continuous distributions equations - memphis ii equations.pdf · 2011. 11. 15. · confidence...
TRANSCRIPT
Continuous Distributions Exponential:
Standard Normal:
Confidence Intervals
CONFIDENCE INTERVAL ON A MEAN (σ KNOWN)
where is the critical point corresponding to a tail area of
CONFIDENCE INTERVAL ON A MEAN (σ UNKNOWN)
where is the critical point corresponding to a tail area of
CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (σ1 and σ2 KNOWN)
where is the critical point corresponding to a tail area of
CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (σ1 = σ2 UNKNOWN)
where Sp is a “pooled” estimator of the unknown standard deviation and is calculated as
But this can only be used if both populations are normally distributed.
CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (PAIRED SAMPLES)
But this can only be used if both populations are normally distributed.
CONFIDENCE INTERVAL ON A VARIANCE
But this can only be used if the population is normally distributed.
Hypothesis Testing
HYPOTHESIS TEST ON A MEAN (σ KNOWN)
Ho:
TS: ← uses known σ
Ha: Ha: Ha:
RR: RR: RR:
HYPOTHESIS TEST ON A MEAN (σ UNKNOWN, n LARGE)
Ho:
TS: ← uses s instead of σ
Ha: Ha: Ha:
RR: RR: RR:
HYPOTHESIS TEST ON A MEAN (σ UNKNOWN, n SMALL)
Ho:
TS: ← uses t instead of z
Ha: Ha: Ha:
RR: RR: RR:
DIFFERENCE BETWEEN MEANS (σ KNOWN)
Ho:
TS: ← uses known σ
Ha: Ha: Ha:
RR: RR: RR:
DIFFERENCE BETWEEN MEANS (σ UNKNOWN, n LARGE)
Ho:
TS: ← uses s instead of σ
Ha: Ha: Ha:
RR: RR: RR:
DIFFERENCE BETWEEN MEANS (σ UNKNOWN BUT EQUAL, n SMALL)
Ho:
TS: ← uses t instead of z
Ha: Ha: Ha:
RR: RR: RR:
DIFFERENCE BETWEEN MEANS (σ UNKNOWN AND UNEQUAL)
Ho:
TS:
⇐ round down!
Ha: Ha: Ha:
RR: RR: RR:
DIFFERENCE BETWEEN MEANS (PAIRED SAMPLES)
Ho:
TS:
Ha: Ha: Ha:
RR: RR: RR:
45ENGINEERING PROBABILITY AND STATISTICS
UNIT NORMAL DISTRIBUTION
x f(x) F(x) R(x) 2R(x) W(x)
0.00.10.20.30.4
0.50.60.70.80.9
1.01.11.21.31.4
1.51.61.71.81.9
2.02.12.22.32.4
2.52.62.72.82.93.0
Fractiles1.28161.64491.96002.05372.32632.5758
0.39890.39700.39100.38140.3683
0.35210.33320.31230.28970.2661
0.24200.21790.19420.17140.1497
0.12950.11090.09400.07900.0656
0.05400.04400.03550.02830.0224
0.01750.01360.01040.00790.00600.0044
0.17550.10310.05840.04840.02670.0145
0.50000.53980.57930.61790.6554
0.69150.72570.75800.78810.8159
0.84130.86430.88490.90320.9192
0.93320.94520.95540.96410.9713
0.97720.98210.98610.98930.9918
0.99380.99530.99650.99740.99810.9987
0.90000.95000.97500.98000.99000.9950
0.50000.46020.42070.38210.3446
0.30850.27430.24200.21190.1841
0.15870.13570.11510.09680.0808
0.06680.05480.04460.03590.0287
0.02280.01790.01390.01070.0082
0.00620.00470.00350.00260.00190.0013
0.10000.05000.02500.02000.01000.0050
1.00000.92030.84150.76420.6892
0.61710.54850.48390.42370.3681
0.31730.27130.23010.19360.1615
0.13360.10960.08910.07190.0574
0.04550.03570.02780.02140.0164
