In: Statistics and Probability
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 | ||||||
15 | 32 | 20 | ||||||
13 | 32 | 26 | ||||||
14 | 32 | 24 | ||||||
19 | 35 | 26 | ||||||
16 | 35 | 25 | ||||||
The Excel output of a one-way ANOVA of the Bottle Design Study Data is shown below.
SUMMARY | ||||
Groups | Count | Sum | Average | Variance |
Design A | 5 | 77 | 15.4 | 5.3 |
Design B | 5 | 166 | 33.2 | 2.7 |
Design C | 5 | 121 | 24.2 | 6.2 |
ANOVA | ||||||
Source of Variation | SS | df | MS | F | P-Value | F crit |
Between Groups | 792.1333 | 2 | 396.0667 | 83.68 | 3.23E-06 | 3.88529 |
Within Groups | 56.8 | 12.0 | 4.7333 | |||
Total | 848.9333 | 14 | ||||
(a) Test the null hypothesis that μ_{A}, μ_{B}, and μ_{C} 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. Leave no cells blank - be certain to enter "0" wherever required.)
F | |
p-value | |
(Click to select) Do not reject Reject H_{0}: bottle design (Click to select) does not does have an impact on sales.
(b) Consider the pairwise differences
μ_{B} – μ_{A},
μ_{C} – μ_{A} , and
μ_{C} – μ_{B}. 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.)
Point estimate Confidence interval |
μ_{B} –μ_{A}: , [ , ] |
μ_{C} –μ_{A}: , [ , ] |
μ_{C} –μ_{B}: , [ , ] |
Bottle design (Click to select) B A C maximizes sales.
(c) Find a 95 percent confidence interval for each of the treatment means μ_{A}, μ_{B}, and μ_{C}. Interpret these intervals. (Round your answers to 2 decimal places. Negative amounts should be indicated by a minus sign.)
Confidence interval |
μ_{A}: [ , ] |
μ_{B}: [ , ] |
μ_{C}: [ , ] |
a)
Test statistic:
F = 83.68
p-value = F.DIST.RT(83.6761, 2, 12) = 0.0000
Reject H0: bottle design does have an impact on sales.
b)
At α = 0.05, k = 3, N-K = 12, Q value = 3.77
Critical Range, CV = Q*√(MSW/n) = 3.77*√(4.7333/5) = 3.67
Comparison | Diff. = (xi - xj) | Critical Range | Confidence interval | |
(xi - xj) - CV | (xi - xj) + CV | |||
x̅B - x̅A | 17.8 | 3.67 | 14.13 | 21.47 |
x̅C - x̅A | 8.8 | 3.67 | 5.13 | 12.47 |
x̅C - x̅B | -9 | 3.67 | -12.67 | -5.33 |
c)
df = n-p = 15-3 = 12
Critical value, t-crit = T.INV.2T(0.05, 12) = 2.179
95% confidence interval for A:
Lower Bound = x̅ - t-crit*√(MSE/n1) = 15.4 - 2.179 *√(4.7333/5) = 13.28
Upper Bound = x̅ + t-crit*√(MSE/n1) = 15.4 + 2.179 *√(4.7333/5) = 17.52
13.28 < µ < 17.52
95% confidence interval for B:
Lower Bound = x̅ - t-crit*√(MSE/n1) = 33.2 - 2.179 *√(4.7333/5) = 31.08
Upper Bound = x̅ + t-crit*√(MSE/n1) = 33.2 + 2.179 *√(4.7333/5) = 35.32
31.08 < µ < 35.32
95% confidence interval for C:
Lower Bound = x̅ - t-crit*√(MSE/n1) = 24.2 - 2.179 *√(4.7333/5) = 22.08
Upper Bound = x̅ + t-crit*√(MSE/n1) = 24.2 + 2.179 *√(4.7333/5) = 26.32
22.08 < µ < 26.32