10-1
CHAPTER 10. ANALYSIS OF VARIANCE
CHAPTER 10. ANALYSIS OF VARIANCE …………………………………………………………………………1
10.1. Introduction ……………………………………………………………………………………………..……………..1
10.2. General Procedure ……………………………………………………………………………………………………1
10.3. Multiple Comparisons in the ANOVA Test …………………………………………………………………4
The following table provides a summary of the problems with their appropriate sections:
Section Problems
10.1 None
10.2 1 to 13
10.1. Introduction
None
10.2. General Procedure
Problem 10-1.
First, four samples of each brand is a small sample
Problem 10-2.
(a) All factors are not considered; the different brands have the same ingredients – the alcohol.
Problem 10-3.
10-2
Problem 10-4.
Problem 10-5.
The parameter msb represents the variation between the sample means while the denominator
Problem 10-6.
A 5% level of significance is commonly used for decision making even though an engineering
Problem 10-7.
The region of rejection consists of all values of F that are unlikely to occur from sample
Problem 10-8.
The k means are sufficiently far apart.
Problem 10-9.
The first term
P
is the grand mean. The second term )(
P
P
j is the deviation of the mean of
Problem 10-10.
10-10(a):
GROUP MEASUREMENTS
—– ————
1 4.00000 6.00000 6.00000 7.00000 8.00000
10-3
-2.9212 = Computed value of the t statistic
8 = Degrees of freedom
Level of Critical t
significance statistic Decision
———— ———- ———
.1000 1.8600 REJECT Ho
part (b)
GROUP MEASUREMENTS
—– ————
1 4.00000 6.00000 6.00000 7.00000 8.00000
TWO-SAMPLE t TEST
-3.2796 = Computed value of the t statistic
8 = Degrees of freedom
Level of Critical t
significance statistic Decision
———— ———- ———
.1000 1.8600 REJECT Ho
Problem 10-11.
Source
X
SS MS F
D
FDecision
Between 3 1397.564 465.855 10.759 3.05 Reject Ho
GROUP MEASUREMENTS
—– ————
1 78.00000 83.00000 65.00000 74.00000 91.00000 83.00000
2 92.00000 81.00000 87.00000 76.00000 94.00000 85.00000
=================================================
10-4
SUMS OF MEAN COMPUTED
VARIATION df SQUARES SQUARE F
——— —- ——— ——– ——–
Between 3. 1397.560 465.853 10.759
Problem 10-12.
Source
X
SS MS F
D
FDecision
Between 5 4
104575.0
u4
100915.0
u7.01 4.25 Reject Ho
Problem 10-13.
Source
X
SS MS F
D
FDecision
Between 3 347179.75 115726.583 50.556 3.49 Reject Ho
10.3. Multiple Comparisons in the ANOVA Test
Problem 10-14.
Source
X
SS MS F
D
FDecision
Between 2 314.133 157.067 25.609 3.89 Reject Ho
Problem 10-15.
y1 = y2 = 10
Mean St. Dev. Var. 2
Sp
D
ttDecision
(a) y1 = y2 = 15
Mean St. Dev. Var. 2
Sp
D
ttDecision
(a) for y1 = y2 = 10
Source SS
X
MS F
D
FDecision
10-5
(b) for y1 = y2 = 15
Source SS
X
MS F
D
FDecision
Between 18.75 1 18.75 1.76 4.96 Accept Ho
Problem 10-16.
GROUP MEASUREMENTS
—– ————
1 13.50000 13.60000 13.40000
2 13.30000 12.90000 13.00000
5 13.40000 13.60000 13.00000
sample group standard coeff of
group size mean deviation variation Se of mean
—– —— ———- ——— ——— ———-
1 3 13.50000 .10000 .0074 .0577
=================================================
SUMS OF MEAN COMPUTED
VARIATION df SQUARES SQUARE F
——— —- ——— ——– ——–
Between 4. 1.729 .432 7.719
Problem 10-17.
GROUP MEASUREMENTS
—– ————
1 66.00000 71.00000 70.00000 65.00000
sample group standard coeff of
group size mean deviation variation Se of mean
—– —— ———- ——— ——— ———-
1 4 68.00000 2.94392 .0433 1.4720
10-6
5 4 67.00000 2.16025 .0322 1.0801
ANOVA TABLE
=================================================
SUMS OF MEAN COMPUTED
VARIATION df SQUARES SQUARE F
Problem 10-18.
(see the ANOVA output for problem 10-17)
RESULTS OF DUNCAN TEST ON EACH PAIR
==========================================================
PAIR MEAN MEAN COMPUTED CRITICAL
I J I J DIFFERENCE DIFFERENCE DECISION
– – ——– ——– ———- ———- ——–
1 2 68.000 74.500 6.500 4.295 REJECT Ho
1 3 68.000 69.500 1.500 4.295 ACCEPT Ho
Problem 10-19.
(see 10-16 for ANOVA table)
GROUP MEASUREMENTS
—– ————
1 13.50000 13.60000 13.40000
2 13.30000 12.90000 13.00000
Reject the null hypothesis of equal means
COMPUTED F VALUE IS SIGNIFICANT
RESULTS OF DUNCAN TEST ON EACH PAIR
==========================================================
PAIR MEAN MEAN COMPUTED CRITICAL
I J I J DIFFERENCE DIFFERENCE DECISION
– – ——– ——– ———- ———- ——–
1 2 13.500 13.067 .433 .469 ACCEPT Ho
1 3 13.500 13.133 .367 .469 ACCEPT Ho
Problem 10-20.
