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

16 33 23

18 31 27

19 37 21

17 29 28

13 34 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 - 83 - 16.6 - 5.3

Design B - 5 - 164 - 32.8 - 9.2

Design C - 5 - 124 - 24.8 - 8.2

ANOVA Source of Variation - SS - df - MS - F - P-Value - F crit

Between Groups 656.1333 - 2 - 328.0667 - 43.35683 - 3.23E-06 - 3.88529

Within Groups 90.8 - 12 - 7.566667

Total 746.9333 - 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 answer to 2 decimal places.)

F =

p-value =

_ 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: , [, ]

μC –μA: , [, ]

μC –μB: , [, ]

Bottle design (Click to select)ABC maximizes sales.

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

Confidence interval

μA: [, ]

μB: [, ]

μC: [, ]

Answer 1

(a)

One-way ANOVA: No. of bottles versus Design

Source DF      SS      MS      F      P
Design   2 656.13 328.07 43.36 0.000
Error    12   90.80    7.57
Total    14 746.93

F =43.36

p-value =0.00

Since p-value<0.05 we conclude that bottle designs A, B, and C have different effects on mean daily sales.

(b)

Grouping Information Using Tukey Method

Design N    Mean Grouping
   B       5 32.800 A
   C       5 24.800    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 Design

Individual confidence level = 97.94%

Design = A subtracted from:

Design   Lower Center   Upper ---+---------+---------+---------+------
B       11.562 16.200 20.838                           (---*----)
C        3.562   8.200 12.838                   (---*----)
                                ---+---------+---------+---------+------
                                 -10         0        10        20

Design = B subtracted from:

Design    Lower Center   Upper ---+---------+---------+---------+------
C       -12.638 -8.000 -3.362 (----*----)
                                 ---+---------+---------+---------+------
                                  -10         0        10        20

HB HA 11.56, 20.84| C HA 3.56, 12.84] C HB 12.64, -3.36]

(c)

95% C.I. for LA MSE MSE (13.92, 19.28) Ato.025,12 Ato.025,12 where A 16.6, MSE = mean sum 7.57 of square due to error 2.1788 to.025,12 95% C.I. for HB= MSE MSE (30.12, 35.48) to.025,12 TBto.025,12 5 where, B 32.8 95% C.I. for c MSE MSE cto.025,12 (22.12, 27.48) C to.025,12 5 5 where, c 24.8