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DAE702S - DESIGN AND ANALYSIS OF EXPERIMENT - 2ND OPP - JANUARY 2024 |
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1 Pages 1-10 |
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1.1 Page 1 |
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nAmlBIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Department of Mathematics,
Statistics and Actuarial Science
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
I QUALIFICATION:
QUALIFICATION
CODE:
IICOURSE:
I DATE:
I DURATION:
BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND
STATISTICS
07BAMS
I LEVEL: 7
DESIGN AND ANALYSIS OF EXPERI-1 COURSE DAE702S
MENT
CODE:
JAN 2024
I SESSION: 2
3 HOURS
I MARKS: 100
I SUPPLEMENTARY/
I EXAMINER
I MODERATOR
2ND OPPORTUNITY
EXAMINATION QUESTION PAPER I
I Dr. Jacob Ong'ala
J
I Dr Petrus Iiyambo
I
INSTRUCTION
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS
1. Non-Programmable Calculator
ATTACHEMENTS
1. F distribution table
2. OC Curves for the Fixed model Analysis of Variance
3. t distribution table
THIS QUESTION PAPER CONSISTS OF 4 PAGES
(including the front page)
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1.2 Page 2 |
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QUESTION ONE - 28 MARKS
(a) Consider an experiment of two factor (A and B). Each factors has two levels (low and High).
the response for the experiment is shown in the table below.
Factor A
Facto,r B
Low High
Lovv 20 40
High 50 12
(i) Compute main effect of factor A
(ii) Compute main effect of factor B
(iii) Compute the interaction effect
(iv) Draw an interaction plot for the experiment
[1 mks]
[1 mks]
[2 mks]
[2 mks]
(b) Several ovens in a metal working shop are used to heat metal specimens. All the ovens are
supposed to operate at the same temperature, although it is suspected that this may not
be true. Three ovens are selected at random, and their temperatures on successive heats
are noted. The data collected are as follows:(Use o: = 0.05)
Oven
Temperature
491.50 498.30 498.10 493.50 493.60
2 488.50 484.65 479.90 477.35
3 490.10 484.80 488.25 473.00 471.85 478.65
(i) Construct ANOVA table
(ii) Formulate the hypothesis when random effect model is assumed
(iii) What do you conclude from the ANOVA table above?
(iv) Estimate variance component
[12 mks]
[2 mks]
[3 mks]
[5 mks]
QUESTION TWO - 22 MARKS
Suppose that four normal populations have means of µ 1 = 50, µ2 = 60, µ3 = 50, and µ4 = 60.
(a) How many observations should be taken from each population so that the probability of
rejecting the null hypothesis of equal population means is at least 0.90? Assume that
o: = 0.05 and that a reasonable estimate of the error variance is a 2 = 25.
[10 mks]
(b) How would your answer change if a reasonable estimate of the experimental error variance
were a 2 = 36
[5 mks]
(c) How would your answer change if a reasonable estimate of the experimental error variance
were a 2 = 49
[5 mks]
2
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1.3 Page 3 |
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(d) Can you draw any conclusions about the sensitivity of your answer in this particular situ-
ation about how your estimate of CJ affects the decision about sample size?
[2 mks]
QUESTION THREE - 28 MARKS
(a) The ANOVA from a randomized complete block experiment output is shown below.
Source
Treatment
Block
Error
Total
DF
ss
MS
Fp
4 1010.56
? 29.84 ?
?
? 64.765
??
20 169.33
?
29 1503.71
(i) Fill in the blanks
(ii) How many blocks were used in this experiment?
(iii) What conclusions can you draw?
[4 mks]
[1 mks]
[2 mks]
(b) A chemist wishes to test the effect of four chemical agents on the strength of a particular
type of cloth. Because there might be variability from one bolt to another, the chemist
decides to use a randomized block design, with the bolts of cloth considered as blocks. She
selects five bolts and applies all four chemicals in random order to each bolt. The resulting
tensile strengths are as follow
Chemical
2
3
4
Bolt
1
2
3
4
5
73 68 74 71 67
73 67 75 72 70
75 68 78 73 68
73 71 75 75 69
(i) Analyze the data from this experiment.
[12 mks]
(ii) Use the Fisher LSD method to make comparisons among the four Chemicals
[9 mks]
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QUESTION FOUR - 22 MARKS
The survival times of group of four beetles randomly allocated to twelve treatments groups
obtained by crossing the levels of four insecticides (A,B,C,D) at each of three concentrations
of the insecticides (l=low, 2=Medium, 3=High) is shown in the table below.This is a balanced
4-by-3 factorial design (two-factor design) that is replicated 4 times
Dose Insecticide
:~:i Survival Times
J.43; 1 :.4i5~ .6 .71~3.
.45 .71 .6 .6 .
. 36 .29 .40 .23
.92 ,6! .49 1.24
:.~22i
1:~~:1l
.21 .18
:ig
.23
...-3i:03i0 ...3-i:7si6 ..,3.~,814 ..,2.i,932
(a) Formulate the hypotheses for this experiment
(b) Test the hypothesis above using Analysis of Variance
(c) What do you conclude from the A.NOVAresults above
[2 mks]
[16 mks]
[4 mks]
- END OF QUESTIONS -
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1.5 Page 5 |
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lT-12 Tables
Table entry for p is the
critical value f• with
probability plying to
its right.