Test on variances:
Group Mean Var.
F, W 120.2 24.7
2
10
2
10
1780.0log4log)420(
¦
ipa
SSV
Test on means:
Source
X
SS MS F
D
FDecision
Between 3 1380.146 460.049 13.671 3.25 1: 321
P
P
P
o
Hreject
594.2)5/65.33()/(8.114)4.1092.120(
2
1
)(
)(
5.05.0
nMSSwmean
a
jw
x
783.73.112)4.1042.120(
2
1
)(
)(
?
HrejectRFmean
b
om
10-8
)(
c
Problem 10-21.
(see ANOVA results of problem 10-11)
RESULTS OF DUNCAN TEST ON EACH PAIR
==========================================================
PAIR MEAN MEAN COMPUTED CRITICAL
I J I J DIFFERENCE DIFFERENCE DECISION
– – ——– ——– ———- ———- ——–
1 2 79.000 86.429 7.429 8.570 ACCEPT Ho
Problem 10-22.
06.3
22
3
/1/19.1291/1/1 2
1
22
D
X
X
F
nn
XX
knnMS
XX
F
ji
ji
jiw
ji
Class n i
X
1 6 79.0000
Pair F Decision
1-2 1.372 Accept Ho
1-3 1.701 Accept Ho
Problem 10-23.
(see solution for 10-16 for ANOVA test)
RESULTS OF SCHEFFE TEST ON EACH PAIR
======================================================
PAIR MEAN MEAN COMPUTED CRITICAL
I J I J F F DECISION
1 3 13.500 13.133 .900 3.480 ACCEPT Ho
1 4 13.500 12.500 6.696 3.480 REJECT Ho
1 5 13.500 13.333 .186 3.480 ACCEPT Ho
Problem 10-24.
GROUP MEASUREMENTS
—– ————
1 114.00000 132.00000 109.00000 116.00000 108.00000
sample group standard coeff of
group size mean deviation variation Se of mean
—– —— ———- ——— ——— ———-
1 5 115.80000 9.65401 .0834 4.3174
=================================================
SUMS OF MEAN COMPUTED
VARIATION df SQUARES SQUARE F
——— —- ——— ——– ——–
Between 3. 1542.613 514.204 6.838
Reject the null hypothesis of equal means
COMPUTED F VALUE IS SIGNIFICANT
RESULTS OF SCHEFFE TEST ON EACH PAIR
======================================================
PAIR MEAN MEAN COMPUTED CRITICAL
I J I J F F DECISION
– – —- —- ——– ——– ——–
1 2 115.800 100.800 2.493 3.240 ACCEPT Ho
———————————————–
Problem 10-25.
BARTLETT TEST FOR EQUALITY OF VARIANCES
Data Matrix
———–
12.2000 12.7000 12.3000 12.8000 13.0000
Group Mean Std Dev Variance
—– ——– ——– ——–
1 12.60000 .33912 .11500
2 11.74000 .33615 .11300
10.4. Test of Population Variances
Problem 10-26.
Ao
SpSSS
differofpaironeleastatHH
?
667.2967.65,667.21,67.1
:;:
2
2
3
2
2
2
1
2
2
3
2
2
2
1
VVVV
Problem 10-27.
Group 1 2 3 4
22
1.31
1
¦
ip
SS
Problem 10-28.
Group Mean Var.
1 24.8 5.7
10-11
Problem 10-29.
Test on means:
Source
X
SS MS F
D
FDecision
Between 2 187.93 93.97 23.608 6.93 Reject Ho
Test on variances:
Group Mean Var.
1 9.52 0.737
2 15.58 9.092
o
D
Summary: inequality of population means but equality of population variances.
Problem 10-30.
HacceptXVaSpSiset
o
001
3
5
:1
2
22
? ? ?
10.5. Randomized Block Design
Problem 10-31.
Source
X
SS MS F
D
FDecision
Treatment 2 3
101548.0
u4
107740.0
u15.8 4.46 Reject Ho
10-12
104096.0
u
Conclusion: significant treatment and block effects
Problem 10-32.
Source
X
SS MS F
D
FDecision
Treatment 3 192.7 64.2 10.8 6.99 Reject Ho
Problem 10-33.
Source
X
SS MS F
D
FDecision
Treatment 3 270.3 90.1 7.56 9.78 Accept Ho
10.6. Two-Way Analysis of Variance
Problem 10-34.
Source
X
SS MS F
D
FDecision
Bet. Cells — 24 —
Bet column 1 21.33 21.33 4.266 11.26
Bet row 1 1.33 1.33 0.27 11.26
Conclusion: no significant differences
Problem 10-35.
Source
X
SS MS F
D
FDecision
Bet. Cells — 35 — —
10-13
Bet column 1 9 9 2.45 4.75 Accept Ho
Bet row 1 25 25 6.82 4.75 Reject Ho
Interaction 1 1 1 0.27 4.75 Accept Ho
Problem 10-36.
Source
X
SS MS F
D
FDecision
Bet. Cells — 336 — — —
Bet column 2 228 114 20.52 8.02 Reject Ho
Problem 10-38.
Source
X
SS MS F
D
FDecision
Bet. Cells — 3244 — —
Bet column 3 984 328 21.4
Bet row 2 592 296 19.3
One-way: row One-way: column
Source
X
SS MS F
D
FSource
X
SS MS F
D
F
Between 2 592 296 2.19 3.47 B/W 2 984 492 4.23 3.47