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DAE702S - DESIGN AND ANALYSIS OF EXPERIMENT - 2ND OPP - JAN 2023 |
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1.1 Page 1 |
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n Am I B I A u nl VE Rs I TY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS
I QUALIFICATION:
PROGRAMME
I CODE:
I COURSE CODE:
I DATE
I DURATION:
BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND
STATISTICS
07BAMS
I LEVEL: 7
DAE702S
Jan-2023
3 HOURS
COURSE
I NAME:
I PAPER:
I MARKS
DESIGN ANDANALYSIS OFEXPER-1
IMENT
THEORY
I
100
I
SUPPLEMENTARY/ 2ND OPPORTUNITY EXAMINATION
QUESTION PAPER
I EXAMINER
I MODERATOR
I Dr. Jacob Ong'ala
I Prof Peter Njuho
INSTRUCTION
1. Answer ALL the questions in the booklet provided
2. Scan your answer sheet and upload it in the e-learning
2. Show clearly all the steps used in the calculation
3. All written work must be shown in the answer sheet.
THIS QUESTION
PERMISSIBLE MATERIALS
Non-programmable calculator without cover
PAPER CONSISTS OF 15 PAGES (including the front page and
attachments)
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QUESTION ONE - 27 MARKS
(a) A computer ANOVA output is shown below. Fill in the blanks. ( You may give bounds on
the P-value)
[5 mks]
One-way ANOVA
Source
DF
ss
MS
f
p
Factor
?
246.93
?
?
Error
25
186.53
?
Tota L
29
1174.24
(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 :5uspected 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 a:= 0.05)
Oven
Temperature
1 491.50 498.30 498.10 493.50 493.60
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, µ 3 = 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
a:= 0.05 and that a reasonable estimate of the error variance is cr2 = 25.
[10 mks]
(b) How would your answer change if a reasonable estimate of the experimental error variance
were cr2 = 36
[5 mks]
(c) How would your answer change if a reasonable estimate of the experimental error variance
were cr2 = 49
[5 mks]
(d) Can you draw any conclusions about the sensitivity of your answer in this particular situ-
ation about how your estimate of er 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.
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1.3 Page 3 |
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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) V/hat conclusions can you draw?
[4 mks]
[1 mks]
[2 mks]
(b) A chemist wishes to test the effect of four chemical agents 011 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 - 23 MARKS
Suppose you want to determine whether the brand of laundry detergent used and the tempera-
ture affects the amount of dirt removed from your laundry. To this end, you buy two different
brand of detergent (" Super" and "Best") and choose three different temperature levels ("cold",
"warm", and "hot"). Then you divide your laundry randomly into 6xr piles of equal size
and assign each r=4 piles into the combination of ("Super" and "Best") and ("cold","warm",
and "hot"). The amounts 41'ijk of dirt removed when washing sub pile k(k = 1, 2, 3, 4) with
detergent i(i = 1, 2) at temperature j(j = 1, 2, 3) are recorded in the table below.
Super
Best
Cold
4,5.6.5
6,6.4,4
Warm
Hot
7,9,8,12 10,12.11.9
13,15.12,12 12,13,10,13
(a) Formulate the hypotheses for this experiment
(b) Test the hypothesis above using Analysis of Variance
(c) What do you conclude from the ANOVA results above
[2 mks]
[17 mks]
[4 mks]
- END OF QUESTIONS -
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l T-12 Tables
~-.
Table entry for p is the
critical value F* with
probability p lying to
its right.
F critical values
p
.100
.050
1 .025
.010
.001
.100
.050
2 .025
.010
.001
....
