Samsung is designing a new smartphone. They have narrowed their choices to two s
ID: 3350767 • Letter: S
Question
Samsung is designing a new smartphone. They have narrowed their choices to two sizes (5 and 7 inches) and two colors (red and blue) and want to decide between the four options. At this stage of the development, Samsung is using the response of a sample of 20 people on how much they each liked the one design shown to them to decide. Use the data below to run a factorial Anova. For the size, the color, and their interaction: Provide the hypothesis tested, the p-value, the decision rule, whether you accept or reject the chosen Ho at a confidence level of 95%. Provide also a verbal interpretation of the overall outcome. Finally, as a manager with Samsung, which phone design should you continue to develop and why? Additional Computer output: provide the anova table and the profile plot. Respondent Size Color Like 10 4 10 10 4 15 4 4 20Explanation / Answer
ANOVA – 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:
Let xijk = like score for kth respondent of ith size and jth colour – i = 1 for size 5, 2 for size 7; j = 1 for red and 2 for blue.
xijk are tabulated below:
Size (i)
Colour (j)
Red (1)
Blue (2)
5 (1)
7
6
7
3
10
9
10
10
9
9
7(2)
4
6
7
4
7
6
1
4
9
4
G
132
C
871.2
Sumxijk^2
1002
Sumxij.^2
4658
Sumxi..^2
910.4
Sumx.j.^2
876.2
SST
130.8
SSB
60.4
SSR
39.2
SSC
5
SSI
16.2
SSE
70.4
ANOVA TABLE
Source
DF
SS
MSS
Fobs
Fcrit
Row
1
39.2
39.2
8.909091
4.493998
Column
1
5
5
1.136364
4.493998
Interaction
1
16.2
16.2
3.681818
4.493998
Between
3
60.4
20.13333
Error
16
70.4
4.4
Total
19
130.8
6.884211
Since Fcal is greater than Fcrit for Row only, only Row (size) is significant.
=> there is evidence to suggest that only size has an impact on the respondent rating score. ANSWER
DONE
Size (i)
Colour (j)
Red (1)
Blue (2)
5 (1)
7
6
7
3
10
9
10
10
9
9
7(2)
4
6
7
4
7
6
1
4
9
4
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