==BmD
F critical values
1
2
:..
·e;0;;
C:
3
0
C:
"-0
-5
·= 4
0
-0...,
J.".:..
0
<I)
"5
04
Q"
6
7
p
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
,025
.010
.001
.JOO
.050
.025
.010
.001
.100
.050
.025
.010
.001
1
39.86
161.45
647.79
4052.2
405284
8.53
18.51
38.51
98.50
998.50
5.54
10.13
17.44
34.12
167.03
4.54
7.71
12.22
21.20
74.14
4.06
6.61
10.01
16.26
47.18
3.78
5.99
8.81
13.75
35.51
3.59
5.59
8.07
12.25
29.25
2
49.50
199.50
799.50
4999.5
500000
9.00
19.00
39.00
99.00
999.00
5.46
9.55
16.04
30.82
148.50
4.32
6.94
10.65
18.00
61.25
3.78
5.79
8.43
13.27
37.12
3.46
5.14
7.26
10.92
27.00
3.26
4.74
6.54
9.55
21.69
Probability p
/_
F'
Degrees of freedom in the numerator
3
4
5
6
7
53.59
215.71
864.16
5403.4
540379
55.83
224.58
899.58
5624.6
562500
57.24
230.16
921.85
5763.6
576405
58.20
233.99
937.11
5859.0
585937
58.91
236.77
948.22
5928.4
592873
9.16
19.16
39.17
99.17
999.17
9.24
19.25
39.25
99.25
999.25
9.29
19.30
39.30
99.30
999.30
9.33
19.33
39.33
99.33
999.33
9.35
19.35
39.36
99.36
999.36
5.39
9.28
15.44
29.46
141.1 I
4.19
6.59
9.98
16.69
56.18
5.34
9.12
15.10
28.71
137.10
4.11
6.39
9.60
15.98
53.44
5.31
9.01
14.88
28.24
134.58
4.05
6.26
9.36
15.52
51.71
5.28
8.94
14.73
27.91
132.85
4.01
6.16
9.20
15.21
50.53
5.27
8.89
14.62
27.67
131.58
3.98
6.09
9.07
14.98
49.66
3.62
5.41
7.76
12.06
33.20
3.52
5.19
7.39
11.39
31.09
3.45
5.05
7.15
10.97
29.75
3.40
4.95
6.98
10.67
28.83
3.37
4.88
6.85
10.46
28.16
3.29
4.76
6.60
9.78
23.70
3.18
4.53
6.23
9.15
21.92
3.1 I
4.39
5.99
8.75
20.80
3.05
4.28
5.82
8.47
20.03
3.01
4.21
5.70
8.26
19.46
3.07
4.35
5.89
8.45
18.77
2.96
4.12
5.52
7.$5
17.20
2.88
3.97
5.29
7.46
16.21
2.83
3.87
5.12
7.19
15.52
2.78
3.79
4.99
6.99
15.02
8
59.44
238.88
956.66
5981.J
598144
9.37
19.37
39.37
99.37
999.37
5.25
8.85
14.54
27.49
130.62
3.95
6.04
8.98
14.80
49.00
3.34
4.82
6.76
10.29
27.65
2.98
4.15
5.60
8.10
19.03
2.75
3.73
4.90
6.84
14.63
9
59.86
240.54
963.28
6022.5
602284
9.38
19.38
39.39
99.39
999.39
5.24
8.81
14.47
27.35
129.86
3.94
6.00
8.90
14.66
48.47
3.32
4.77
6.68
10.16
27.24
2.96
4.10
5.52
7.98
18.69
2.72
3.68
4.82
6.72
14.33
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l Tables
T-13
Table entry for p is the
critical value F' with
probability p lying to
its right.
