> 49 4. A statistics professor conducts an experiment to compare the effectivene
ID: 3266233 • Letter: #
Question
> 49 4. A statistics professor conducts an experiment to compare the effectiveness of two methods of teaching his course. Method l is the usual way he teaches the course: lectures, homework assignments, and a final exam. Method Il is the same as method I, except the students receiving method Il get 1 additional hour per week in which they solve illustrative problems under the guidance of the professor. Since the professor is also interested in how the methods affect students of differing mathematical abilities, volunteers for the experiment are subdivided according to mathematical ability into superior, average, and poor groups. Five students from each group are randomly assigned to method I and five students from each group to method Il. At the end of the course, all 30 students take the final exam. Scores are the number of points received out of a total of 50 possible points. The final exam scores are shown below. (25 points):Explanation / Answer
Solution
This is a case of ANOVA 2-way classification with equal number of observations per cell.
Back-up Theory
Suppose we have data of a 2-way classification ANOVA, with r rows, c columns and n observations per cell.
Let xijk represent the kth observation in the ith row-jth column, k = 1,2,…,n; i = 1,2,……,r ; j = 1,2,…..,c.
Then the ANOVA model is: xijk = µ + i + j + ij + ijk, where µ = common effect, i = effect of ith row, j = effect of jth column, ij = row-column interaction and ijk is the error component which is assumed to be Normally Distributed with mean 0 and variance 2.
Now, to work out the solution,
Terminology:
Cell total = xij. = sum over k of xijk
Row total = xi..= sum over j of xij.
Column total = x.j. = sum over i of xij.
Grand total = G = sum over i of xi.. = sum over j of x.j.
Correction Factor = C = G2/N, where N = total number of observations = r x c x n =
Total Sum of Squares: SST = (sum over i,j and k of xijk2) – C
Row Sum of Squares: SSR = {(sum over i of xi..2)/(cxn)} – C
Column Sum of Squares: SSC = {(sum over j of x.j.2)/(rxn)} – C
Between Sum of Squares: SSB = {(sum over i and jof xij.2)/n} – C
Interaction Sum of Squares: SSI = SSB – SSR – SSC
Error Sum of Squares: SSE = SST – SSB
Mean Sum of Squares = Sum of squares/Degrees of Freedom
Degrees of Freedom:
Total: N (i.e., rcn) – 1;
Between: rc – 1;
Within(Error): DF for Total – DF for Between;
Rows: (r - 1);
Columns: (c - 1);
Interaction: DF for Between – DF for Rows – DF for Columns;
Fobs:
for Rows: MSSR/MSSE;
for Columns: MSSC/MSSE;
for Interaction: MSSI/MSSE
Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for the numerator MSS and n2 is the DF for the denominator MSS of Fobs
Significance: Fobs is significant if Fobs > Fcrit
Calculations:
Excel Calculations
r
3
c
2
n
5
N
30
i
j
xijk; k =
xij.
Xijk square
sum
xij. square
Row sum
xi..
Row sum
sq/cn
Col sum
x.j.
1
2
3
4
5
1
1
39
41
48
42
44
214
9206
45796
448
20070.4
585
2
49
47
47
48
43
234
10972
54756
634
2
1
43
36
40
35
42
196
7734
38416
411
16892.1
49612.07
2
38
46
45
42
44
215
9285
46225
3
1
30
33
29
36
47
175
6335
30625
360
12960
2
37
41
34
40
33
185
6895
34225
Excel Summary
G
1219
C
49532.03
Sumxijk^2
50427
Sumxij.^2
250043
Sumxi..^2
49922.5
Sumx.j.^2
49612.07
SST
894.9667
SSB
476.5667
SSR
390.4667
SSC
80.03333
SSI
6.066667
SSE
418.4
ANOVA TABLE
Source
DF
SS
MSS
Fobs
Fcrit
Row
2
390.4667
195.2333
11.19885
3.402826
Column
1
80.03333
80.03333
4.590822
4.259677
Interaction
2
6.066667
3.033333
0.173996
3.402826
Between
5
476.5667
95.31333
Error
24
418.4
17.43333
Total
29
894.9667
30.86092
Since Fobs > Fcrit, for row and column, at 5% level of significance, both row effect and column effect are significant => we conclude that teaching method has an effect on the test score and the average scores of the three groups of students are significantly different.
However, since Fobs < Fcrit, interaction effect does not exist.
r
3
c
2
n
5
N
30
i
j
xijk; k =
xij.
Xijk square
sum
xij. square
Row sum
xi..
Row sum
sq/cn
Col sum
x.j.
1
2
3
4
5
1
1
39
41
48
42
44
214
9206
45796
448
20070.4
585
2
49
47
47
48
43
234
10972
54756
634
2
1
43
36
40
35
42
196
7734
38416
411
16892.1
49612.07
2
38
46
45
42
44
215
9285
46225
3
1
30
33
29
36
47
175
6335
30625
360
12960
2
37
41
34
40
33
185
6895
34225
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