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 19 32 31 31 29 32 26 19 18 23 The Excel output of a one-way ANOVA of the Bottle Design Study Data is shown below SUMMARY Count Sum Average Variance Design A Design B Design C 83 155 120 16.6 31.0 24.0 8.3 6.0 ANOVA Source of Variation Between Groups Within Groups Total F crit 3.88529 MS P.Value 3.23E-06 518.5333 259.2667 49.23 63.2 12.0 5.2667 581.7333 (a) Test the null hypothesis that μΑ, μΒ, and μ'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 "O" wherever required.) Click to selecti Ho: bottle design (Click to select) " have an impact on sales b) Consider the pair ise differences μΒ-μΑ, μ^-MA and 4c-Mg 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 C-A Bottle design (Click to select) maximizes sales. c) Find a 95 percent confidence interval for each of the treatment means μΑ, μΒ and μc Interpret these intervals Negative amounts should be indicated by a minus sign.) Round your answers to 2 decimal places. Confidence interval

Answer 1

One-way ANOVA: Data versus Bottle design

Source DF SS MS F P
Bottle design 2 518.53 259.27 49.23 0.000
Error 12 63.20 5.27
Total 14 581.73

S = 2.295 R-Sq = 89.14% R-Sq(adj) = 87.33%

Individual 95% CIs For Mean Based on Pooled StDev
Level N Mean StDev -+---------+---------+---------+--------
A 5 16.600 2.881 (---*----)
B 5 31.000 1.225 (---*---)
C 5 24.000 2.449 (---*---)
-+---------+---------+---------+--------
15.0 20.0 25.0 30.0

Pooled StDev = 2.295

Grouping Information Using Tukey Method

Bottle
design N Mean Grouping
B 5 31.000 A
C 5 24.000 B
A 5 16.600 C

Means that do not share a letter are significantly different.

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Bottle design

Individual confidence level = 97.94%

Bottle design = A subtracted from:

Bottle
design Lower Center Upper ----+---------+---------+---------+-----
B 10.531 14.400 18.269 (----*----)
C 3.531 7.400 11.269 (----*----)
----+---------+---------+---------+-----
-8.0 0.0 8.0 16.0

Bottle design = B subtracted from:

Bottle
design Lower Center Upper ----+---------+---------+---------+-----
C -10.869 -7.000 -3.131 (----*----)
----+---------+---------+---------+-----
-8.0 0.0 8.0 16.0

$(a)~F=49.23\\ p-value=0.00\\ Reject~H_0:~bottle~design~significantly~have~an~impact~on~scales.\\ (b)~\mu_B-\mu_A:~14.40,~[10.53,~ 18.27]\\ \mu_C-\mu_A:~7.40,~[ 3.53,~11.27 ]\\ \mu_C-\mu_B:~ -7.00,~[-10.87,~ -3.13]\\ Bottle~design~B~maximizes~sales.\\ (Since~C.I.~of~\mu_C-\mu_B~contains~all~negative~values~so,~\mu_B>\mu_C~and~from~other~two~C.I.s~we~infer\\ ~\mu_B>\mu_A,~\mu_C>\mu_A,~so~\mu_B>\mu_C>\mu_A).\\$$(c)~\bar{x}_A=16.6,~\bar{x}_B=31,~\bar{x}_C=24,~MSE=5.27,~t_{0.025,12}=2.1788\\~95\%~C.I.~for~\mu_A~is\\ \left[\bar{x}_A-t_{0.025,12}\sqrt{MSE/5},~ \bar{x}_A+t_{0.025,12}\sqrt{MSE/5}\right ]=[14.36,~18.84]\\ 95\%~C.I.~for~\mu_B~is\\ \left[\bar{x}_B-t_{0.025,12}\sqrt{MSE/5},~ \bar{x}_B+t_{0.025,12}\sqrt{MSE/5}\right ]=[28.76,~33.24]\\ 95\%~C.I.~for~\mu_C~is\\ \left[\bar{x}_C-t_{0.025,12}\sqrt{MSE/5},~ \bar{x}_C+t_{0.025,12}\sqrt{MSE/5}\right ]=[21.76,~26.24]\\$