Question

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 14 31 24 17 32 25 13 29 28 14 30 27 15 34 28 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 73 14.6 2.3 Design B 5 156 31.2 3.7 Design C 5 132 26.4 3.3 ANOVA Source of Variation SS df MS F P-Value F crit Between Groups 729.7333 2 364.8667 117.70 3.23E-06 3.88529 Within Groups 37.2 12.0 3.1000 Total 766.9333 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. Leave no cells blank - be certain to enter "0" wherever required.) F 117.70 p-value 0.00 H0: bottle design have an impact on sales. (b) Consider the pairwise differences μB – μA, μC – μA , and μC – μB. 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 μB –μA: 16.60 , [ 13.63 , 19.57 ] μC –μA: 11.80 , [ 8.83 , 14.77 ] μC –μB: -4.80 , [ -7.77 , -1.83 ] Bottle design maximizes sales. (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.) Confidence interval μA: [ 13.12 , 16.08 ] μB: [ 29.32 , 33.08 ] μC: [ 24.63 , 28.17 ].

Answer 1

(a) Since the p-value is 0.00 which is less than the level of significance 0.05 we can reject the null hypothesis and conclude that at least one bottle design has a different effect on mean daily sales.

(b) The difference between mean sales of B and A is 16.60, the difference between mean sales of C and A is 11.80, and the difference between mean sales of C and B is -4.80. The 95% confidence interval for the difference between mean sales of B and A is 13.63 to 19.57 that means if we take more samples then 95% of the times the difference between mean sales of B and A will lie in this range. The 95% confidence interval for the difference between mean sales of C and A is 8.83 to 14.77 that means if we take more samples then 95% of the times the difference between mean sales of C and A will lie in this range. The 95% confidence interval for the difference between mean sales of C and B is -7.77 to -1.83 that means if we take more samples then 95% of the times the difference between mean sales of C and B will lie in this range.

The bottle design B maximizes mean daily sales because the mean difference between B and A is maximum.

(c) The 95% confidence interval for the mean effect of daily sales by design A is 13.12 to 16.08 that means if we take more samples then 95% of the times the mean effect of daily sales by design A will lie in this range.

The 95% confidence interval for the mean effect of daily sales by design B is 29.32 to 33.08 that means if we take more samples then 95% of the times the mean effect of daily sales by design B will lie in this range.

The 95% confidence interval for the mean effect of daily sales by design C is 24.63 to 28.17 that means if we take more samples then 95% of the times the mean effect of daily sales by design C will lie in this range.