0.01240.00930.00690.00510.00370.0027
0.20000.10000.05000.04000.02000.0100
0.00000.07970.15850.23580.3108
0.38290.45150.51610.57630.6319
0.68270.72870.76990.80640.8385
0.86640.89040.91090.92810.9426
0.95450.96430.97220.97860.9836
0.98760.99070.99310.99490.99630.9973
0.80000.90000.95000.96000.98000.9900
x x x x−xx−x
f
46 ENGINEERING PROBABILITY AND STATISTICS
STUDENT'S t-DISTRIBUTION
VALUES OF
α,n
α
t
tα,n
n
12345
6789
10
1112131415
1617181920
2122232425
26272829
∞30
0.10
3.0781.8861.6381.5331.476
1.4401.4151.3971.3831.372
1.3631.3561.3501.3451.341
1.3371.3331.3301.3281.325
1.3231.3211.3191.3181.316
1.3151.3141.3131.311
1.2821.310
0.05
6.3142.9202.3532.1322.015
1.9431.8951.8601.8331.812
1.7961.7821.7711.7611.753
1.7461.7401.7341.7291.725
1.7211.7171.7141.7111.708
1.7061.7031.7011.699
1.6451.697
0.025
12.7064.3033.1822.7762.571
2.4472.3652.3062.2622.228
2.2012.1792.1602.1452.131
2.1202.1102.1012.0932.086
2.0802.0742.0692.0642.060
2.0562.0522.0482.045
1.9602.042
0.01
31.8216.9654.5413.7473.365
3.1432.9982.8962.8212.764
2.7182.6812.6502.6242.602
2.5832.5672.5522.5392.528
2.5182.5082.5002.4922.485
2.4792.4732.4672.462
2.3262.457
n
12345
6789
10
1112131415
1617181920
2122232425
26272829
∞30
0.15
1.9631.3861.3501.1901.156
1.1341.1191.1081.1001.093
1.0881.0831.0791.0761.074
1.0711.0691.0671.0661.064
1.0631.0611.0601.0591.058
1.0581.0571.0561.055
1.0361.055
0.20
1.3761.0610.9780.9410.920
0.9060.8960.8890.8830.879
0.8760.8730.8700.8680.866
0.8650.8630.8620.8610.860
0.8590.8580.8580.8570.856
0.8560.8550.8550.854
0.8420.854
0.25
1.0000.8160.7650.7410.727
0.7180.7110.7060.7030.700
0.6970.6950.6940.6920.691
0.6900.6890.6880.6880.687
0.6860.6860.6850.6850.684
0.6840.6840.6830.683
0.6740.683
0.005
63.6579.9255.8414.6044.032
3.7073.4993.3553.2503.169
3.1063.0553.0122.9772.947
2.9212.8982.8782.8612.845
2.8312.8192.8072.7972.787
2.7792.7712.7632.756
2.5762.750
47ENGINEERING PROBABILITY AND STATISTICS
CR
ITIC
AL
VA
LU
ES
OF
TH
EF
DIS
TR
IBU
TIO
N–
TA
BL
E
For
ap
art
icu
lar
com
bin
ati
on
of
nu
mer
ato
ra
nd
den
om
ina
tor
deg
rees
of
free
dom
,en
try
rep
rese
nts
the
crit
ical
va
lues
of
F c
orr
esp
on
din
g
toa
spec
ifie
du
pp
erta
il a
rea
(α).
Nu
mer
ato
rdf
1D
enom
ina
tor
df2
12
34
56
78
91
01
21
52
02
43
04
06
01
20
1 2 3 4 5 6 7 8 91
0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120
161.
418
.51
10.1
37.
716
.61
5.99
5.59
5.32
5.12
4.9
6
4.84
4.75
4.67
4.60
4.5
4
4.49
4.45
4.41
4.38
4.3
5
4.32
4.30
4.28
4.26
4.2
4
4.23
4.21
4.20
4.18
4.1
7
4.08
4.00
3.92
3.84
199.
519
.00
9.55
6.94
5.7
9
5.14
4.74
4.46
4.26
4.1
0
3.98
3.89
3.81
3.74
3.6
8
3.63
3.59
3.55
3.52
3.4
9
3.47
3.44
3.42
3.40
3.3
9
3.37
3.35
3.34
3.33
3.3
2
3.23
3.15
3.07
3.00
215.
719
.16
9.28
6.59
5.4
1
4.76
4.35
4.07
3.86
3.7
1
3.59
3.49
3.41
3.34
3.2
9
3.24
3.20
3.16
3.13
3.1
0
3.07
3.05
3.03
3.01
2.9
9
2.98
2.96
2.95
2.93
2.9
2
2.84
2.76
2.68
2.60
224.
