 |
DAE702S- DESIGN AND ANALYSIS OF EXPERIMENTS - JAN 2020 |
|
|
1 Pages 1-10 |
▲back to top |
|
|
1.1 Page 1 |
▲back to top |
‘4
NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BAMS
LEVEL: 7
COURSE CODE: DAE702S
CEOXUPRESRIEMENNATMSE: DESIGN AND ANALYSIS OF
SESSION: JANUARY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SECOND OPPORTUNITY/SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINERS
DR C.R KIKAWA
MODERATOR:
PROF PETER NJUHO
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
ATTACHMENTS
Standard Normal Distribution table, Inverse Cum Freq Distribution table, t-table, f-tables
(T-12 to T 19)
THIS QUESTION PAPER CONSISTS OF 6 PAGES (including this front page)
|
|
1.2 Page 2 |
▲back to top |
NAMIBIA UNIVERSITY OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF MATHEMATICS AND STATISTICS
DESIGN & ANALYSIS OF EXPERIMENTS: DAE702S
SECOND/SUPPLEMENTARY EXAMINATION: JANUARY 2020
Time-3 Hrs
Attempt all Questions
Maximum Marks - 100
1. Question
(a) Discuss the following concepts as used in experimental design:
1. Treatment
Experimental units
Responses
Randomization
Confounding
(15 marks, (3@))
(b) Briefly discuss two examples that could best distinguish an experimental unit and a
measurement unit.
(Hint: One example should be agricultural and the other educational)
(5 marks)
2. Question
The viscosity of a liquid detergent is supposed to average 800 units at 25°c. A random sample
of 16 batches of detergent is collected, and the average viscosity is 812 units. Suppose we know
that the standard deviation of viscosity is g = 25 units.
(a) State the hypotheses that should be tested.
(2 marks)
(b) Test these hypotheses using a = 0.05. What are your conclusions?
(10 marks)
(c) What is the p-value for the test in part (b)’?
(3 marks)
(d) Find a 95% confidence interval on the mean.
(5 marks)
|
|
1.3 Page 3 |
▲back to top |
3. Question
The tensile strength of portland cement is being studied. Four different mixing techniques can
be used economically. The data in Table 1 have been collected
Table 1: Results of Tensile strength and Mixing Techniques
Mixing Technique
1
2
3
4
Tensile Strength
3129 3000 2865 2890
3200 3300 2975 3150
2800 2900 2985 3050
2600 2700 2600 2765
(a) Construct an appropriate ANOVA table.
(b) Test the hypothesis that mixing techniques affect the strength of the cement.
0.05. What are your conclusions?
(13 marks)
Use a =
(7 marks)
4. Question
An experiment is conducted to compare four different mixtures of the components oxidizer,
binder, and fuel used in the manufacturing of rocket propellant. To compare the four mixtures,
five different samples of propellant are prepared from each mixture and readied for testing.
Each of five investigators is randomly assigned one sample of each of the four mixtures and
asked to measure the propellant thrust. These data are summarized in the table below. Use
a = 0.05.
Mixture
1
2
3
4
1
2,340
2,658
2,449
2,403
2
2,355
2,650
2,458
2,410
Investigator
3
2,362
2,665
2,432
2,418
4
2,350
2,640
2,437
2,397
5
2,348
2,653
2,445
2,405
Figure 1: Table of values
(a) Construct a general ANOVA display for the randomized complete block design (RCBD),
clearly defining all variables being used. (No calculation required).
(8 marks)
(b) Suppose that there are a treatments (factor levels) and 6 blocks. Write an effects model
for the RCBD.
(5 marks)
(c) State the appropriate hypotheses for the experiment.
(2 marks)
|
|
1.4 Page 4 |
▲back to top |
(d) Using the information in the output given in the ANOVA table below, test the hypotheses
stated in part (c). What are your conclusions?