E
.100
.050
"<l' 3 .025
'§
.010
0
<l
.001
"-0
-".s
.5 4
.100
.050
.025
E
0
.010
-0
J""::
.001
"0"'
.100
V)
.050
"Ie'n:! 5
Q"
.025
.010
.001
.100
.050
6 .025
.010
.001
.100
.050
7 .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
141.11
5.34
9.12
15.10
28.71
137.10
5.31
9.01
14.88
28.24
134.58
5.28
8.94
14.73
27.91
132.85
5.27
8.89
14.62
27.67
131.58
4.19
6.59
9.98
16.69
56.18
4.11
6.39
9.60
15.98
53.44
4.05
6.26
9.36
15.52
51.71
4.01
6.16
9.20
15.21
50.53
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
3.11
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
7.85
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
5981.1
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
5
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l Tables T-13
Table entry for p is the
critical value F• with
probability p lying to
its right.
r--119'•1:H=---
F critical values (continued)
Probability p
F*
Degrees of freedom in the numerator
10
12
15
20
25
30
40
so
60
60.19
241.88
968.63
6055.8
605621
60.71
243.91
976.71
6106.3
610668
61.22
245.95
984.87
6157.3
615764
61.74
248.01
993.10
6208.7
620908
62.05
249.26
998.08
6239.8
624017
62.26
250.10
1001.4
6260.6
626099
62.53
251.14
1005.6
6286.8
628712
62.69
251.77
1008.1
6302.5
630285
62.79
252.20
1009.8
6313.0
631337
9.39
19.40
39.40
99.40
999.40
9.41
19.41
39.41
99.42
999.42
9.42
19.43
39.43
99.43
999.43
9.44
19.45
39.45
99.45
999.45
9.45
19.46
39.46
99.46
999.46
9.46
19.46
39.46
99.47
999.47
9.47
19.47
39.47
99.47
999.47
9.47
19.48
39.48
99.48
999.48
9.47
19.48
39.48
99.48
999.48
5.23
8.79
14.42
27.23
129.25
5.22
8.74
14.34
27.05
128.32
5.20
8.70
14.25
26.87
127.37
5.18
8.66
14.17
26.69
126.42
5.17
8.63
14.12
26.58
125.84
5.17
8.62
14.08
26.50
125.45
5.16
8.59
14.04
26.41
124.96
5.15
8.58
14.01
26.35
124.66
5.15
8.57
13.99
26.32
124.47
3.92
5.96
8.84
14.55
48.05
3.90
5.91
8.75
14.37
47.41
3.87
5.86
8.66
14.20
46.76
3.84
5.80
8.56
14.02
46.10
3.83
5.77
8.50
13.91
45.70
3.82
5.75
8.46
13.84
45.43
3.80
5.72
8.41
13.75
45.09
3.80
5.70
8.38
13.69
44.88
3.79
5.69
8.36
13.65
44.75
3.30
4.74
6.62
10.05
26.92
3.27
4.68
6.52
9.89
26.42
3.24
4.62
6.43
9.72
25.91
3.21
4.56
6.33
9.55
25.39
3.19
4.52
6.27
9.45
25.08
3.17
4.50
6.23
9.38
24.87
3.16
4.46
6.18
9.29
24.60
3.15
4.44
6.14
9.24
24.44
3.14
4.43
6.12
9.20
24.33
2.94
4.06
5.46
7.87
18.41
2.90
4.00
5.37
7.72
17.99
2.87
3.94
5.27
7.56
17.56
2.84
3.87
5.17
7.40
17.12
2.81
3.83
5.11
7.30
16.85
2.80
3.81
5.07
7.23
16.67
2.78
3.77
5.01
7.14
16.44
2.77
3.75
4.98
7.09
16.31
2.76
3.74
4.96
7.06
16.21
2.70
3.64
4.76
6.62
14.08
2.67
3.57
4.67
6.47
13.71
2.63
3.51
4.57
6.31
13.32
2.59
3.44
4.47
6.16
12.93
2.57
3.40
4.40
6.06
12.69
2.56
3.38
4.36
5.99
12.53
2.54
3.34
4.31
5.91
12.33
2.52
3.32
4.28
5.86
12.20
2.51
3.30
4.25
5.82
12.12
6
120
1000
63.06
253.25
1014.0
6339.4
633972
63.30
254.19
1017.7
6362.7
636301
9.48
19.49
39.49
99.49
999.49
9.49
19.49
39.50
99.50
999.50
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
9.11
24.06
3.11
4.37
6.02
9.03
23.82
2.74
3.70
4.90
6.97
15.98
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
;..-.- lr&11
:=_
F critical values (continued)
8
9
10
....