619
.25
9.12
6.39
5.1
9
4.53
4.12
3.84
3.63
3.4
8
3.36
3.26
3.18
3.11
3.0
6
3.01
2.96
2.93
2.90
2.8
7
2.84
2.82
2.80
2.78
2.7
6
2.74
2.73
2.71
2.70
2.6
9
2.61
2.53
2.45
2.37
230.
219
.30
9.01
6.26
5.0
5
4.39
3.97
3.69
3.48
3.3
3
3.20
3.11
3.03
2.96
2.9
0
2.85
2.81
2.77
2.74
2.7
1
2.68
2.66
2.64
2.62
2.6
0
2.59
2.57
2.56
2.55
2.5
3
2.45
2.37
2.29
2.21
234.
019
.33
8.94
6.16
4.9
5
4.28
3.87
3 .58
3.37
3.2
2
3.09
3.00
2.92
2.85
2.7
9
2.74
2.70
2.66
2.63
2.6
0
2.57
2.55
2.53
2.51
2.4
9
2.47
2.46
2.45
2.43
2.4
2
2.34
2.25
2.17
2.10
236.
819
.35
8.89
6.09
4.8
8
4.21
3.79
3.50
3.29
3.1
4
3.01
2.91
2.83
2.76
2.7
1
2.66
2.61
2.58
2.54
2.5
1
2.49
2.46
2.44
2.42
2.4
0
2.39
2.37
2.36
2.35
2.3
3
2.25
2.17
2.09
2.01
238.
919
.37
8.85
6.04
4.8
2
4.15
3.73
3.44
3.23
3.0
7
2.95
2.85
2.77
2.70
2.6
4
2.59
2.55
2.51
2.48
2.4
5
2.42
2.40
2.37
2.36
2.3
4
2.32
2.31
2.29
2.28
2.2
7
2.18
2.10
2.02
1.94
240.
519
.38
8.81
6.00
4.7
7
4.10
3.68
3.39
3.18
3.0
2
2.90
2.80
2.71
2.65
2.5
9
2.54
2.49
2.46
2.42
2.3
9
2.37
2.34
2.32
2.30
2.2
8
2.27
2.25
2.24
2.22
2.2
1
2.12
2.04
1.96
1.88
241.
919
.40
8.79
5.96
4.7
4
4.06
3.64
3.35
3.14
2.9
8
2.85
2.75
2.67
2.60
2.5
4
2.49
2.45
2.41
2.38
2.3
5
2.32
2.30
2.27
2.25
2.2
4
2.22
2.20
2.19
2.18
2.1
6
2.08
1.99
1.91
1.83
243.
919
.41
8.74
5.91
4.6
8
4.00
3.57
3.28
3.07
2.9
1
2.79
2.69
2.60
2.53
2.4
8
2.42
2.38
2.34
2.31
2.2
8
2.25
2.23
2.20
2.18
2.1
6
2.15
2.13
2.12
2.10
2.0
9
2.00
1.92
1 .83
1.75
245.
919
.43
8.70
5.86
4.6
2
3.94
3.51
3.22
3.01
2.8
5
2.72
2.62
2.53
2.46
2.4
0
2.35
2.31
2.27
2.23
2.2
0
2.18
2.15
2.13
2.11
2.0
9
2.07
2.06
2.04
2.03
2.0
1
1.92
1.84
1.75
1.67
248.
019
.45
8.66
5.80
4.5
6
3.87
3.44
3.15
2.94
2.7
7
2.65
2.54
2.46
2.39
2.3
3
2.28
2.23
2.19
2.16
2.1
2
2.10
2.07
2.05
2.03
2.0
1
1.99
1.97
1.96
1.94
1.9
3
1.84
1.75
1.66
1.57
249.
119
.45
8.64
5.77
4.5
3
3.84
3.41
3.12
2.90
2.7
4
2.61
2.51
2.42
2.35
2.2
9
2.24
2.19
2.15
2.11
2.0
8
2.05
2.03
2.01
1.98
1.9
6
1.95
1.93
1.91
1.90
1.8
9
1.79
1.70
1.61
1.52
250.
