Question

The Kenton Food Company wished to test four different package designs

for a new breakfast cereal. Twenty stores, with approximately equal sales volumes, were

selected as the experimental units. Each store was randomly assigned one of the package

designs, with each package design assigned to five stores. A fire occurred in one store during

the study period, so this store had to be dropped from the study. Hence, one of the designs

were tested in only four stores. The stores were chosen to be comparable in location and

sales volume. Other relevant conditions that could affect sales, such as price, amount and

location of shelf space, and special promotional efforts, were kept the same for all of the

stores in the experiment. Sales, in number of cases, were obtained for the study period.

Each of Four Package Designs
			Store (replication)
		Package Design	1	2	3	4	5
		1	11	17	16	14	15
		2	12	10	15	19	11
		3	23	20	18	17
		4	27	33	22	26	28

Conduct a test to determine whether or not the mean response rates for the packages differ. Solve the hypothesis testing about the equivalence of the package designs. Use level of significance a=.10. State the alternatives, decision rule, and conclusion. What is the p-value of the test.

Answer 1

Since, the effecting variables are all controlled for different stores, this problem can be modeled using one way ANOVA, where -

Null hypothesis - There is no significant difference in the mean response rate for all the pairs of package design

Alternate hypothesis - There is significant difference in the mean response rate for at least one pair of package design

Package Design	1	2	3	4	5
1	11	17	16	14	15
2	12	10	15	19	11
3	23	20	18	17
4	27	33	22	26	28

Summary of Data
	Treatments
	1	2	3	4	Total
N	5	5	4	5	20
∑X	73	67	78	136	370
Mean	14.6	13.4	19.5	27.2	18.5
∑X2	1087	951	1352	3762	7598
Std.Dev.	2.3	3.6	2.3	4	6.3

Result Details
Source	SS	df	MS
Between-treatments	585	3	195	F = 18.57143
Within-treatments	158	15	10.5
Total	753	18

Where SS = sum of squared errors and df = degrees of freedom

F = Ratio variance between between-treatments variance and within-treatments variance

Decision rule -

The p-value for F value = 18.57 and numerator df =3 and denominator df = 15 from the normal distribution table is .000018

The p value is less than 0.1 (significance level)

Hence, the result is significant

There is significant difference in the mean response rate for at least one pair of package design