---•~1:11111
F critical values (continued}
Probability p
F*
Degrees of freedom in the numerator
10
12
15
20
25
30
40
so
60
120
l000
60.19
241.88
968.63
6055.8
605621
9.39
19.40
39.40
99.40
999.40
60.71
243.91
976.71
6106.3
610668
9.41
19.41
39.41
99.42
999.42
61.22
245.95
984.87
6157.3
615764
9.42
19.43
39.43
99.43
999.43
61.74
248.01
993.10
6208.7
620908
9.44
19.45
39.45
99.45
999.45
62.05
249.26
998.08
6239.8
624017
9.45
19.46
39.46
99.46
999.46
62.26
250.10
1001.4
6260.6
626099
9.46
19.46
39.46
99.47
999.47
62.53
251.14
1005.6
6286.8
628712
9.47
19.47
39.47
99.47
999.47
62.69
251.77
1008.1
6302.5
630285
9.47
19.48
39.48
99.48
999.48
62.79
252.20
1009.8
6313.0
631337
9.47
19.48
39.48
99.48
999.48
63.06
253.25
1014.0
6339.4
633972
9.48
19.49
39.49
99.49
999.49
63.30
254.19
1017.7
6362.7
636301
9.49
19.49
39.50
99.50
999.50
5.23
8.79
14.42
27.23
129.25
3.92
5.96
8.84
14.55
48.05
3.30
4.74
6.62
10.05
26.92
2.94
4.06
5.46
7.87
16.41
2.70
3.64
4.76
6.62
14.08
5.22
8.74
14.34
27.05
128.32
3.90
5.91
8.75
14.37
47.41
3.27
4.6S
6.52
9.89
26.42
2.90
4.00
5.37
7.72
17.99
2.67
3.57
4.67
6.47
13.71
5.20
S.70
14.25
26.S7
127.37
3.87
5.86
8.66
14.20
46.76
3.24
4.62
6.43
9.72
25.91
2.S7
3.94
5.27
7.56
17.56
2.63
3.51
4.57
6.31
13.32
5.18
S.66
14.17
26.69
126.42
3.84
5.80
6.56
14.02
46.10
3.21
4.56
6.33
9.55
25.39
2.S4
3.87
5.17
7.40
17.12
2.59
3.44
4.47
6.16
12.93
5.17
S.63
14.12
26.58
125.84
3.83
5.77
6.50
13.91
45.70
3.19
4.52
6.27
9.45
25.08
2.81
3.83
5.11
7.30
16.85
2.57
3.40
4.40
6.06
12.69
5.17
S.62
14.08
26.50
125.45
3.82
5.75
8.46
13.84
45.43
3.17
4.50
6.23
9.38
24.87
2.SO
3.Sl
5.07
7.23
16.67
2.56
3.38
4.36
5.99
12.53
5.16
8.59
14.04
26.41
124.96
3.80
5.72
8.41
13.75
45.09
3.16
4.46
6.18
9.29
24.60
2.7S
3.77
5.01
7.14
16.44
2.54
3.34
4.31
5.91
12.33
5.15
8.58
14.01
26.35
124.66
3.80
5.70
8.36
13.69
44.8S
3.15
4.44
6.14
9.24
24.44
2.77
3.75
4.98
7.09
16.31
2.52
3.32
4.28
5.S6
12.20
5.15
8.57
13.99
26.32
124.47
3.79
5.69
8.36
13.65
44.75
3.14
4.43
6.12
9.20
24.33
2.76
3.74
4.96
7.06
16.21
2.51
3.30
4.25
5.82
12.12
5.14
8.55
13.95
26.22
123.97
5.13
8.53
13.91
26.14
123.53
3.78
5.66
8.31
13.56
44.40
3.76
5.63
8.26
13.47
44.09
3.12
4.40
6.07
9.l l
24.06
3.l 1
4.37
6.02
9.03
23.82
2.74
3.70
4.90
6.97
15.96
2.72
3.67
4.86
6.89
15.77
2.49
3.27
4.20
5.74
11.91
2.47
3.23
4.15
5.66
11.72
(Continued)
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l T-14
Tables
FIi.·=. ,.
F critical values (continued)
8
9
10
:..
11
;0 ;
'§C
0
C
"-0
-5 12
·E =
0
-J0",::,
o
13
"(; '
":..
0J)
Q"
14
15
16
17
p
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.DIO
.001
.100
.050
.025
.010
.001
1
3.46
5.32
7.57
11.26
25.41
3.36
5.12
7.21
10.56
22.86
3.29
4.96
6.94
10.04
21.04
3.23
4.84
6.72
9.65
19.69
3.18
4.75
6.55
9.33
18.64
3.14
4.67
6.41
9.07
17.82
3.10
4.60
6.30
8.86
17.14
3.07
4.54
6.20
8.68
16.59
