The marketing manager of a company producing a new cereal aimed for children wan
ID: 3218265 • Letter: T
Question
The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the color and shape of the box's logo on the approval rating of the cereal. He combined 4 colors and 3 shapes to produce a total of 12 designs. Each logo was presented to 2 different groups (a total of 24 groups) and the approval rating for each was recorded and is shown below. The manager analyzed these data using the = 0.01 level of significance for all inferences, and a partially completed 2-way ANOVA table is provided.
COLORS
SHAPES
Red
Green
Blue
Yellow
Circle
54
67
36
45
44
61
44
41
Square
34
56
36
21
36
58
30
25
Diamond
46
60
34
31
48
60
38
33
Analysis of Variance
Source df SS MS F
Colors
Shapes 579.00
Interaction 150.33
Error 150.00
Total 3590.50
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1) What is the sum of squares due to factor Colors?
2) What are the degrees of freedom for the factor Shapes and the Error ?
3) How many treatment combinations are there in the experiment and how many replicates for each treatment combination?
4) Perform the F- test on 'Colors'. Is the factor 'Colors' significant ( = 0.01)?
5) Perform the F- test on 'Interaction'. Is the factor 'Interaction' significant (=0.01)?
COLORS
SHAPES
Red
Green
Blue
Yellow
Circle
54
67
36
45
44
61
44
41
Square
34
56
36
21
36
58
30
25
Diamond
46
60
34
31
48
60
38
33
Explanation / Answer
Part (1)
Relation between different sum of squares (SS)
Total: SST = SSB (between) + SSW (within or error)
SSB = SSC (color) + SSS (shape) + SSI (interaction)
So, SSB = SST – SSW = 3590.5 – 150 = 3440.5
And hence, SSC = SSB - SSS (shape) - SSI (interaction)
= 3440.5 – 579 – 150.33 = 2708.17 ANSWER
Part (2)
Degrees of freedom for shapes = Number of shapes – 1 = 3 – 1 = 2
Degrees of freedom for colors = Number of colors – 1 = 4 – 1 = 3 ANSWER
Part (3)
Number treatment combinations = 12 (3 shapes x 4 colors)
Number replicates per treatment combinations = 2 ANSWER
Part (4)
To test for significance of color effect,
Test Statistic:
F = Mean sum of squares for Colors/Mean sum of squares for Error
= (SSC/DF)/(SSE/DF) = (2708.17/3)/(150/12) = 451.362/12.5 = 35.263.
Upper 1% point (given = 0.01) of F3, 12 = 5.95 being less than 35.263, the F is significant implying that different colors have different effects.ANSWER
[Degrees of freedom for Total is total number of observations – 1 = 24 – 1 = 23.
Degrees of freedom for Between is total number of treatment combinations – 1 = 12 – 1 = 11.
Degrees of freedom for Error is Total – Between = 23 – 11 = 12.]
Part (5)
To test for significance of (colorxshape) interaction effect,
Test Statistic:
F = Mean sum of squares for Interaction/Mean sum of squares for Error
= (SSI/DF)/(SSE/DF) = (150.3/6)/(150/12) = 25.05/12.5 = 2.004.
Upper 1% point (given = 0.01) of F6, 12 = 4.82 being greater than 2.004, the F is not significant implying that there is no sufficient evidence of color-shape interaction effect. ANSWER
[Degrees of freedom for Interaction = Degrees of freedom for Between - Degrees of freedom for color - Degrees of freedom for shape = 11 – 3 – 2 = 6]
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