Question

The Castle Bakery Company supplies wrapped Italian bread to a large number of supermarkets in a metropolitan area. An experimental study was made of the effects of heights of the shelf display (Factor A: bottom, middle, top) and the width of the shelf display (Factor B: regular, wide) on sales of this bakery's bread. Twelve supermarkets, similar in terms of sales volume and clientele, were utilized in the study. The six treatments were assigned at random to two stores.

Factor B

Sales (Factor A) Regular Wide

Bottom 47, 43 46, 40

Middle 62, 68 67, 71

Top 41, 39 42, 46

a) Test whether or not the two factors interact; use level of significance a=0.05. State the alternatives, decision rule, and conclusion.

b) Test for display height (factor A) main effects; use level of significance a=0.05. State the alternatives,decision rule, and conclusion.

c) Test for display height (factor B) main effects; use level of significance a=0.05. State the alternatives,decision rule, and conclusion.

d). Test simultaneously all pairwise differences among the shelf height means, using the Tukey multiple comparison procedure with family significance level a=0.05.

Answer 1

MINITAB used.

The Castle Bakery Company supplies wrapped Italian bread to a large number of supermarkets in a metropolitan area. An experimental study was made of the effects of heights of the shelf display (Factor A: bottom, middle, top) and the width of the shelf display (Factor B: regular, wide) on sales of this bakery's bread. Twelve supermarkets, similar in terms of sales volume and clientele, were utilized in the study. The six treatments were assigned at random to two stores.

Factor B

Sales (Factor A) Regular Wide

Bottom 47, 43 46, 40

Middle 62, 68 67, 71

Top 41, 39 42, 46

a) Test whether or not the two factors interact; use level of significance a=0.05. State the alternatives, decision rule, and conclusion.

Ho: there is no interaction between Factor A and Factor B

H1: there is an interaction between Factor A and Factor B

Calculated F=1.16, P=0.375 which is > 0.05 level of significance. Ho is not rejected.

We conclude that interaction is not significant.

b) Test for display height (factor A) main effects; use level of significance a=0.05. State the alternatives,decision rule, and conclusion.

Ho: there is No Factor A effect on sales

H1: there is Factor A effect on sales

Calculated F=74.71, P=0.000 which is < 0.05 level of significance. Ho is rejected.

We conclude that Factor effect A significant.

c) Test for display height (factor B) main effects; use level of significance a=0.05. State the alternatives,decision rule, and conclusion.

Ho: there is No Factor B effect on sales

H1: there is Factor B effect on sales

Calculated F=1.16, P=0.323 which is > 0.05 level of significance. Ho is not rejected.

We conclude that Factor effect B is not significant.

d). Test simultaneously all pairwise differences among the shelf height means, using the Tukey multiple comparison procedure with family significance level a=0.05.

Pairwise differences among the shelf height means shows that Middle is significantly different from top and bottom.

Top and bottom are not significant.

MINITAB output:

General Linear Model: sales versus A, B

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
A	Fixed	3	bottom, middle, top
B	Fixed	2	regular, wide

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
A	2	1544.00	772.00	74.71	0.000
B	1	12.00	12.00	1.16	0.323
A*B	2	24.00	12.00	1.16	0.375
Error	6	62.00	10.33
Total	11	1642.00

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
3.21455	96.22%	93.08%	84.90%

Comparisons for sales

Tukey Pairwise Comparisons: A

Grouping Information Using the Tukey Method and 95% Confidence

A	N	Mean	Grouping
middle	4	67	A
bottom	4	44		B
top	4	42		B

Means that do not share a letter are significantly different.

Tukey Simultaneous Tests for Differences of Means

Difference of A Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
middle - bottom	23.00	2.27	(16.02, 29.98)	10.12	0.000
top - bottom	-2.00	2.27	(-8.98, 4.98)	-0.88	0.671
top - middle	-25.00	2.27	(-31.98, -18.02)	-11.00	0.000

Individual confidence level = 97.80%