3.05
4.49
6.12
8.53
16.12
3.03
4.45
6.04
8.40
15.72
2
3.11
4.46
6.06
8.65
18.49
3.01
4.26
5.71
8.02
16.39
2.92
4.10
5.46
7.56
14.91
2.86
3.98
5.26
7.21
13.81
2.81
3.89
5.10
6.93
12.97
2.76
3.81
4.97
6.70
12.31
2.73
3.74
4.86
6.51
11.78
2.70
3.68
4.77
6.36
11.34
2.67
3.63
4.69
6.23
10.97
2.64
3.59
4.62
6.11
10.66
Degrees of freedom in the numerator
3
4
5
6
2.92
4.07
5.42
7.59
15.83
2.81
3.86
5.08
6.99
13.90
2.73
3.71
4.83
6.55
12.55
2.66
3.59
4.63
6.22
1I .56
2.61
3.49
4.47
5.95
10.80
2.56
3.41
4.35
5.74
10.21
2.52
3.34
4.24
5.56
9.73
2.49
3.29
4.15
5.42
9.34
2.46
3.24
4.08
5.29
9.01
2.44
3.20
4.01
5.19
8.73
2.81
3.84
5.05
7.01
14.39
2.69
3.63
4.72
6.42
12.56
2.61
3.48
4.47
5.99
11.28
2.54
3.36
4.28
5.67
10.35
2.48
3.26
4.12
5.41
9.63
2.43
3.18
4.00
5.21
9.07
2.39
3.11
3.89
5.04
S.62
2.36
3.06
3.80
4.89
8.25
2.33
3.01
3.73
4.77
7.94
2.31
2.96
3.66
4.67
7.68
2.73
3.69
4.82
6.63
13.48
2.61
3.48
4.48
6.06
11.71
2.52
3.33
4.24
5.64
10.48
2.45
3.20
4.04
5.32
9.58
2.39
3.11
3.89
5.06
8.89
2.35
3.03
3.77
4.86
8.35
2.31
2.96
3.66
4.69
7.92
2.27
2.90
3.58
4.56
7.57
2.24
2.85
3.50
4.44
7.27
2.22
2.81
3.44
4.34
7.02
2.67
3.58
4.65
6.37
12.86
2.55
3.37
4.32
5.80
11.13
2.46
3.22
4.07
5.39
9.93
2.39
3.09
3.88
5.07
9.05
2.33
3.00
3.73
4.82
8.38
2.28
2.92
3.60
4.62
7.86
2.24
2.85
3.50
4.46
7.44
2.21
2.79
3.41
4.32
7.09
2.18
2.74
3.34
4.20
6.80
2.15
2.70
3.28
4.10
6.56
7
2.62
3.50
4.53
6.18
12.40
2.51
3.29
4.20
5.61
J0.70
2.41
3.14
3.95
5.20
9.52
2.34
3.01
3.76
4.89
8.66
2.28
2.91
3.61
4.64
8.00
2.23
2.83
3.48
4.44
7.49
2.19
2.76
3.38
4.28
7.08
2.16
2.71
3.29
4.14
6.74
2.13
2.66
3.22
4.03
6.46
2.10
2.61
3.16
3.93
6.22
8
2.59
3.44
4.43
6.03
12.05
2.47
3.23
4.10
5.47
10.37
2.38
3.07
3.85
5.06
9.20
2.30
2.95
3.66
4.74
8.35
2.24
2.85
3.51
4.50
7.71
2.20
2.77
3.39
4.30
7.21
2.15
2.70
3.29
4.14
6.80
2.12
2.64
3.20
4.00
6.47
2.09
2.59
3.12
3.89
6.19
2.06
2.55
3.06
3.79
5.96
9
2.56
3.39
4.36
5.91
11.77
2.44
3.18
4.03
5.35
10.11
2.35
3.02
3.78
4.94
8.96
2.27
2.90
3.59
4.63
8.12
2.21
2.80
3.44
4.39
7.48
2.16
2.71
3.31
4.19
6.98
2.12
2.65
3.21
4.03
6.58
2.09
2.59
3.12
3.89
6.26
2.06
2.54
3.05
3.78
5.98
2.03
2.49
2.98
3.6S
5.75
|
|
1.8 Page 8 |
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1 Tables T-15
F critical values (continued)
10
2.54
3.35
4.30
5.81
11.54
2.42
3.14
3.96
5.26
9.89
2.32
2.98
3.72
4.85
8.75
2.25
2.85
3.53
4.54
7.92
2.19
2.75
3.37
4.30
7.29
2.14
2.67
3.25
4.10
6.80
2.10
2.60
3.15
3.94
6.40
2.06
2.54
3.06
3.60
6.08
2.03
2.49
2.99
3.69
5.81
2.00
2.45
2.92
3.59
5.58
12
2.50
3.28
4.20
5.67
11.19
2.38
3.07
3.87
5.11
9.57
2.28
2.91
3.62
4.71
8.45
2.21
2.79
3.43
4.40
7.63
2.15
2.69
3.28
4.16
7.00
2.10
2.60
3.15
3.96
6.52
2.05
2.53
3.05
3.80
6.13
2.02
2.48
2.96
3.67
5.81
1.99
2.42
2.89
3.55
5.55
l.96
2.38
2.82
3.46
5.32
15
2.46
3.22
4.10
5.52
10.84
2.34
3.01
3.77
4.96
9.24
2.24
2.85