(5 marks)
Tests of Between-Subjects Effects
Dependent Variable: Thrust
Source
treatment
Type Ill Sum
df
of Squares | Mean Square
3 | 261260.9500
87086.9833
block
4
452.5000
113.1250
Error
12
826.3000
68.8583
Corrected Total
19 | 262539.7500
F
|1264.7269
1.6429
Figure 2: General ANOVA table for a RCBD
Sig.
0000
2273
|
|
1.5 Page 5 |
▲back to top |
5. Question
Here we quote an experiment that had been designed as a Latin square. The skins of rabbits’
backs were inoculated with a diffusing factor in six separate sites. Six rabbits were therefore
used and the order in which the sites were inoculated was done six different ways. The outcome
measured was area of blister (cm). The overall objective was to see whether or not the order
of administration affected this outcome. The experimental design and data are represented in
the Latin square below.
Rabbit
4 2 3 4 8s 6
a iil Vv iv of vi il
79 87 74 74 7.1 82
b Wii
vi ov lil i
75 81 6 64 62 75
Position
d vi i
ii ii IVov
69 85 68 77 85 85
e ii ivi
ill Vv vi
67 99 73 64 64 7.3
fv
ow ii ivi
iil
73 83 73 58 64 7.7
Figure 3: Latin Square Design and Data
(a) You are required to construct a test workbook (ANOVA worksheet: Observations; Rabbit;
Position; Order).
(8 marks)
|
|
1.6 Page 6 |
▲back to top |
(b) From the Latin square test presented below, discuss the research findings (Strictly stick
to the objective of the research as stated).
(5 marks)
Factors: Rabbit, Position, Order.
Source of Variation Sum Squares DF Mean Squa
Rows
3.833333
5 0.766667
Columns
12.833333
5 2.566667
Treatments
0.563333
5 0.112667
Residual
13.13
20 0.6565
Total
30.36
35
F (rows) = 1.167809, P = .3592
F (columns) = 3.909622, P = .0124
F (treatments) = 0.171617, P = .9701
Figure 4: Results table for the Latin square design
(c) Discuss the concepts of a Latin Square design.
(7 marks)
END
|
|
1.7 Page 7 |
▲back to top |
Z
00
0.0 | .50000
0.1 | .53983
0.2 | .57926
0.3 | .61791
0.4 | .65542
0.5 | .69146
0.6 | .72575
0.7 | .75804
0.8 | .78814
0.9 | .81594
1.0 | .84134
1.1 | .86433
1.2 | .88493
1.3} .90320
1.4 | .91924
1.5 | .93319
1.6 | .94520
1.7 | .95543
1.8 | .96407
1.9 | .97128
2.0 | .97725
2.1 | .98214
2.2 | .98610
2.3 | .98928
2.4 | .99180
2.5 | .99379
2.6 | .99534
2.7 | .99653
2.8 | .99744
2.9 | .99813
3.0 | .99865
3.1 | .99903
3.2 | .99931
3.3 | .99952
3.4 | .99966
3.5 | .99977
3.6 | .99984
3.7 | .99989
3.8 | .99993
3.9 | .99995
01
50399
54380
58317
62172
65910
.69497
72907
76115
.79103
81859
84375
86650
88686
.90490
92073
93448
94630
95637
.96485
97193
.97778
98257
98645
98956
99202
99396
99547
99664
99752
99819
99869
99906
99934
99953
99968
.99978
99985
.99990
99993
99995
02
50798
54776
58706
62552
66276
69847
73237
76424
79389
82121
84614
86864
88877
90658
92220
93574
94738
95728
96562
97257
97831
98300
98679
98983
99224
99413
99560
99674
99760
99825
99874
99910
99936
99955
99969
99978
99985
99990
99993
99996
able Values Represent AREA to the LEFT of the Z score.