11
0
"·Ca:
0
C:
"-0
-"5
12
.5
E
0
-0
..""!::
"-0
13
""".0.0'..
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
.Q25
.010
.001
.100
.050
.Q25
.010
.001
.100
.050
.025
.010
.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
7
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
11.56
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
8.62
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
2.62
3.50
4.53
6.18
12.40
2.51
3.29
4.20
5.61
10.70
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
7
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
3.68
5.75
|
|
1.8 Page 8 |
▲back to top |
l T-16 Tables
F critical values (continued)
Degrees of freedom in the numerator
p
1
2
3
4
5
6
7
8
9
.100
3.01
2.62
2.42
2.29
2.20
2.13
2.08
2.04
2.00
.050
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
18
.025
5.98
4.56
3.95
3.61
3.38
3.22
3.10
3.01
2.93
.010
8.29
6.01
5.09
4.58
4.25
4.01
3.84
3.71
3.60
.001
15.38
10.39
8.49
7.46
6.81
6.35
6.02
5.76
5.56
.100
.050
19
.025
.010
.001
2.99
4.38
5.92
8.18
15.08
2.61
2.40
2.27
2.18
3.52
3.13
2.90
2.74
4.51
3.90
3.56
3.33
5.93
5.01
4.50
4.17
10.16
8.28
7.27
6.62
2.11
2.06
2.02
1.98
2.63
2.54
2.48
2.42
3.17
3.05
2.96
2.88
3.94
3.77
3.63
3.52
6.18
5.85
5.59
5.39
.100
.050
20
.025
.010
.001
2.97
4.35
5.87
8.10
14.82
2.59
2.38
2.25
2.16
2.09
2.04
2.00
1.96
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
4.46
3.86
3.51
3.29
3.13
3.01
2.91
2.84
5.85
4.94
4.43
4.10
3.87
3.70
3.56
3.46
9.95
8.10
7.10
6.46
6.02
5.69
5.44
5.24
.100
.050
... 21
.025
B
.010
""i:§:
.001
0
i::
.100
""O
.050
-"s 22
.025
.010
.5
.001
E
0
"O
."... 23
.100
.050
.025
0
<ll
.010
"
.001
Oil
C" l
.100
.050
24
.025
.010
.001
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
14.20
2.93
4.26
5.72
7.82
14.03
2.57
2.36
2.23
2.14
2.08
2.02
1.98
1.95
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
4.42
3.82
3.48
3.25
3.09
2.97
2.87
2.80
5.78
4.87
4.37
4.04
3.81
3.64
3.51
3.40
9.77
7.94
6.95
6.32
5.88
5.56
5.31
5.11
2.56
2.35
2.22
2.13
2.06
2.01
1.97
1.93
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
4.38
3.78
3.44
3.22
3.05
2.93
2.84
2.76
5.72
4.82
4.31
3.99
3.76
3.59
3.45
3.35
9.61
7.80
6.81
6.19
5.76
5.44
5.19
4.99
2.55
2.34
2.21
2.11
2.05
1.99
1.95
1.92
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
4.35
3.75
3.41
3.18
3.02
2.90
2.81
2.73
5.66
4.76
4.26
3.94
3.71
3.54
3.41
3.30
9.47
7.67