3.52
4.56
8.13
2.17
2.72
3.33
4.25
7.32
2.10
2.62
3.18
4.01
6.71
2.05
2.53
3.05
3.82
6.23
2.01
2.46
2.95
3.66
5.85
1.97
2.40
2.86
3.52
5.54
1.94
2.35
2.79
3.41
5.27
l.91
2.31
2.72
3.31
5.05
D<!gr<!<o!fSfr<!edomin th<!num<!rator
20
25
30
40
2.42
3.15
4.00
5.36
10.48
2.30
2.94
3.67
4.81
8.90
2.20
2.77
3.42
4.41
7.80
2.12
2.65
3.23
4.10
7.01
2.06
2.54
3.07
3.86
6.40
2.01
2.46
2.95
3.66
5.93
1.96
2.39
2.84
3.51
5.56
1.92
2.33
2.76
3.37
5.25
1.89
2.28
2.68
3.26
4.99
1.86
2.23
2.62
3.16
4.78
2.40
3.11
3.94
5.26
10.26
2.27
2.89
3.60
4.71
8.69
2.17
2.73
3.35
4.31
7.60
2.10
2.60
3.16
4.01
6.81
2.03
2.50
3.01
3.76
6.22
1.98
2.41
2.88
3.57
5.75
1.93
2.34
2.78
3.41
5.38
1.89
2.28
2.69
3.28
5.07
1.86
2.23
2.61
3.16
4.82
1.63
2.18
2.55
3.07
4.60
2.38
3.08
3.89
5.20
10.11
2.25
2.86
3.56
4.65
8.55
2.16
2.70
3.31
4.25
7.47
2.08
2.57
3.12
3.94
6.68
2.01
2.47
2.96
3.70
6.09
1.96
2.38
2.84
3.51
5.63
1.91
2.31
2.73
3.35
5.25
1.87
2.25
2.64
3.21
4.95
1.84
2.19
2.57
3.10
4.70
I.SI
2.15
2.50
3.00
4.48
2.36
3.04
3.84
5.12
9.92
2.23
2.83
3.51
4.57
8.37
2.13
2.66
3.26
4.17
7.30
2.05
2.53
3.06
3.86
6.52
1.99
2.43
2.91
3.62
5.93
1.93
2.34
2.78
3.43
5.47
1.89
2.27
2.67
3.27
5.10
1.85
2.20
2.59
3.13
4.80
I.SI
2.15
2.51
3.02
4.54
1.78
2.10
2.44
2.92
4.33
50
2.35
3.02
3.81
5.07
9.80
2.22
2.80
3.47
4.52
8.26
2.12
2.64
3.22
4.12
7.19
2.04
2.51
3.03
3.81
6.42
J.97
2.40
2.87
3.57
5.83
1.92
2.31
2.74
3.38
5.37
1.87
2.24
2.64
3.22
5.00
1.83
2.18
2.55
3.08
4.70
1.79
2.12
2.47
2.97
4.45
1.76
2.08
2.41
2.87
4.24
60
2.34
3.01
3.78
5.03
9.73
2.21
2.79
3.45
4.48
8.19
2.11
2.62
3.20
4.08
7.12
2.03
2.49
3.00
3.78
6.35
1.96
2.38
2.85
3.54
5.76
I.90
2.30
2.72
3.34
5.30
1.86
2.22
2.61
3.18
4.94
l.62
2.16
2.52
3.05
4.64
1.78
2.l I
2.45
2.93
4.39
1.75
2.06
2.38
2.83
4.18
120
1000
2.32
2.97
3.73
4.95
9.53
2.18
2.75
3.39
4.40
8.00
2.08
2.58
3.14
4.00
6.94
2.00
2.45
2.94
3.69
6.18
1.93
2.34
2.79
3.45
5.59
1.88
2.25
2.66
3.25
5.14
1.83
2.18
2.55
3.09
4.77
1.79
2.11
2.46
2.96
4.47
1.75
2.06
2.38
2.84
4.23
1.72
2.01
2.32
2.75
4.02
2.30
2.93
3.68
4.87
9.36
2.16
2.71
3.34
4.32
7.84
2.06
2.54
3.09
3.92
6.78
1.98
2.41
2.89
3.61
6.02
1.91
2.30
2.73
3.37
5.44
1.85
2.21
2.60
3.18
4.99
l.80
2.14
2.50
3.02
4.62
1.76
2.07
2.40
2.88
4.33
1.72
2.02
2.32
2.76
4.08
1.69
l.97
2.26
2.66
3.87
(Continued)
|
|
1.9 Page 9 |
▲back to top |
l T-16
Tables
F critical values (continued)
18
19
20
21
·§ =
"O""
-5 22
·E -
0
"O
"
..... 23
0
""''
00
C" i
24
25
26
27
p
.100
.050
.Q25
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.Q25
.010
.001
.100
.050
.025
.DID
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
.100
.050
.025
.010
.001
1
3.01
4.41
5.98
S.29
15.3S
2.99
4.38
5.92
8.18
15.08
2.97
4.35
5.87
8.10
14.82
2.96
4.32
5.83
8.02
14.59
2.95
4.30
5.79
7.95
14.38
2.94
4.28
5.75
7.88
14.ZO
2.93
4.26
5.72
7.82
14.03
2.92
4.24
5.69
7.77
13.S8
2.9!
