Applied Statistics and Probability for Engineers

476 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE

EXAMPLE 13-1 Consider the paper tensile strength experiment described in Section 13-2.1. We can use the

analysis of variance to test the hypothesis that different hardwood concentrations do not affect

the mean tensile strength of the paper.

The hypotheses are

.

We will use 0.01. The sums of squares for the analysis of variance are computed from

Equations 13-8, 13-9, and 13-10 as follows:

The ANOVA is summarized in Table 13-4. Since f0.01,3,204.94, we reject H 0 and conclude

that hardwood concentration in the pulp significantly affects the mean strength of the paper.

We can also find a P-value for this test statistic as follows:

Since is considerably smaller than 0.01, we have strong evidence to

conclude that H 0 is not true.

Minitab Output

Many software packages have the capability to analyze data from designed experiments using

the analysis of variance. Table 13-5 presents the output from the Minitab one-way analysis of

variance routine for the paper tensile strength experiment in Example 13-1. The results agree

closely with the manual calculations reported previously in Table 13-4.

P3.59 

10 ^6

PP 1 F3,20 19.60 2 3.59 

10 ^6

512.96382.79130.17

SSESSTSSTreatments



16022  19422  110222  112722

6



138322

24

382.79

SSTreatments a

4

i 1

y^2 i.

n 

y^2 ..

N

 1722  1822 p 12022 

138322

24

512.96

SST a

4

i 1

(^) a
6
j 1
y^2 ij
y..^2
N
H 1 : i 0 for at least one i
H 0 :  1  2  3  4  0
Table 13-4 ANOVA for the Tensile Strength Data
Source of Degrees of
Variation Sum of Squares Freedom Mean Square f 0 P-value
Hardwood
concentration 382.79 3 127.60 19.60 3.59 E-6
Error 130.17 20 6.51
Total 512.96 23
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