-03
.04
05
-06
07
08
09
51197
51595
51994
52392
52790
53188
53586
55172
55567
55962
56356
56749
57142
57535
59095
59483
59871
.60257
.60642
.61026
.61409
.62930
.63307
.63683
-64058
64431
64803
.65173
.66640
.67003
67364
.67724
.68082
68439
.68793
.70194
70540
.70884
.71226
.71566
.71904
.72240
73565
73891
74215
74537
74857
«75175
.75490
.76730
77035
77337
-77637
77935
.78230
78524
79673
79955
80234
80511
80785
81057
81327
82381
82639
82894
83147
83398
83646
83891
84849
85083
85314
85543
.85769
85993
86214
87076
87286
87493
.87698
.87900
.88100 88298
89065
89251
89435
89617
.89796
89973
90147
90824
90988
91149
91309
.91466
91621
91774
92364
92507
92647
92785
92922
93056
93189
93699
93822
93943
94062
94179
94295
94408
94845
94950
95053
95154
95254
95352
95449
95818
95907
95994
.96080
.96164
96246
.96327
.96638
.96712
.96784
96856
.96926
96995
.97062
.97320 97381
9744]
.97500 97558
.97615 .97670
97882
97932
97982
.98030
.98077
98124
98169
98341
98382
98422
.98461
98500
98537
98574
98713
98745
.98778
.98809 98840
.98870
98899
99010
.99036
99061
.99086
99111
99134
99158
99245
99266
99286
99305
99324
99343
99361
99430
99446
99461
99477
99492
99506
99520
99573
99585
99598
.99609
99621
99632
99643
99683
99693
99702
99711
99720
99728
.99736
.99767 .99774 99781
99788
99795
99801
.99807
99831
99836
99841
99846
99851
99856
99861
99878
99882
99886
99889
99893
99896
99900
99913
99916
99918
99921
99924
99926
99929
99938
99940
99942
99944
.99946
99948
99950
99957
99958
99960
99961
99962
99964
99965
99970
99971
99972
99973
99974
99975
.99976
99979
99980
99981
99981
99982
99983
99983
99986
99986
99987
99987
99988
99988
99989
99990
99991
99991
99992
99992
99992
99992
99994
99994
99994
99994
99995
99995
99995
99996
99996
99996
99996
99996
99997
99997
|
|
1.8 Page 8 |
▲back to top |
Statistical Tables for Students
Normal
Table 5 Normal distribution — inverse cumula
0.50 0.0000
0.51 0.0251
0.52 0.0502
0.53 0.0753
0.54 0.1004
0.55 0.1257
0.56 0.1510
0.57 0.1764
0.58 0.2019
0.59 0.2275
0.60 0.2533
0.61 0.2793
0.62 0.3055
0.63 0.3319
0.64 0.3585
0.65 0.3853
0.66 0.4125
0.67 0.4399
0.68 0.4677
0.69 0.4958
0.70 0.5244
0.71 0.5534
0.72 0.5828
0.73 0.6128
0.74 0.6433
0.75 0.6745
0.76 0.7063
0.77 0.7388
0.78 0.7722
0.79 0.8064
0.80 0.8416
0.81 0.8779
0.82 0.9154
0.83 0.9542
0.84 0.9945
0.85 1.0364
0.86 1.0803
0.87 1.1264
0.88 1.1750
0.89 1.2265
0.90 1.2816
0.91 1.3408
0.92 1.4051
0.93 1.4758
0.94 1.5548
0.95 1.6449
0.96 1.7507
0.97 1.8808
0.975 1.9600
0.98 2.0537
0.99 2.3263
0.991 2.3656
0.992 2.4089
0.993 2.4573
0.994 2.5121
0.995 2.5758
0.996 2.6521
0.997 2.7478
0.998 2.8782
0.999 3.0902
|
|
1.9 Page 9 |
▲back to top |
t Table
cum. prob
t 50
tzs
t a0
t ss
t 30
t 95
t 975
t a9
t 995
t s99
t g995
one-tail 0.50
0.25
0.20
0.15
0.10
0.05 0.025
0.01 0.005 0.001 0.0005
two-tails
1.00
0.50
0.40
0.30
0.20
0.10
0.05
0.02
0.01 0.002 0.001
df
1) 0.000
2} 0.000
3} 0.000
4) 0.000
5} 0.000
6] 0.000
7) 0.000.