4.23
5.66
7.72
13.74
2.90
4.21
5.63
7.6S
13.61
2
2.62
3.55
4.56
6.01
10.39
2.61
3.52
4.51
5.93
10.16
2.59
3.49
4.46
5.85
9.95
2.57
3.47
4.42
5.78
9.77
2.56
3.44
4.38
5.72
9.61
2.55
3.42
4.35
5.66
9.47
2.54
3.40
4.32
5.61
9.34
2.53
3.39
4.29
5.57
9.22
2.52
3.37
4.27
5.53
9.12
2.51
3.35
4.24
5.49
9.02
Degrt!<'Sof fr.,c::dornin the num.,rator
3
4
5
6
2.42
3.16
3.95
5.09
8.49
2.40
3.13
3.90
5.01
8.28
2.38
3.10
3.86
4.94
8.10
2.36
3.07
3.82
4.S7
7.94
2.35
3.05
3.78
4.82
7.80
2.34
3.03
3.75
4.76
7.67
2.33
3.01
3.72
4.72
7.55
2.32
2.99
3.69
4.68
7.45
2.31
2.98
3.67
4.64
7.36
2.30
2.96
3.65
4.60
7.27
2.29
2.93
3.61
4.58
7.46
2.27
2.90
3.56
4.50
7.27
2.25
2.87
3.51
4.43
7.10
2.23
2.84
3.48
4.37
6.95
2.22
2.82
3.44
4.31
6.S1
2.21
2.80
3.41
4.26
6.70
2.19
2.78
3.38
4.22
6.59
2.18
2.76
3.35
4.1S
6.49
2.17
2.74
3.33
4.14
6.41
2.17
2.73
3.31
4.1 I
6.33
2.20
2.77
3.38
4.25
6.81
2.18
2.74
3.33
4.17
6.62
2.16
2.71
3.29
4.10
6.46
2.14
2.68
3.25
4.04
6.32
2.13
2.66
3.22
3.99
6.19
2.11
2.64
3.18
3.94
6.08
2.10
2.62
3.15
3.90
5.98
2.09
2.60
3.13
3.S5
5.89
2.0S
2.59
3.10
3.82
5.80
2.07
2.57
3.08
3.78
5.73
2.13
2.66
3.22
4.01
6.35
2.11
2.63
3.17
3.94
6.18
2.09
2.60
3.13
3.87
6.02
2.08
2.57
3.09
3.81
5.88
2.06
2.55
3.05
3.76
5.76
2.05
2.53
3.02
3.71
5.65
2.04
2.51
2.99
3.67
5.55
2.02
2.49
2.97
3.63
5.46
2.01
2.47
2.94
3.59
5.38
2.00
2.46
2.92
3.56
5.31
7
2.08
2.58
3.10
3.84
6.02
2.06
2.54
3.05
3.77
5.85
2.04
2.51
3.01
3.70
5.69
2.02
2.49
2.97
3.64
5.56
2.01
2.46
2.93
3.59
5.44
1.99
2.44
2.90
3.54
5.33
1.98
2.42
2.87
3.50
5.23
1.97
2.40
2.S5
3.46
5.15
1.96
2.39
2.82
3.42
5.07
1.95
2.37
2.80
3.39
5.00
8
2.04
2.51
3.01
3.71
5.76
2.02
2.48
2.96
3.63
5.59
2.00
2.45
2.91
3.56
5.44
l.98
2.42
2.87
3.51
5.31
1.97
2.40
2.S4
3.45
5.19
1.95
2.37
2.81
3.41
5.09
1.94
2.36
2.78
3.36
4.99
1.93
2.34
2.75
3.32
4.91
1.92
2.32
2.73
3.29
4.83
1.91
2.31
2.71
3.26
4.76
9
2.00
2.46
2.93
3.60
5.56
1.98
2.42
2.88
3.52
5.39
1.96
2.39
2.84
3.46
5.24
l.95
2.37
2.80
3.40
5.11
1.93
2.34
2.76
3.35
4.99
1.92
2.32
2.73
3.30
4.S9
1.91
2.30
2.70
3.26
4.80
1.89
2.28
2.68
3.22
4.71
1.88
2.27
2.65
3.18
4.64
J.87
2.25
2.63
3.15
4.57
|
|
1.10 Page 10 |
▲back to top |
-·•'IDIJ
F critical values (continued)
10
1.98
2.41
2.87
3.51
5.39
1.96
2.38
2.82
3.43
5.22
1.94
2.35
2.77
3.37
5.08
1.92
2.32
2.73
3.31
4.95
1.90
2.30
2.70
3.26
4.83
1.89
2.27
2.67
3.21
4.73
1.88
2.25
2.64
3.17
4,64
1.87
2.24
2.61
3.13
4.56
1.86
2.22
2.59
3.09
4.48
1.85
2.20
2.57
3.06
4.41
12
1.93
2.34
2.77
3.37
5.13
1.91
2.31
2.72
3.30
4.97
1.89
2.28
2.68
3.23
4.82
1.87
2.25
2.64
3.17
4.70
1.86
2.23
2.60
3.12
4.58
1.84
2.20
2.57
3.07
4.48
1.83
2.16
2.54
3.03
4.39
1.82
2.16
2.51
2.99
4.31
1.81
2.15
2.49
2.96
4.24
I.SO
2.13
2.47
2.93
4.17
15
1.89
2.27
2.67
3.23
4.87
1.86
2.23
2.62
3.15
4.70
1.84
2.20
2.57
3.09
4.56
1.83
2.18
2.53
3.03
4.44
1.81
2.15
2.50
2.98
4.33
I.SO
2.13
2.47
2.93
4.23
1.78
2.11
2.44
2.89
4.14
1.77
2.09
2.41
2.85
4.06
1.76
2.07
2.39
2.81
3.99
J.75
2.06
2.36
2.78
3.92
Degrees of freedom in the numerator