6.70
6.08
5.65
5.33
5.09
4.89
2.54
2.33
2.19
2.10
2.04
1.98
1.94
1.91
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
4.32
3.72
3.38
3.15
2.99
2.87
2.78
2.70
5.61
4.72
4.22
3.90
3.67
3.50
3.36
3.26
9.34
7.55
6.59
5.98
5.55
5.23
4.99
4.80
.100
.050
25
.025
.010
.001
2.92
4.24
5.69
7.77
13.88
2.53
2.32
2.18
2.09
2.02
1.97
1.93
1.89
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
4.29
3.69
3.35
3.13
2.97
2.85
2.75
2.68
5.57
4.68
4.18
3.85
3.63
3.46
3.32
3.22
9.22
7.45
6.49
5.89
5.46
5.15
4.91
4.71
.100
2.91
.050
4.23
26
.025
5.66
.010
7.72
.001
13.74
2.52
2.31
2.17
2.08
2.01
1.96
1.92
1.88
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
4.27
3.67
3.33
3.10
2.94
2.82
2.73
2.65
5.53
4.64
4.14
3.82
3.59
3.42
3.29
3.18
9.12
7.36
6.41
5.80
5.38
5.07
4.83
4.64
.JOO
2.90
.050
4.21
27
.025
5.63
.010
7.68
.001
13.61
2.51
2.30
2.17
2.07
2.00
1.95
1.91
1.87
3.35
2.96
2.73
2.57
2.46
2.37
2.31
2.25
4.24
3.65
3.31
3.08
2.92
2.80
2.7 J
2.63
5.49
4.60
4.11
3.78
3.56
3.39
3.26
3.15
9.02
7.27
6.33
5.73
5.31
5.00
4.76
4.57
9
|
|
1.9 Page 9 |
▲back to top |
l Tables
T-15
F critical values (continued)
Degrees of freedom in the numerator
10
12
15
20
25
30
40
so
60
120
1000
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
3.80
6.08
2.03
2.49
2.99
3.69
5.81
2.00
2.45
2.92
3.59
5.58
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
1.96
2.38
2.82
3.46
5.32
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
1.91
2.31
2.72
3.31
5.05
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
2.38
2.36
2.35
3.11
3.08
3.04
3.02
3.94
3.89
3.84
3.81
5.26
5.20
5.12
5.07
10.26
JO.JI
9.92
9.80
2.27
2.25
2.23
2.22
2.89
2.86
2.83
2.80
3.60
3.56
3.51
3.47
4.71
4.65
4.57
4.52
8.69
8.55
8.37
8.26
2.17
2.16
2.13
2.12
2.73
2.70
2.66
2.64
3.35
3.31
3.26
3.22
4.31
4.25
4.17
4.12
7.60
7.47
7.30
7.19
2.10
2.08
2.05
2.04
2.60
2.57
2.53
2.51
3.16
3.12
3.06
3.03
4.01
3.94
3.86
3.81
6.81
6.68
6.52
6.42
2.03
2.01
1.99
1.97
2.50
2.47
2.43
2.40
3.01
2.96
2.91
2.87
3.76
3.70
3.62
3.57
6.22
6.09
5.93
5.83
1.98
1.96
1.93
1.92
2.41
2.38
2.34
2.31
2.88
2.84
2.78
2.74
3.57
3.51
3.43
3.38
5.75
5.63
5.47
5.37
1.93
1.91
1.89
1.87
2.34
2.31
2.27
2.24
2.78
2.73
2.67
2.64
3.41
3.35
3.27
3.22
5.38
5.25
5.10
5.00
1.89
1.87
1.85
1.83
2.28
2.25
2.20
2.18
2.69
2.64
2.59
2.55
3.28
3.21
3.13
3.08
5.07
4.95
4.80
4.70
1.86
1.84
1.81
1.79
2.23
2.19
2.15
2.12
2.61
2.57
2.51
2.47
3.16