1.000
0.816
0.765
0741
0.727
0718
0711
1.376 1.963
1.061 1.386
0978 1.250
0.941 1190
0.920 1.156
§ 0906 1134
(0896. 1119
3.078 6314 12.71
1.886 2.920
1.638 2.353
1.533 2.132
1476 2.015 2571
21 4409643 | 3447
1415 |. 15057
31.82
6.965
4.541
3.747
3.365
8 344s)
2.998
6366
9.925
5841
4.604
4.032
S707)
3499
31831
22.327
10.215
7.173
5893
6 208
4.785
636.62
31.599
12.924
8.610
6.869
6.659)
5.408
8) 20000. 0706
0.889 1408". 1397. 1.860 -
32.896 23/055).
= 91 0000. 0.703. 0883 1100 1.383 1.833
#2 10| 220.000. 0,700 0.879. 711093.) 1.372 1.812
11} 0.000 0697 0876 1.088 1.363 41.796
2824) 43.260 9 4207 4
(21643169. 44
2.718 3.106
12} 0.000 0695 0.873 1.083 1.356 1.782
2.681 3.055
13} 0.000 0694 0870 1.079 1.350 1.771
2.650 3.012
14, 0.000 0692 0.868 1.076 1.345 1.761
2.624 2.977
15} 0.000 0.691 0866 1.074 1.341 1.753
2.602 2.947
PAG |i 0000) 7 0.600 0.865 071 wiser et 46 5 0 ease loa
v i7\\ 0.000. 0.689. 0863. 1069 11333211740. 2410 = 2567, 2.808)
18) 20000 0688 0862.) 1.067. 133041734 2401. 2552. 21878©
910.000" 0688. 0861. 1066 1328 1720 2003 2539 | 296] 33
= 2210])
00,.000000"
1006.86867
00..886509)
11..0066345
1325
1.323
51.725 2.086.
1.721 2080
2.528.
2518
2.845
2.831
22) 0.000 0686 0.858 1.061 1.321 1.717 2.074 2.508 2.819
23) 0.000 0685 0.858 1.060 1.319 1.714 2.069 2.500 2.807
244 0.000 0685 0.857 1.059 1.318 1.711 2.064 2492 2.797
25) 0.000 0684 0856 1.058 1.316 1.708 2.060 2485 2.787
261, 0.000; 0.684 | 0.856 = 1058) 1315 4) 706s 20bee 476 3770
27|-- 0.000: 5 0.684 0.855. 1.057. 1.314% 4.703, 2.052 ...2473 2774
28), 0.000.) 0.683 | 0.855. 41.056 1318 4) 4701. 2048 2467, 2763 |
/29|) 0.000 0.683 0854) 1055. 4 1311 1609. 32:045.4 2460. 21756
0
40)
60}
80}
[20.000
0.000
0,683.7
0681
0.854
0.851
1055
1.050
0
AS10
1.303
eSr 2.04250
1.684 2.021
0.000 0679 0848 1.045 1.296 1.671 2.000
0.000 0678 0.846 1.043 1.292 1.664 1.990
2457 2.750
2423 2.704
38:
3307
2390 2660 3.232
2374 2639 3.195
646
3.551
3.460
3.416
100} 0.000 0.677 0.845 1.042 1.290 1.660 1.984 2364 2626 3.174 3.390
1000} 0.000 0.675 = 0.842, 1.037 = 1.282 1.646 = 1.962 =. 2.330 2.581 = 3.098 -~—s 3.300
Z\\7 0.000. 0674. 0.842. 1036 1,282 © 16455 41.960., 2326 2576 8000 3.201
0% 50% 60% 70% ~—=80%_~— 90% ~=— 95% ~— 98% ~— 99% 99.8% 99.9%
Confidence Level
t-table.xls 7/14/2007
|
|
1.10 Page 10 |
▲back to top |
T-12
Tables
Table entry for p is the
critical value F* with
probability p lying to
its right.
NS TABLE E ees
F critical values
Pp
.100
-050
1
.025
.010
.001
.100
.050
2
.025
.010
-001
.