20
25
30
40
1.84
2.19
2.56
3.08
4.59
I.SI
2.16
2.51
3.00
4.43
1.79
2.12
2.46
2.94
4.29
1.78
2.10
2.42
2.88
4.17
1.76
2.07
2.39
2.83
4.06
1.74
2.05
2.36
2.78
3.96
1.73
2.03
2.33
2.74
3.87
1.72
2.01
2.30
2.70
3.79
1.71
1.99
2.28
2.66
3.72
1.70
1.97
2.25
2.63
3.66
1.80
2.14
2.49
2.98
4.42
1.78
2.11
2.44
2.91
4.26
J.76
2.07
2.40
2.84
4.12
1.74
2.05
2.36
2.79
4.00
1.73
2.02
2.32
2.73
3.89
1.71
2.00
2.29
2.69
3.79
1.70
1.97
2.26
2.64
3.71
1.68
1.96
2.23
2.60
3.63
1.67
1.94
2.21
2.57
3.56
1.66
1.92
2.18
2.54
3.49
1.78
2.11
2.44
2.92
4.30
1.76
2.07
2.39
2.84
4 ..14
1.74
2.04
2.35
2.78
4.00
1.72
2.01
2.31
2.72
3.88
1.70
1.98
2.27
2.67
3.78
1.69
1.96
2.24
2.62
3.68
1.67
1.94
2.21
2.58
3.59
1.66
1.92
2.18
2.54
3.52
1.65
1.90
2.16
2.50
3.44
1.64
1.88
2.13
2.47
3.38
1.75
2.06
2.38
2.84
4.15
1.73
2.03
2.33
2.76
3.99
1.71
1.99
2.29
2.69
3.86
1.69
1.96
2.25
2.64
3.74
1.67
1.94
2.21
2.58
3.63
1.66
1.91
2.18
2.54
3.53
1.64
1.89
2.15
2.49
3.45
1.63
1.87
2.12
2.45
3.37
1.61
1.85
2.09
2.42
3.30
1.60
1.84
2.07
2.38
3.23
so
1.74
2.04
2.35
2.78
4.06
1.71
2.00
2.30
2.71
3.90
1.69
1.97
2.25
2.64
3.77
1.67
1.94
2.21
2.58
3.64
1.65
1.91
2.17
2.53
3.54
1.64
1.88
2.14
2.48
3.44
1.62
1.86
2.11
2.44
3.36
1.61
1.84
2.08
2.40
3.28
1.59
1.82
2.05
2.36
3.21
1.58
1.81
2.03
2.33
3.14
60
1.72
2.02
2.32
2.75
4.00
1.70
1.98
2.27
2.67
3.84
1.68
1.95
2.22
2.61
3.70
1.66
1.92
2.18
2.55
3.58
1.64
1.89
2.14
2.50
3.48
1.62
1.86
2.11
2.45
3.38
1.61
1.84
2.08
2.40
3.29
1.59
1.82
2.05
2.36
3.22
1.58
1.80
2.03
2.33
3.15
1.57
1.79
2.00
2.29
3.08
1 Tables
T-17
120
1.69
1.97
2.26
2.66
3.84
1.67
1.93
2.20
2.58
3.68
1.64
1.90
2.16
2.52
3.54
1.62
1.87
2.11
2.46
3.42
1.60
1.84
2.08
2.40
3.32
1.59
1.81
2.04
2.35
3.22
1.57
1.79
2.01
2.31
3.14
1.56
1.77
1.98
2.27
3.06
1.54
1.75
1.95
2.23
2.99
1.53
1.73
1.93
2.20
2.92
1000
1.66
1.92
2.20
2.58
3.69
1.64
1.88
2.14
2.50
3.53
1.61
1.85
2.09
2.43
3.40
1.59
1.82
2.05
2.37
3.28
1.57
1.79
2.01
2.32
3.17
1.55
1.76
1.98
2.27
3.08
1.54
1.74
1.94
2.22
2.99
1.52
1.72
1.91
2.18
2.91
I.SI
1.70
1.89
2.14
2.84
I.SO
1.68
1.86
2.11
2.78
(Continued)
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2 Pages 11-20 |
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|
2.1 Page 11 |
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Appendix 693
V Operating Characteristic Curves for the Fixed Effects Model Analysis of Variance•
1.00
0.80
0.70
0.60
·".;;,';
0.50
0.40
.r:
0 0.30
.C:r,.:.
-"C=l' 0.20
.!:
C.
0""""''
0.10
0.08
0.07
0.06
0.05
:c
."0 '
0.04
E!
Q,.
0.03
0.02
0.01
1.5
1.00
0.80
0.70
0.60
·";;';
."r:'
0.50
0.40
0 0.30
C:,..
.r:
-"5' 0.20
Cl
.!:
C.
0 "" "" ' '
0.10
0.08
0.07
-~ 0.06
0.05
:0
.0"'
0.04
.0,: 0.03
0.02
2
2.5
3
<I>(for a= 0.01) -
2
3.5 -<I>
3
(for a= 0.05)
4
5
2
3 -<I>
(for a= 0.05)
<l>(fora=0.01)-
1
2
3
4
5
u, Numerator degrees of freedom, u2 Denominator degrees of freedom
'Adapted with permission from Bfomrtrika Tablesfor Statisticians. Vol. 2. by E. S. Pearson Md H. 0. Honlcy. Cambridge
University Press. C:imbridgc, 1972.