3.10
3.02
2.97
4.82
4.70
4.54
4.45
1.83
1.81
1.78
1.76
2.18
2.15
2.10
2.08
2.55
2.50
2.44
2.41
3.07
3.00
2.92
2.87
4.60
4.48
4.33
4.24
8
2.34
2.32
2.30
3.01
2.97
2.93
3.78
3.73
3.68
5.03
4.95
4.87
9.73
9.53
9.36
2.21
2.18
2.16
2.79
2.75
2.71
3.45
3.39
3.34
4.48
4.40
4.32
8.19
8.00
7.84
2.11
2.08
2.06
2.62
2.58
2.54
3.20
3.14
3.09
4.08
4.00
3.92
7.12
6.94
6.78
2.03
2.00
1.98
2.49
2.45
2.41
3.00
2.94
2.89
3.78
3.69
3.61
6.35
6.18
6.02
1.96
1.93
1.91
2.38
2.34
2.30
2.85
2.79
2.73
3.54
3.45
3.37
5.76
5.59
5.44
1.90
1.88
1.85
2.30
2.25
2.21
2.72
2.66
2.60
3.34
3.25
3.18
5.30
5.14
4.99
1.86
1.83
1.80
2.22
2.18
2.14
2.61
2.55
2.50
3.18
3.09
3.02
4.94
4.77
4.62
1.82
1.79
1.76
2.16
2.11
2.07
2.52
2.46
2.40
3.05
2.96
2.88
4.64
4.47
4.33
1.78
1.75
1.72
2.11
2.06
2.02
2.45
2.38
2.32
2.93
2.84
2.76
4.39
4.23
4.08
1.75
1.72
1.69
2.06
2.01
1.97
2.38
2.32
2.26
2.83
2.75
2.66
4.18
4.02
3.87
(Continued)
|
|
1.10 Page 10 |
▲back to top |
l Tables
T-17
F critical values (continued)
Degrees of freedom in the numerator
10
12
15
20
25
30
40
so
60
120
1000
1.98
1.93
1.89
1.84
1.80
1.78
1.75
1.74
1.72
1.69
1.66
2.41
2.34
2.27
2.19
2.14
2.11
2.06
2.04
2.02
1.97
1.92
2.87
2.77
2.67
2.56
2.49
2.44
2.38
2.35
2.32
2.26
2.20
3.51
3.37
3.23
3.08
2.98
2.92
2.84
2.78
2.75
2.66
2.58
5.39
5.13
4.87
4.59
4.42
4.30
4.15
4.06
4.00
3.84
3.69
1.96
1.91
1.86
1.81
1.78
1.76
1.73
1.71
1.70
1.67
1.64
2.38
2.31
2.23
2.16
2.11
2.07
2.03
2.00
1.98
1.93
1.88
2.82
2.72
2.62
2.51
2.44
2.39
2.33
2.30
2.27
2.20
2.14
3.43
3.30
3.15
3.00
2.91
2.84
2.76
2.71
2.67
2.58
2.50
5.22
4.97
4.70
4.43
4.26
4.14
3.99
3.90
3.84
3.68
3.53
1.94
1.89
1.84
1.79
1.76
1.74
1.71
1.69
1.68
1.64
1.61
2.35
2.28
2.20
2.12
2.07
2.04
1.99
1.97
1.95
1.90
1.85
2.77
2.68
2.57
2.46
2.40
2.35
2.29
2.25
2.22
2.16
2.09
3.37
3.23
3.09
2.94
2.84
2.78
2.69
2.64
2.61
2.52
2.43
5.08
4.82
4.56
4.29
4.12
4.00
3.86
3.77
3.70
3.54
3.40
1.92
1.87
1.83
1.78
1.74
1.72
1.69
1.67
1.66
1.62
1.59
2.32
2.25
2.18
2.10
2.05
2.01
1.96
1.94
1.92
1.87
1.82
2.73
2.64
2.53
2.42
2.36
2.31
2.25
2.21
2.18
2.11
2.05
3.31
3.17
3.03
2.88
2.79
2.72
2.64
2.58
2.55
2.46
2.37
4.95
4.70
4.44
4.17
4.00
3.88
3.74
3.64
3.58
3.42
3.28
1.90
1.86
1.81
1.76
1.73
1.70
1.67
1.65
1.64
1.60
1.57
2.30
2.23
2.15
2.07
2.02
1.98
1.94
1.91
1.89
1.84
1.79
2.70
2.60
2.50
2.39
2.32
2.27
2.21
2.17
2.14
2.08
2.01
3.26
3.12
2.98
2.83
2.73
2.67
2.58
2.53
2.50
2.40
2.32