.100
3
.050
g
3
.025
‘g
.010
g
.001
;)o
a
.100
3
.050
oe
4
.025
5
.010
3
.001
oO
&
est
.100
8
.050
2
5
.025
e
.010
A
.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
3712
3.46
5.14
7.26
10.92
27.00
3.26
4.74
6.54
9.55
21.69
Probability p
P*
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
3.29
4.76
6.60
9.78
23.70
3.07
4.35
5.89
8.45
18.77
3.52
5.19
7.39
11.39
31.09
3.18
4.53
6.23
9.15
21.92
2.96
4.12
5.52
7.85
17.20
3.45
5.05
7.15
10.97
29.75
3.11
4.39
5.99
8.75
20.80
2.88
3.97
5.29
746
16.21
3.40
4.95
6.98
10.67
28.83
3.05
4.28
5.82
8.47
20.03
2.83
3.87
5.12
7TA9
15.52
3.37
4.88
6.85
10.46
28.16
3.01
4.21
5.70
8.26
19.46
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
|
|
2 Pages 11-20 |
▲back to top |
|
|
2.1 Page 11 |
▲back to top |
T-14
Tables
F critical values (continued)
Pp
.100
.050
8
.025
-010
.001
.100
.050
9
.025
.010
.001
.100
.050
10
.025
.010
.001
-100
-050
Le
11
.025
2
.010
g
-001
&
2
.100
g
.050
ig
12
.025
=
.010
:
.001
3
.100
i
.050
cri
13
.025
e
.010
®
.001
on
A
.100
-050
14
.025
.010
001
.100
.050
15
.025
.010
001
-100
.050
16
025
.010
001
.100
.050
17
.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
2063:
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
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
|
|
2.2 Page 12 |
▲back to top |
T-16
Tables
ESN TABLE 3 ROLE etal ARON
F critical values (continued)
Pp
1
-100
.050
18
.025
.010
001
3.01
4.41
5.98
8.29
15.38
-100
-050
19
-025
.010
.001
2.99
4.38
5.92
8.18
15.08
.100
.050
20
.025
-010
.001
2.97
4.35
5.87
8.10
14.82
.100
2.96
-050
4.32
he
21
-025
5.83
2
.010
8.02
4
.001
14.59
g
.100
2.95
3
-050
4.30
2
22
.025
5.79
3
.010
7.95
:
001
14.38
3
.100
2
-050
a
23
.025
ec
.010
®
-001
oD
a
100
.050
24
.025
.010
001
2.94
4.28
5.75
7.88
14.20
2.93
4.26
5.72
7.82
14.03
.100
.050
25
.025
.010
001
2.92
4.24
5.69
7.77
13.88
-100
-050
26
-025
.010
001
2.91
4.23
5.66
7.72
13.74
-100
-050
27
-025
.010
001
2.90
4.21
5.63
7.68
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
Degrees of freedom in the numerator
3
4
5
6
7
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
4.87
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
6.81
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
4.18
6.49
2.17
2.74
3.33
4.14
6.41
2.17
2.73
3.31
4.11
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
3.85
5.89
2.08
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
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
2.85
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
1.98
2.42
2.87
3.51
5.31
1.97
2.40
2.84
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