119
.46
8.62
5.75
4.5
0
3.81
3.38
3.08
2.86
2.7
0
2.57
2.47
2.38
2.31
2.2
5
2.19
2.15
2.11
2.07
2.0
4
2.01
1.98
1.96
1.94
1.9
2
1.90
1.88
1.87
1.85
1.8
4
1.74
1.65
1.55
1.46
251.
119
.47
8.59
5.72
4.4
6
3.77
3.34
3.04
2.83
2.6
6
2.53
2.43
2.34
2.27
2.2
0
2.15
2.10
2.06
2.03
1.9
9
1.96
1.94
1.91
1.89
1.8
7
1.85
1.84
1.82
1.81
1.7
9
1.69
1.59
1.50
1.39
252.
219
.48
8.57
5.69
4.4
3
3.74
3.30
3.01
2.79
2.6
2
2.49
2.38
2.30
2.22
2.1
6
2.11
2.06
2.02
1.98
1.9
5
1.92
1.89
1.86
1.84
1.8
2
1.80
1.79
1.77
1.75
1.7
4
1.64
1.53
1.43
1.32
253.
319
.49
8.55
5.66
4.4
0
3.70
3.27
2.97
2.75
2.5
8
2.45
2.34
2.25
2.18
2.1
1
2.06
2.01
1.97
1.93
1.9
0
1.87
1.84
1.81
1.79
1.7
7
1.75
1.73
1.71
1.70
1.6
8
1.58
1.47
1.35
1.22
254.
319
.50
8.53
5.63
4.3
6
3.67
3.23
2.93
2.71
2.5
4
2.40
2.30
2.21
2.13
2.0
7
2.01
1.96
1.92
1.88
1.8
4
1.81
1.78
1.76
1.73
1.7
1
1.69
1.67
1.65
1.64
1.6
2
1.51
1.39
1.25
1.00
∞
∞
F(α,
df 1, d
f 2)F
α =
0.0
5
0
48 ENGINEERING PROBABILITY AND STATISTICS
critical values of x2 distributionC
RIT
ICA
L V
AL
UE
S O
FX
2 D
IST
RIB
UT
ION
f (X 2
)
0X
2
X2
α
α,n
Deg
rees
of
Fre
edom
.9
95
.990
.975
.950
.900
.100
.050
.025
.010
X2
.005
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
50
60
70
80
90
100
0.
0000
393
0.
0100
251
0.
0717
212
0.
2069
90
0.41
1740
0.67
5727
0.98
9265
1.34
4419
1.73
4926
2.15
585
2.
6032
1
3.07
382
3.
5650
3
4.07
468
4.
6009
4
5.14
224
5.
6972
4
6.26
481
6.
8439
8
7.43
386
8.
0336
6
8.64
272
9.
2604
2
9.88
623
10
.519
7
11.1
603
11
.807
6
12.4
613
13
.121
1
13.7
867
20
.706
5
27.9
907
35
.534
6
43.2
752
51
.172
0
59.1
963
67
.327
6
0.
0001
571
0.
0201
007
0.
1148
32
0.
2971
10
0.
5543
00
0.
8720
85
1.
2390
43
1.
6464
82
2.
0879
12
2.
5582
1
3.05
347
3.
5705
6
4.10
691
4.
6604
3
5.22
935
5.
8122
1
6.40
776
7.
0149
1
7.63
273
8.
2604
0
8.89
720
9.
5424
9
10.1
9567
10.8
564
11
.524
0
12.1
981
12
.878
6
13.5
648
14
.256
5
14.9
535
22
.164
3
29.7
067
37
.484
8
45.4
418
53
.540
0
61.7
541
70
.064
8
0.00
0982
10.
0506
356
0.21
5795
0.48
4419
0.83
1211
1.23
7347
1.68
987
2.17
973
2.70
039
3.24
697
3.81
575
4.40
379
5.00
874
5.62
872
6.26
214
6.90
766
7.56
418
8.23
075
8.90
655
9.59
083
10.2
8293
10.9
823
11.6
885
12.4
011
13.1
197
13.8
439
14.5
733
15.3
079
16.0
471
16.7
908
24.4
331
32.3
574
40.4
817
48.7
576
57.1
532
65.6
466
74.2
219
0.
0039
321
0.
1025
87
0.
3518
46
0.
7107
21
1.
1454
76
1.
6353
9
2.16
735
2.