4.83
4.58
4.33
4.06
3.89
3.78
3.63
3.54
3.48
3.32
3.17
1.89
1.84
1.80
1.74
1.71
1.69
1.66
1.64
1.62
1.59
1.55
2.27
2.20
2.13
2.05
2.00
1.96
1.91
1.88
1.86
1.81
1.76
2.67
2.57
2.47
2.36
2.29
2.24
2.18
2.14
2.11
2.04
1.98
3.21
3.07
2.93
2.78
2.69
2.62
2.54
2.48
2.45
2.35
2.27
4.73
4.48
4.23
3.96
3.79
3.68
3.53
3.44
3.38
3.22
3.08
1.88
1.83
1.78
1.73
1.70
1.67
1.64
1.62
1.61
1.57
1.54
2.25
2.18
2.11
2.03
1.97
1.94
1.89
1.86
1.84
1.79
1.74
2.64
2.54
2.44
2.33
2.26
2.21
2.15
2.11
2.08
2.01
1.94
3.17
3.03
2.89
2.74
2.64
2.58
2.49
2.44
2.40
2.31
2.22
4.64
4.39
4.14
3.87
3.71
3.59
3.45
3.36
3.29
3.14
2.99
1.87
1.82
1.77
1.72
1.68
1.66
1.63
1.61
1.59
1.56
1.52
2.24
2.16
2.09
2.01
1.96
1.92
1.87
1.84
1.82
1.77
1.72
2.61
2.51
2.41
2.30
2.23
2.18
2.12
2.08
2.05
1.98
1.91
3.13
2.99
2.85
2.70
2.60
2.54
2.45
2.40
2.36
2.27
2.18
4.56
4.31
4.06
3.79
3.63
3.52
3.37
3.28
3.22
3.06
2.91
1.86
1.81
1.76
1.71
1.67
1.65
1.61
1.59
1.58
1.54
I.SI
2.22
2.15
2.07
1.99
1.94
1.90
1.85
1.82
1.80
1.75
1.70
2.59
2.49
2.39
2.28
2.21
2.16
2.09
2.05
2.03
1.95
1.89
3.09
2.96
2.81
2.66
2.57
2.50
2.42
2.36
2.33
2.23
2.14
4.48
4.24
3.99
3.72
3.56
3.44
3.30
3.21
3.15
2.99
2.84
1.85
I.SO
1.75
1.70
1.66
1.64
1.60
1.58
1.57
1.53
I.SO
2.20
2.13
2.06
1.97
1.92
1.88
1.84
1.81
1.79
1.73
1.68
2.57
2.47
2.36
2.25
2.18
2.13
2.07
2.03
2.00
1.93
1.86
3.06
2.93
2.78
2.63
2.54
2.47
2.38
2.33
2.29
2.20
2.11
4.41
4.17
3.92
3.66
3.49
3.38
3.23
3.14
3.08
2.92
2.78
10
(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 Variancea
1.00 ,---,------,-----,-------,-----,------,-----,----,
0.80 l----+-----+------+-----+-----+-----1-----+----i
0.70 I----+-----+------+-----+,----+-----+----+----<
0.60
0.50
·";;';
Q)
0.40
.c
.0a>c.- 0.30 1-------,1---
Q)
-5
Cl
.!:
~==~========~=====~~~~~~~~~;;:~= 0.
Q)
0.10 l-----+----t-----'~~~~,---+--Wt-\\\\3i'\\\\l~-'\\;,l-
.""".'.. 0.087
0 0.06 1--1-----+----;>,M;J~cW'rl..-----\\ll,\\'\\:\\-\\-W\\-\\-Hr----+---l
0.05 l----+-----+---
:c
e.0"'
a..
0.04 1------+-----+---~
0.03 I---+-----+---
-::=i"?~i&'~M.-'1.---i\\'\\-\\-WK\\-\\-l\\--\\----+-----;
1.5
2
2.5
3
3.5 --<I>
(for et= 0.05)
<l>(foret=0.01)-
2
3
4
5
1.00 ,------""T'"-------,------,------,---------,,-----,
8:~18---:'.:_-=.-=.-=.-=.-=.-=.t-=--=-~
0.60
0.50
·".;';
Q)
0.40
.c
0a. 0.30
.>c -
Q)
-5
Cl
.!:
0. 0.10 1-------j\\\\\\,W.),\\\\;~=
8:8~ .""..'.
0 0.06 1------t--\\\\-ffi¼\\'i\\,,lf"=
:c
.a0"e..'
