Question

A consumer preference study involving three different bottle designs (A, B, and C) for the jumbo size of a new liquid laundry detergent was carried out using a randomized block experimental design, with supermarkets as blocks. Specifically, four supermarkets were supplied with all three bottle designs, which were priced the same. The following table gives the number of bottles of each design sold in a 24-hour period at each supermarket. If we use these data, SST, SSB, and SSE can be calculated to be 586.1667, 421.6667, and 1.8333, respectively.

Results of a Bottle Design Experiment
	Supermarket, j
Bottle Design, i	1	2	3	4
A	16	14	1	6
B	33	30	19	23
C	23	21	8	12

(a&b) Test the null hypothesis H₀ that no differences exist between the effects of the bottle designs and supermarkets on mean daily sales. Set α = .05. Can we conclude that the different bottle designs have different effects on mean sales? (Round F to 2 decimal places and SS, MS to 3 decimal places. Leave no cells blank - be certain to enter "0" wherever required.)

Analysis of Variance for factorl

Tukey q_.05 = 4.34, MSE = .306, s = .553, b = 4

Source	DF	SS	MS	F	P
Bottle
Market
Error
Total

(c) Use Tukey simultaneous 95 percent confidence intervals to make pairwise comparisons of the bottle design effects on mean daily sales. Which bottle design(s) maximize mean sales? (Round your answers 2 decimal places. Negative amounts should be indicated by a minus sign.)

AB:	[ ,  ]
AC:	[ ,  ]
BC:	[ ,  ]

  Bottle design (A, B, OR C) maximizes sales.

Answer 1

a & b)

using excel data analysis tool for two factor anova, following o/p Is obtained :

write data>menu>data>data analysis>anova :two factor without replication>enter required labels>ok, and following o/p Is obtained,

Anova: Two-Factor Without Replication
SUMMARY	Count	Sum	Average	Variance
A	4	37	9	48.917
B	4	105	26	40.917
C	4	64	16	51.333
sample 1	3	72	24	73.000
`SAMPLE 2	3	65	142	64.333
SAMPLE 3	3	28	134	82.333
SAMPLE 4	3	41	13.667	74.333
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Rows	586.17	2	293.08	959.18	0.00	5.14
Columns	421.67	3	140.56	460.00	0.00	4.76
Error	1.83	6	0.31
Total	1009.67	11

ANOVA
Source of Variation	SS	df	MS	F-stat	p-value
Rows	586.167	2	293.083	959.18	0.0000
Columns	421.667	3	140.556	460.00	0.0000
Error	1.833	6	0.306
Total	1009.667	11

c)

	A	B	C
count, ni =	4	4	4
mean , x̅ i =Σxi / ni	9.250	26.250	16.000

Level of significance	0.05
no of treatments	3
df error	6
MSE	0.3056
q-statistic value	4.34

Tukey Kramer test
critical value = q*√(MSE/2*(1/ni+1/nj))

confidence interval = mean difference ± critical value

if confidence interval contans zero, then means are not different.

			confidence interval
population mean difference		critical value	lower limit	upper limit	result
AB	-17.0	1.1995	-18.20	-15.80	means are different
AC	-6.8	1.1995	-7.95	-5.55	means are different
BC	10.3	1.1995	9.05	11.45	means are different

  Bottle design ( B ) maximizes sales.