Question

In: Statistics and Probability

A consumer preference study compares the effects of three different bottle designs (A, B, and C)...

A consumer preference study compares the effects of three different bottle designs (A, B, and C) on sales of a popular fabric softener. A completely randomized design is employed. Specifically, 15 supermarkets of equal sales potential are selected, and 5 of these supermarkets are randomly assigned to each bottle design. The number of bottles sold in 24 hours at each supermarket is recorded. The data obtained are displayed in the following table.

Bottle Design Study Data
A B C
19 29 24
18 33 23
16 32 26
17 33 23
18 32 24
SUMMARY
Groups Count Sum Average Variance
Design A 5 88 17.6 1.3
Design B 5 159 31.8 2.7
Design C 5 120 24.0 1.5
ANOVA
Source of Variation SS df MS F P-Value F crit
Between Groups 505.7333 2 252.8667 137.93 5.25E-09 3.88529
Within Groups 22.0 12.0 1.8330
Total 527.7333 14

(a) Test the null hypothesis that μA, μB, and μC are equal by setting α = .05. Based on this test, can we conclude that bottle designs A, B, and C have different effects on mean daily sales? (Round your answers to 2 decimal places.)

F=______ ; P-Value = ______; (reject/do not reject) H0: bottle design (Does/Does not) have an impact on sales.

(b) Consider the pairwise differences μBμA, μCμA , and μCμB. Find a point estimate of and a Tukey simultaneous 95 percent confidence interval for each pairwise difference. Interpret the results in practical terms. Which bottle design maximizes mean daily sales? (Round your answers to 2 decimal places. Negative amounts should be indicated by a minus sign.)

Ub-Ua: (confidence interval) [    ]
Uc-Ua [ ]
Uc-Ub [ ]

(c) Find a 95 percent confidence interval for each of the treatment means μA, μB, and μC. Interpret these intervals. (Round your answers to 2 decimal places. Negative amounts should be indicated by a minus sign.)

Ua:[      
Ub:[
Uc:[

Solutions

Expert Solution

a)
F =137.93

p value =0.000

reject Ho, bottle design Does have an impact on sales.

b)

pooled standard deviation        =Sp =√MSE      =                                                                      1.354
critical q with 0.05 level and at k=3 and N-k=12 degree of freedom= 3.77
Tukey's (HSD) for group i and j =                  (q/√2)*(sp*√(1/ni+1/nj)         = 2.28
Confidence interval
Lower bound Upper bound
(xi-xj ) (xi-xj)-ME (xi-xj)+ME
μB-μA 14.20 11.92 16.48
μC-μA 6.40 4.12 8.68
μB-μA -7.80 -10.08 -5.52

c)

lower Upper
bound bound
A 16.28 18.92
B 30.48 33.12
C 22.68 25.32

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