7326
4
3.32
511
3.
9403
0
4.57
481
5.
2260
3
5.89
186
6.
5706
3
7.26
094
7.
9616
4
8.67
176
9.
3904
6
10.1
170
10
.850
8
11.5
913
12
.338
0
13.0
905
13
.848
4
14.6
114
15
.379
1
16.1
513
16
.927
9
17.7
083
18
.492
6
26.5
093
34
.764
2
43.1
879
51
.739
3
60.3
915
69
.126
0
77.9
295
0.
0157
908
0.
2107
20
0.
5843
75
1.
0636
23
1.
6103
1
2.20
413
2.
8331
1
3.48
954
4.
1681
6
4.86
518
5.
5777
9
6.30
380
7.
0415
0
7.78
953
8.
5467
5
9.31
223
10
.085
2
10.8
649
11
.650
9
12.4
426
13
.239
6
14.0
415
14
.847
9
15.6
587
16
.473
4
17.2
919
18
.113
8
18.9
392
19
.767
7
20.5
992
29
.050
5
37.6
886
46
.458
9
55.3
290
64
.277
8
73.2
912
82
.358
1
2.
7055
4
4.60
517
6.
2513
9
7.77
944
9.
2363
5
10.6
446
12
.017
0
13.3
616
14
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7
15.9
871
17
.275
0
18.5
494
19
.811
9
21.0
642
22
.307
2
23.5
418
24
.769
0
25.9
894
27
.203
6
28.4
120
29
.615
1
30.8
133
32
.006
9
33.1
963
34
.381
6
35.5
631
36
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2
37.9
159
39
.087
5
40.2
560
51
.805
0
63.1
671
74
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0
85.5
271
96
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2 1
07.5
65
118
.498
3.
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6
5.99
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0
18.3
070
19
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1
21.0
261
22
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1
23.6
848
24
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8
26.2
962
27
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1
28.8
693
30
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5
31.4
104
32
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5
33.9
244
35
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5
36.4
151
37
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5
38.8
852
40
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3
41.3
372
42
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9
43.7
729
55
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5
67.5
048
79
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9
90.5
312
101
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1
13.1
45
124
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5.
0238
9
7.37
776
9.
3484
0
11.1
433
12
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5
14.4
494
16
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8
17.5
346
19
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8
20.4
831
21
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0
23.3
367
24
.735
6
26.1
190
27
.488
4
28.8
454
30
.191
0
31.5
264
32
.852
3
34.1
696
35
.478
9
36.7
807
38
.075
7
39.3
641
40
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5
41.9
232
43
.194
4
44.4
607
45
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2
46.9
792
59
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7
71.4
202
83
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6
95.0
231
106
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1
18.1
36
129
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6.
6349
0
9.21
034
11
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9
13.2
767
15
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3
16.8
119
18
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3
20.0
902
21
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0
23.2
093
24
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0
26.2
170
27
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3
29.1
413
30
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9
31.9
999
33
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7
34.8
053
36
.190
8
37.5
662
38
.932
1
40.2
894
41
.638
4
42.9
798
44
.314
1
45.6
417
46
.963
0
48.2
782
49
.587
9
50.8
922
63
.690
7
76.1
539
88
.379
4 1
00.4
25
112
.329
1
24.1
16
135
.807
7.
8794
4
10.5
966
12
.838
1
14.8
602
16
.749
6
18.5
476
20
.277
7
21.9
550
23
.589
3
25.1
882
26
.756
9
28.2
995
29
.819
4
31.3
193
32
.801
3
34.2
672
35
.718
5
37.1
564
38
.582
2
39.9
968
41
.401
0
42.7
956
44
.181
3
45.5
585
46
.927
8
48.2
899
49
.644
9
50.9
933
52
.335
6
53.6
720
66
.765
9
79.4
900
91
.951
7 1
04.2
15
116
.321
1
28.2
99
140.
169
X2
X2
X2
X2
X2
X2
X2
X2
X2
Sour
ce: T
hom
pson
, C. M
., “T
able
s of
the
Perc
enta
ge P
oint
s of
the
X2 –Dis
trib
utio
n,”
Biom
etri
ka, ©
1941
, 32,
188
-189
. Rep
rodu
ced
by p
erm
issi
on o
f th
e Bi
omet
rika
Tru
stee
s.