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Two-way analysis of variance (ANOVA) with replication

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An experiment is conducted to study the effects of two sales approaches-high-pressure (H) and low-pressure (L)-and to study the effects of two sales pitches (1 and 2) on the weekly sales of a product. The data in Table (1) below are obtained by using a completely randomized design, and Figure (2) gives the Excel output of a two-way ANOVA of the sales experiment data.

a Perform graphical analysis to check for interaction between sales pressure and sales pitch.

b Test for interaction by setting a = .05.

c Test for differences in the effects of the levels of sales pressure by setting a _ .05. That is, test the significance of sales pressure effects with a = .05.

d Calculate and interpret a 95 percent individual confidence interval for
uH - uL

e Test for differences in the effects of the levels of sales pitch by setting a = .05. That is, test the significance of sales pitch effects with a = .05.

f Calculate and interpret a 95 percent individual confidence interval for
u1 - u2.

g Calculate a 95 percent (individual) confidence interval for mean sales when using high sales pressure and sales pitch 1. Interpret this interval.

(1) Results of the Sales Approach Experiment

Sales Pressure
1 2
H
32 32
29 30
30 28

L 28 25
25 24
23 23

(2) Excel Output of a Two-Way ANOVA of the
Sales Approach Data

Anova: Two-Factor with Replication

SUMMARY One Two Total
H
Count 3 3 6
Sum 91 90 181
Average 30.33333 30 30.16667p
Variance 2.333333 4 2.566667

L
Count 3 3 6
Sum 76 72 148
Average 25.33333 24 24.66667q
Variance 6.333333 1 3.466667

Total
Count 6 6
Sum 167 162
Average 27.83333r 27s
Variance 10.96667 12.8

ANOVA
Source of
Variation SS df MS F p-Value F Crit
Pressure 90.75 (a) 1 90.75 (f) 26.56098 (j) 0.00087 (k) 5.317645
Pitch 2.083333 (b) 1 2.083333 (g) 0.609756 (l) 0.457362 (m) 5.317645
Interaction 0.75 (c) 1 0.75 (h) 0.219512 (n) 0.651913 (o) 5.317645
Within 27.33333 (d) 8 3.416667(i)
Total 120.9167 (e) 11

(a) SS(1)
(b) SS(2)
(c) SS(int)
(d) SSE
(e) SSTO
(f) MS(1)
(g) MS(2)
(h) MS(int)
(i) MSE
(j) F(1)
(k) p-value for F(1)
(l) F(2)
(m) p-value for F(2)
(n) F(int)
(o) p-value for F(int)

Solution Summary

In this solution we analyze and interpret a two-way analysis of variance (ANOVA). We show a graphical analysis to check for interaction along with a formal test for interaction. We calculate a confidence interval for the difference in means (main effects) and confidence interval for a cell mean.

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