Applied Statistics and Probability for Engineers

Notice that the contrast constants are orthogonal. Using the data from Table S13-1, we find the

numerical values of the contrasts and the sums of squares as follows:

These contrast sums of squares completely partition the treatment sum of squares; that is,

SSTreatmentsSScSSdSSe. These tests on the contrasts are usually incorporated in the

analysis of variance, such as is shown in Table S13-1. From this analysis, we conclude that

there are significant differences between hardwood concentration 3 and 4, and 1 and 2, but

that the average of 1 and 4 does not differ from the average of 2 and 3. Also, the average of 1

and 3 differs from the average of 2 and 4.

Tukey’s Method

The Tukey procedure for comparing pairs of means makes use of the studentized range

statistic

where and are the largest and smallest sample means, respectively, out of a group

of psample means. For equal sample sizes, the Tukey procedure would indicate that the

two means iand jare different if the absolute value of the observed difference

exceeds

where g(a,f) is the upper percentage point of the studentized range statistic, ais the num-

ber of treatments, and fis the number of even degrees of freedom. Tables of g(a, f) are

Tg 1 a, f 2

B

MSE

n

0 yi. yj. 0

Ymax Ymin

Q

Ymax Ymin

1 MSEn

e 60  94 102  127  59 SSe

15922

6142

145.04

d 60 94  102  127  75 SSd

17522

6142

234.38

c 60 94 102  127  9 SSc

1   922

6142

3.38

13-4

Table S13-1 Analysis of Variance for the Tensile Strength Data

Sum of Degrees of Mean

Source of Variation Squares Freedom Square f 0

Hardwood concentration 382.79 3 127.60 19.61

c(1, 4 vs. 2, 3) 3.38 1 3.38 0.52

d(1, 2 vs. 3, 4) 234.38 1 234.38 36.00

e(1, 3 vs. 2, 4) 145.04 1 145.04 22.28

Error 130.17 20 6.51

Total 512.96 23

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