0.05 1-------t--Wffi-\\W~~
0.04 1-------+--'M~c\\\\'l~F
0.03 1---------l---\\\\'"4,.\\-\\','1-'~=~
IJ.~;,9~i,~x;;c;_t¥~;~;-::::~~-=--=--=--=--=--=--=--=-
2
3 -<I>
(for ex= 0.05)
<I>(for ex= 0.01)-
1
2
3
4
5
v, = Numerator degrees of freedom, v2 = Denominator degrees of freedom
'Adapted with pcnnission from Biometrika TablesforS1a1isticia11s, Vol. 2. by E. S. Pearson and H. 0. Hanley, Cambridge
University Press, Cambridge, 1972.
11
|
|
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
.,·e.n;
0.50
0.40
.c
0 0.30
.,.C>c.-
0.20
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C)
·1.',=i 0.10
0
0ro 0.08
0 0.07
&" 0.06
:rco
.0
0.05
0.04
0
,t 0.03
0.02
2
3 -<1>
(for a; 0.05)
<1>(for a; 0.01)-
1
2
3
4
5
0.80
0.70
0.60
.,·e.n;
0.50
0.40f-~~~'<'c'-l---*W'<l'<'..,._~oe-'--+-----+-----+---l
.c
0
.,.C>c.-
-5
C)
.s
1.,i 0.10 t----~l-\\-\\--'l-\\-',-\\-',+-\\--
t--------H--c-+--+-+--+-.-..-.-.. 00 0.087
>-+---+-.-+ ___+_... ___ ......._____________________
_.
&° 0.06
:C 0.05 t--------lrtt+++\\"1-\\-\\--lrt--+-t-\\-+-+--+---\\--\\--lr--~------t-----l
o.o4 E
t------+t-1-H.-+'.+-1-+--+---+-+-\\--lt-\\---tt--+-+--+-~-+--+----1-----1
ae.
2
3 -<1>
(for a; 0.05)
<1>(for a; 0.01)-
1
2
3
4
5
12
|
|
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
"'·";;'; 0.40
.,;;;
0 0.30
.>,Q;;.;
-"5' 0.20
-~Cl
C
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0.10
0.08
0 0.07
0.06
0.05
:.r.0co 0.04
a:0
0.03
0.02
2
3 -4>
(for a= 0.05)
<l>(for a= 0.01 )-
1
2
3
4
5
1.00 ------------------------------~
0.80
0.70
0.60
0.50
0.40 H™~M~+-----+--+'~,._,_,~~n__,,,..----+-----+-----1
2
3-
<l>(fora=0:05)
<l>(fora=0.01)-
1
2
3
4
13
|
|
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.------...------r-------.--------r-------r-----,
000...186000 ~I ~==_=_=-jt=_=_=_.=:.-.;_...-==_===_==:_:i=r=_===_=t=~=\\\\=!~=~=t~=l=\\\\=\\t=~=
0.50
1/)
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.Qc)
.0a>c.- 0.30 1-Wr\\\\-W.:'ttr----+---\\-+'t-\\-"""'A'~,...,,.__,'<+------+------l
Q)
£
C')
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C., . 0.10 r----;-Tt"Tt"T'<T\\-'\\-ti,'""7''t:=
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o:;l 0.08 l-------'~..4-'l+\\....\\.\\-1..,..\\-,A-,,.""-,..-1==-
0.07 f---H-\\-\\--\\4-\\.\\-\\-¼A.L_.-F=-
:;; 00..0065 l-----\\-1H-'IH--\\--\\~~---
0.04 f-------+t+-t-\\c-\\-'\\':\\-'H,44::=
ae..
2
3-
<I>(for a= 0.05)
<I>(for a= 0.01)-
1
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3
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1.00.-------r-------.--------r-------r------.-----,
00.·7800 1...:,:-_.!..._-_-_-_-~_-++----_...-..:_c--_1-,t_=--_----_+--_----_-+-i-l-l-l-l1:'l..:..J""~t~
0.60
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.c"'
0.401-\\\\:~~~~t--------l---\\-''rt-'n~t'<:''rl~~-+----+----l
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<l>{fora=0.01)-
1
2
3
4
14
|
|
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•.
t•
t distribution critical values
df
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
40
so
60
80
100
1000
z·
.25
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
0.857
0.856
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
1.088
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
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
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 level C
.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
2.887
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
3.261
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%
15