1.95
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
4.89
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
1.87
2.25
2.63
3.15
4.57
|
|
2.3 Page 13 |
▲back to top |
T-18
Tables
F critical values (continued)
Degrees of freedom in the numerator
p
1
2
3
4
5
6
7
100
.050
28
025
.010
001
2.89
4.20
2.50
3.34
2.29
2.16
2.95
2.71
2.06
2.56
2.00
2.45
1.94
2.36
5.61
4.22
3.63
3.29
3.06
2.90
2.78
7.64
13.50
5.45
8.93
4.57
7A9
4.07
6.25
3.75
5.66
3.53
5.24
3.36
4.93
.100
.050
29
.025
010
001
2.89
2.50
2.28
2.15
2.06
1.99
1.93
4.18
5.59
3333
4.20
2.93
3.61
2.70
3:27
2,55
3.04
2.43
2.88
2.35
2.76
7.60
13.39
5.42
8.85
4.54
7.12
4.04
6.19
3.73
5.59
3.50
5.18
3.33
4.87
.100
-050
30
.025
.010
-001
2.88
4.17
5.57
7.56
13.29
2.49
3.32
4.18
5.39
8.77
2.28
2.92
3.59
4.51
7.05
2.14
2.69
3.25
4.02
6.12
2.05
2.53
3.03
3.70
5.53
1.98
2.42
2.87
3.47
5.12
1.93
2.33
2.75
3.30
4.82
te
=
a
B
40
-100
.050
.025
.010
.001
2.84
4.08
5.42
7.31
12.61
2.44
3.23
4.05
5.18
8.25
2.23
2.84
3.46
4.31
6.59
2.09
2.00
1.93
2.61
2.45
2.34
3.13
2.90
2.74
3.83
3.51
3.29
5.70
5.13
4.73
1.87
2.25
2.62
3.12
4.44
g
y
2
E
e
-100
-050
50
.025
.010
.001
2.81
4.03
5.34
7.17
12,22
2.41
2.20
2.06
1.97
1.90
1.84
3.18
2.79
2.56
2.40
2.29
2.20
3.97
3.39
3.05
2.83
2.67
2.55
5.06
4.20
3.72
3.41
3.19
3.02
7.96
6.34
5.46
4.90
4.51
4.22
3
H
as
°
o
8b
-100
.050
60
025
.010
.001
2.79
4.00
5.29
7.08
11.97
2.39
3.15
3.93
4.98
7.77
2.18
2.76
3.34
4.13
6.17
2.04
2.93
3.01
3.65
5.31
1.95
2.37
2.79
3.34
4.76
1.87
2.25
2.63
3.12
4.37
1.82
2.17
2.51
2.95
4.09
A
.100
-050
100
.025
.010
.001
2.76
3.94
5.18
6.90
11.50
2.36
3.09
3.83
4.82
7.41
2.14
2.70
3.25
3.98
5.86
2.00
2.46
2.92
3.51
5.02
1.91
2.31
2.70
3.21
4.48
1.83
2.19
2.54
2.99
4.11
1.78
2.10
2.42
2.82
3.83
100
-050
200
.025
-010
.001
213
3.89
5.10
6.76
11.15
233
3.04
3.76
4.71
7.15
2.11
2.65
3.18
3.88
5.63
1.97
2.42
2.85
3.41
4.81
1.88
2.26
2.63
3.11
4.29
1.80
2.14
2.47
2.89
3.92
1.75
2.06
2.35
2.73
3.65
.100
.050
1000
-025
.010
-001
2.71
3.85
2.31
3.00
2.09
2.61
1.95
1.85
2.38
2.22
1.78
2.11
1.72
2.02
5.04
6.66
10.89
3.70
4.63
6.96
3.13
3.80
5.46
2.80
3.34
4.65
2.58
3.04
4.14
2.42
2.82
3.78
2.30
2.66
3.51
8
1.90
2.29
2.69
3.23
4.69
1.89
2.28
2.67
3.20
4.64
1.88
2.27
2.65
3.17
4.58
1.83
2.18
2.53
2.99
4.21
1.80
2.13
2.46
2.89
4.00
1.77
2:10
2.41
2.82
3.86
1.73
2.03
2.32
2.69
3.61
1.70
1.98
2.26
2.60
3.43
1.68
1.95
2.20
2.53
3.30
9
1.87
2.24
2.61
3.12
4.50
1.86
2.22
2.59
3.09
4.45
1.85
2.21
2.57
3.07
4.39
1.79
2,12
2.45
2.89
4.02
1.76
2.07
2.38
2.78
3.82
1.74
2.04
2.33
2.72
3.69
1.69
1.97
2.24
2.59
3.44
1.66
1,93
2.18
2.50
3.26
1.64
1.89
2.13
2.43
3.13