Question: A researcher is interested in studying the effect ... Bookmark A resea
ID: 3183571 • Letter: Q
Question
Question: A researcher is interested in studying the effect ... Bookmark A researcher is interested in studying the effect that the amount of fat in the diet and amount of exercise has on the mental acuity of middle-aged women. The researcher used three different treatment levels for the diet and two levels for the exercise. The results of the acuity test for the subjects in the different treatment levels are shown below. Diet Exercise <30% fat 30% - 60% fat >60% fat <60 minutes 4 3 2 4 1 2 2 2 2 4 2 2 3 3 1 60 minutes 6 8 5 or more 5 8 7 4 7 5 4 8 5 5 6 6 Perform a two-way analysis of variance and explain the results. (Show all work to receive full credit) Find the effect size for each factor and the interaction and explain the results. (Show all work to receive full credit)
Diet
Exercise
<30% fat
30% - 60% fat
>60% fat
<60 minutes
4
3
2
4
1
2
2
2
2
4
2
2
3
3
1
60 minutes
6
8
5
or more
5
8
7
4
7
5
4
8
5
5
6
6
Diet
Exercise
<30% fat
30% - 60% fat
>60% fat
<60 minutes
4
3
2
4
1
2
2
2
2
4
2
2
3
3
1
60 minutes
6
8
5
or more
5
8
7
4
7
5
4
8
5
5
6
6
Explanation / Answer
Solution
Back-up Theory
Let xijk represent the result of the acuity test for the kth subject under ith exercise and jth diet, k = 1,2,3,4,5 (n); j = 1,2,3 (c) and I = 1, 2 (r)
Then the ANOVA model is: xijk = µ + i + j + ij + ijk, where µ = common effect, i = effect of ith row (exercise), j = effect of jth column (diet), 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:
In the given question, r = 2, c = 3 and n = 5 and hence N = 30.
Level of significance is not given. It is taken as 5%.
Let, for convenience in explaining and presentation,
E1 and E2 respectively represent exercise of less than 60 minutes and exercise of 60 or more minutes. D1, D2, D3 respectively represent diets with less than 30% fat, 30 to 60% fat and more than 60% fat.
From the given data, cell totals (xij.) are:
Exercise (row) (i)
Diet (column) (j)
Total
(xi..)
D1
D2
D3
E1
17
11
9
37
E2
24
37
28
89
Total (x.j.)
41
48
37
126 (G)
C = 1262/30 = 529.2
SST = (42 + 42 + 22 + ……. + 62) – C = 660 – 529.2 = 130.8
SSR = {(372 + 892)}/15 – C = 619.3333 – 529.2 = 90.1333
SSC = {(412 + 482 + 372)}/10 – C = 535.4 – 529.2 = 6.2
SSB = {(172 + 112 + 92 + 242 + 372 + 282)}/5 – C = 644 – 529.2 = 114.8
SSE = 130.8 – 114.8 = 16
SSI = 114.8 – 90.1333 – 6.2 = 18.4667
ANOVA TABLE
Source of
Variation
Degrees of Freedom
Sum of
Squares
Mean Sum
Of Squares
Fobs
Fcrit
Significance
Row (exercise)
1
90.1333
90.1333
135.19
4.26
Significant
Column (diet)
2
6.2
3.1
4.65
3.40
Significant
Interaction
2
18.4667
9.2334
13.85
3.40
Significant
Between
5
114.8
-
Within
24
16
0.6667
Total
29
130.8
Conclusions
Since all Fobs are significant, there is evidence to suggest that at 5% level of significance, both level of exercise and fat percent in diet have influence on mental acuity of women. Different combinations of these two factors also have varying influence.
Done
Exercise (row) (i)
Diet (column) (j)
Total
(xi..)
D1
D2
D3
E1
17
11
9
37
E2
24
37
28
89
Total (x.j.)
41
48
37
126 (G)
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