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2.2 Page 12 |
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694 Appendix
V Operating Characteristic Curves for the Fixed Effects Model Analysis
of Variance (Continued)
1.00
0.80
0.70
0.60
-.~,
0.50
0.40
,::
0 0.30
.,,:Q>:.-
0.20
-5
a,
.!:
..0.,. 0.10
<<..>> 0.08
0 0.07
..1 0.06
:c
0.05
0.04
e.0
Q. 0.03
0.02
2
3 -<l>
(for ex; 0.05)
<l>(for Cl• 0.011-
1
2
3
4
5
1.00 ,-------.-------,-------.-------,-----~---,
0.80
0.70
0.60
., 0.50
·..;,; 0.40
,::
0 0.30 1--------'I~~~~
Q.
,:>.:,-
-5
a,
C:
-~ 0.10 1-----'W.~W~.A-
"' 0.08 l-----+\\-1,1-\\-\\,.\\+ll+'o-
1>'
0 0.07 1-------+'r-H+\\c-\\-\\-\\+--\\--+-+-l-+--\\-'l-\\-f'r\\-~-'lc-__..,,-....----+----1
1:C 00..0065 1-----~\\-H+\\c-H--\\-\\-+-'l+-\\-+--l-lr+--+-+--\\-+--+----lr'r----+----1
2 0.04 1-------+\\c-l+\\+-l+\\+-+----a1-++--+-'l-+'l--'l---\\-\\-~......,f-'----+----1
£ 0.03 1--------1\\--\\-H--\\-l-+1--\\-\\+'--Hr+-+-\\+--\\-~---\\-l---'lc---+----1
2
3 -<l>
(for Cl; 0.051
<l>(for Cl• 0.01 I-
1
2
3
4
5
|
|
2.3 Page 13 |
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Appendix 695
V Operating Characteristic Curves for the Fixed Effects Model Analysis
of Variance (Continued)
1.00
0.80
0.70
0.60
0.50
"'i"ii' 0.40
..:;
0 0.30
..0>:;.
:" 0.20
C,
·=0.
"8
0"'
0.10
0.08
0.07
0.06
0.05
J5
.0"'
0.04
0
o':
0.03
0.02
2
3 -<I>
(for (X C 0.05)
<I>(fora=0.01)-
1
2
3
4
5
·";;;'
"..:;
0
0.
..>:;
:"
C,
·=°g- 0.10 f-----'H-'l-\\-lr\\-lM--b:\\'-::;,-"";,.-'f--
8
0"'
0.08
0.07
0.06 f------\\-'1:-\\-1,r\\-1~~6-'ii-:,-t=
J5 0.05 l-----\\+lr\\-lr\\-''a-'H-44'=
0.04 f-----\\-\\--\\t\\-~~,¥,~t,-=-""-----+-++-::\\-:-\\--:-\\--:1\\-'.---\\-:---\\---f\\-;
0
o':
2
3-
<I>(for (X C 0.05)
<I>(fora=0.01)-
1
2
3
4
|
|
2.4 Page 14 |
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696 Appendix
V Operating Characteristic Curves for the Fixed Effects Model Analysis
of Variance (Continued)
1.00
0.80
0.70
0.60
-~ 0.50
., 0.40
.c
0C,_. 0.30
.c
-"5
0.20
Cl
.a.!:,. 0.10
<<.,>> 0.08
0
0.07
0.06
0.05
0.04
e.0
0.. 0.03
0.02
2
3 -<I>
(for a= 0.05)
<I>(for a= 0.01)-
1
2
3
4
-"~' 0.40 f-1'.Wr'&1'n'"'c'-+-----+-~-\\->c+''n'l~-'l<-''<-'~---l-------l---i
.c
0C,_.
! 0.20 '--44'..\\-\\-+\\.->,\\->,l+-',__-----1'----4-+-\\c-\\l...\\->.-4.4-+-""4-"------+-----1
-5
2
3 _<!>(fora=0.05)
<!>(fora c 0.01)-
1
2
3
4
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2.5 Page 15 |
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l Tables T-11
Table entry for p and C is
the critical value t· with
probability p lying to its
right and probability C lying
between -t· and t·.
=lm!B
t distribution critical values
df
.25
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
4s0o
60
80
100
1000
z·
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.695
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.686
0.686
0.685
0.685
0.684
0.684
0.684
0.683
0.683
0.683
0.681
0.679
0.679
0.678
0.677
0.675
0.674
50%
.20
1.376
1.061
0.978
0.941
0.920
0.906
0.896
0.889
0.883
0.879
0.876
0.873
0.870
0.868
0.866
0.865
0.863
0.862
0.861
0.860
0.859
0.858
0.858
O.S57
O.S56
0.856
0.855
0.855
0.854
0.854
0.851
0.849
0.848
0.846
0.845
0.842
0.841
60%
.IS
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
J.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
70%
.10
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
80%
Upper-tail probability p
.OS
.025
.02
.01
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
l.753
l.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.70S
1.706
1.703
1.701
l.699
1.697
1.684
1.676
1.67 J
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
90%
95%
96%
98%
Confidence levd C
Probability p
t•
.005
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
99%
.0025
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.8S7
2.871
2.813
2.807
99.5%
.001
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.26]
3.232
3.195
3.174
3.098
3.091
99.8%
.0005
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
99.9%