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Objectives Aner c«npJaÉtg this chapter. you shcndd be abk to n use the ore-way ANOVA technque to 'f there B a differnce •rung three or more rnems. which mems differ, lÄng he Sct•Hfe or test rf he repcted in ttr ANOVA Use the ANOVA to detem•tirE ttzre a differnce n mul eff«ts or 12 Analysis of Variance Outline 1.2—1 One-Way AnosisofVuiance 12—2 and the Test 12—3 Two%yhatysis ofWrOnce Summary. 

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Objectives
Aner c«npJaÉtg this chapter. you shcndd be abk to
n
use the ore-way ANOVA technque to
'f there B a differnce
•rung three or more rnems.
which mems differ, lÄng he
Sct•Hfe or test rf he
repcted in ttr ANOVA
Use the ANOVA to
detem•tirE ttzre a differnce
n mul eff«ts or
12
Analysis of Variance
Outline
1.2—1 One-Way AnosisofVuiance
12—2 and the Test
12—3 Two%yhatysis ofWrOnce
Summary

Figure 9-2 Hypothesis-Testing Situations in the Comparison of Means Sample 1 Sample 2 Population (a) Difference is not significant Do not reject Ho: = since RI — R2 is not significant.. 

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Figure 9-2
Hypothesis-Testing Situations in the Comparison of Means
Sample 1
Sample 2
Population
(a) Difference is not significant
Do not reject Ho: = since RI — R2 is not significant.

Sample 1
Sample 2
Population 1
Population 2
(b) Difference is significant
Reject Ho: since RI —R2 is significant

HYPHOTHESES TESTING. These tests can also be one-tailed, using the following hypotheses: Right-tailed Ho: = or HI: — > O Left-tailed Ho: = or The same critical values used in Section 8—2 are used here. They can be obtained from Table E in Appendix C. If and are not known, the researcher can use the variances from each sample s} and s}, but a t test must be used. This will be explained in Section 9—2. The basic format for hypothesis testing using the traditional method is reviewed here.. 

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These tests can also be one-tailed, using the following hypotheses:
Right-tailed
Ho: =
or
HI: — > O
Left-tailed
Ho: =
or
The same critical values used in Section 8—2 are used here. They can be obtained
from Table E in Appendix C.
If and are not known, the researcher can use the variances from each sample
s} and s}, but a t test must be used. This will be explained in Section 9—2.
The basic format for hypothesis testing using the traditional method is reviewed
here.

What is a variance?. The variance is the average of the squares of the distance each value is from the mean. The symbol for the population variance is (u is the Greek lowercase letter sigma). The formula for the population variance is where X = individual value = population mean N — population size The standard deviation is the square root of the variance. The symbol for the population standard deviation is u. The corresponding formula for the population standard deviation is. 

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The variance is the average of the squares of the distance each value is from the mean.
The symbol for the population variance is (u is the Greek lowercase letter sigma).
The formula for the population variance is
where
X
= individual value
= population mean
N — population size
The standard deviation is the square root of the variance. The symbol for the
population standard deviation is u.
The corresponding formula for the population standard deviation is

Step 1 step 2 step 3 Find the variance and standard deviation for brand B paint data in Example 3—18. The months were 35, 45, 30, 35, 40, 25 Solution —10 Find the mean. EA' 35 + 45 + 30 + 35 + 40+25 210 = — 35 Subtract the mean from each value, and place the result in column B of the table. Square each result and place the squares in column C of the table. 35 45 30 35 25 10 25 25 100. 

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Step 1
step 2
step 3
Find the variance and standard deviation for brand B paint data in Example 3—18.
The months were
35, 45, 30, 35, 40, 25
Solution
—10
Find the mean.
EA' 35 + 45 + 30 + 35 + 40+25 210
= — 35
Subtract the mean from each value, and place the result in column B of the table.
Square each result and place the squares in column C of the table.
35
45
30
35
25
10
25
25
100

step 4
step 5
step 6
Find the sum of the squares in column C.
E(X— = o + 100 + 25 + o + 25 + 100 250
Divide the sum by Nto get the variance.
250
N
41.7
6
Take the square root to get the standard deviation.
= vaj-:ä 6.5
Hence, the standard deviation is 6.5.

Standard Deviation. Formula for the Sample Standard Deviation The standard deviation ofa sample (denoted by s) is s2 = 1 where X = individual value = sample mean n = sample size. 

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Formula for the Sample Standard Deviation
The standard deviation ofa sample (denoted by s) is
s2 =
1
where
X = individual value
= sample mean
n = sample size

Objectives Aner this chapter. you shmald be abk to Use the ore-way ANOVA to detemvne there a difference •nong three or more rnems. which mems differ, uung he Sct•Hfe or Tukey test rf he roll retected in the ANOVA Use the two-way ANOVA to determirE f there is a dtfference rnul effects or Analysis of Variance Outline 12—1 0*Way k'Osis ofVuiance 1.2-2 the TIAeyTest 12—3 Two-Way Etatysis ofWrOnce Summary. 

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Objectives
Aner this chapter. you shmald be abk to
Use the ore-way ANOVA to
detemvne there a difference
•nong three or more rnems.
which mems differ, uung he
Sct•Hfe or Tukey test rf he roll
retected in the ANOVA
Use the two-way ANOVA to
determirE f there is a dtfference
rnul effects or
Analysis of Variance
Outline
12—1 0*Way k'Osis ofVuiance
1.2-2 the TIAeyTest
12—3 Two-Way Etatysis ofWrOnce
Summary

ANALYSIS OF VARIANCE