower Center Upper --+---------+---------+---------+-------
B -8.925 -3.254 2.416 (---------*--------)
C -13.094 -6.812 -0.531 (----------*---------)
D -10.183 -4.183 1.817 (---------*---------)
--+---------+---------+---------+-------
-12.0 -6.0 0.0 6.0

From the above we see that the confidence interval of the difference of mean grades of School C and School A does not contain zero so schools A and C have a significant difference in population means.
School = B subtracted from:

School Lower Center Upper --+---------+---------+---------+-------
C -9.603 -3.558 2.486 (---------*---------)
D -6.680 -0.929 4.823 (--------*---------)
--+---------+---------+---------+-------
-12.0 -6.0 0.0 6.0

School = C subtracted from:

School Lower Center Upper --+---------+---------+---------+-------
D -3.725 2.630 8.985 (---------*----------)
--+---------+---------+---------+-------
-12.0 -6.0 0.0 6.0

Grouping Information Using Fisher Method

School N Mean Grouping
A 20 68.825 A
B 24 65.571 A B
D 19 64.642 A B
C 16 62.012 B

Means that do not share a letter are significantly different.

Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons among Levels of School

Simultaneous confidence level = 79.98%

School = A subtracted from:

School Lower Center Upper ---+---------+---------+---------+------
B -7.549 -3.254 1.040 (-------*--------)
C -11.570 -6.812 -2.055 (--------*---------)
D -8.727 -4.183 0.361 (--------*--------)
---+---------+---------+---------+------
-10.0 -5.0 0.0 5.0

From the above we see that the confidence interval of the difference of mean grades of School C and School A does not contain zero so schools A and C have a significant difference in population means.
School = B subtracted from:

School Lower Center Upper ---+---------+---------+---------+------
C -8.136 -3.558 1.020 (--------*--------)
D -5.284 -0.929 3.427 (--------*--------)
---+---------+---------+---------+------
-10.0 -5.0 0.0 5.0

School = C subtracted from:

School Lower Center Upper ---+---------+---------+---------+------
D -2.183 2.630 7.442 (--------*---------)
---+---------+---------+---------+------
-10.0 -5.0 0.0 5.0

b.

Test for Equal Variances: Student grade versus School

95% Bonferroni confidence intervals for standard deviations

School N Lower StDev Upper
A 20 5.12467 7.23048 11.8412
B 24 4.38298 6.01899 9.3426
C 16 5.44585 7.96592 14.0945
D 19 5.30284 7.54191 12.5571

Bartlett's Test (Normal Distribution)
Test statistic = 1.67, p-value = 0.644

Levene's Test (Any Continuous Distribution)
Test statistic = 0.78, p-value = 0.507

Since p-value>0.05 hence we conclude that the assumption of equal population variances is satisfied in this problem.