Question

Formally assess whether or not the estimated differences in the mean scores (grade) for the three positions (front, middle, and back) are statistically significant,
providing numerical justification (test statistic and P-value) for your conclusion. If you find a
significant difference among the positions, then investigate which positions differ significantly
from each other. You can ignore prevGPA.

DATA two;
INPUT grade position $ prevGPA;
cards;
93.2 front 3.78
88.4 front 3.52
65.4 front 2.67 
56.3 front 2.23 
73.6 front 3.21
44.3 front 1.96
89.4 front 3.45
92.7 front 3.77
75.4 front 2.99
86.5 front 3.26
95.6 middle 4.00
77.3 middle 3.12
91.1 middle 3.59
87.2 middle 3.19
75.3 middle 2.65
97.4 middle 3.82 
67.4 middle 2.87
93.6 middle 3.67
86.2 middle 3.42
88.3 middle 3.44
56.6 middle 1.76
94.3 middle 3.98
99.3 middle 3.93
76.4 middle 3.17
70.8 back 2.43
78.5 back 3.11
98.2 back 3.89
89.3 back 3.24
71.2 back 3.12
46.3 back 2.11
65.5 back 2.34
81.2 back 3.24
85.4 back 3.23
78.3 back 2.93
;
run;

Answer 1

For answering this, I assume that I need not to do it in SAS since it was not asked for.The data required for the analysis is listed below:

	Front	Middle	Back	Total
	93.2	95.6	70.8
	88.4	77.3	78.5
	65.4	91.1	98.2
	56.3	87.2	89.3
	73.6	75.3	71.2
	44.3	97.4	46.3
	89.4	67.4	65.5
	92.7	93.6	81.2
	75.4	86.2	85.4
	86.5	88.3	78.3
		56.6
		94.3
		99.3
		76.4
Total	765.2	1186	764.7	2715.9
n	10	14	10

The grand total of all observations is 2715.9.

Since there are more than 2 groups to be compared, we shall use ANOVA with a single factor(position). The null hypothesis:

Null Hypothesis: There are no difference in estimated mean score between the three positions

Alternate Hypothesis: There is a difference in at lest one pairs.

1. We shall calculate the following quantities for constructing the ANOVA table.

We have the number of times each position occurs :

Correction factor

  

2. Total sum of squares (TSS):

We have to have the individual observations squared and then added. We shall form a table of squares and then add them.

	Front	Middle	Back	Total
	8686.24	9139.36	5012.64
	7814.56	5975.29	6162.25
	4277.16	8299.21	9643.24
	3169.69	7603.84	7974.49
	5416.96	5670.09	5069.44
	1962.49	9486.76	2143.69
	7992.36	4542.76	4290.25
	8593.29	8760.96	6593.44
	5685.16	7430.44	7293.16
	7482.25	7796.89	6130.89
		3203.56
		8892.49
		9860.49
		5836.96
Total	61080.16	102499.1	60313.49	223892.75

3. Sum of squares due to Positions:

where is the total of the ith position.

765.2	1186	764.7
10	14	10
585531.04	1406596	584766.09
58553.104	100471.1429	58476.609

4. Error Sum of squares:

5. Degrees of freedom:

Total =n-1=34-1=33

Position: p-1=3-1=2

Error=Total-position=33-2=31

Source	df	SS	MS	Fratio	Fcritical
Position	2	556.3615	278.1808	1.3491	3.3048
Error	31	6391.8941	206.190
Total	33	6948.2556

Since the F-ratio for Position<Fcritical, we do not reject the Null hypothesis and hence we conclude that the Differences among the positions are not statistically significant.

Since there are no significant difference among the positions, then there is no need to investigate further.