Formulas
Descriptive Statistics
(I)
Ix·
X=_l
n
s =
x
sp =
CL1I(x,I _ x)2
V-;;--
(n 1 - 1)si + (n 2 - 1)s~
(n1 - 1) + (n 2 - 1)
b =
I(xi-x)(yi-:Y)
1
I(\-X)2
r =
-1- I
n -1
-
(x. s- X)(Y. s-:Y]
_1__
x
_I_
y
s
b =r L
1
s
x
I(Yi - yJ2
n-2
~I(Xi - X)2
-48­
Probability
(II)
P(A u B) = P(A) + P(B) - P(A II B)
P(AIB)
= P(A II B)
P(B)
Var(X) = (T2 =
x
L(X.-~
)2 p.
!
X
1
If X has a binomial distribution with parameters nand p , then:
~X
= np
(T
= ~np(1- p)
(T
x
A
p
= ,,~
fP(l n
p)
If x is the mean of a random sample of size n from an infinite
population with mean ~ and standard deviation (T, then:
~x
=~
\
-49­
(III)
Inferential Statistics
Standardized test statistic:
statistic -parameter
standard deviation of statistic
Confidence interval: statistic ± (critical value) • (standard deviation of statistic)
Single-Sample
Sample Mean
Sample Proportion
Two-Sample
Difference of
sample means
Special case when
(Y
(Yl
= (Yz
H01
-.>..
nj
+­1
nz
Difference of
sample proportions
Special case when PJ = P:
b(l _ p)
~1 +
nJ
1
nz
.
. .
(observed - ex ected)z
~
Chi-square test statistic = "
L.J
expecte
-50­
Probability
Table entry for z is the
probability lying below z.
z
Table A Standard normal probabilities
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
-3.4
-3.3
-3.2
-3.1
-3.0
.0003
.0005
.0007
.0010
.0013
.0003
.0005
.0007
.0009
.0013
.0003
.0005
.0006
.0009
.0013
.0003
.0004
.0006
.0009
.0012
.0003
.0004
.0006
.0008
.0012
.0003
.0004
.0006
.0008
.0011
.0003
.0004
.0006
.0008
.0011
.0003
.0004
.0005
.0008
.0011
.0003
.0004
.0005
.0007
.0010
.0002
.0003
.0005
.0007
.0010
-2.4
-2.3
-2.2
-2.1
-2.0
.0082
.0107
.0139
.0179
.0228
.0080
.0104
.0136
.0174
.0222
.0078
.0102
.0132
.0170
.0217
.0075
.0099
.0129
.0166
.0212
.0073
.0096
.0125
.0162
.0207
.0071
.0094
.0122
.0158
.0202
.0069
.0091
.0119
.0154
.0197
.0068
.0089
.0116
.0150
.0192
.0066
.0087
.0113
.0146
.0188
.0064
.0084
.0110
.0143
.0183
-1.4
-1.3
-1.2
-1.1
-1.0
.,-0.9.
.0808
.0968
.1I51
.1357
.1587
;1841 .:
.0793
.0951
.1131
.1335
.1562
.1814
.0778
.0934
.1112
.1314
.1539
.r788:
.0764
.0918
.1093
.1292
.1515
.0749
.0901
.1075
.1271
.1492
.0735
.0885
.1056
.1251
.1469
,.17(e
.0721
.0869
.1038
.1230
.1446
;1685!"
.Q708
.0853
.1020
.1210
.1423
;1660':.'
.0694
.0838
.1003
.1190
.1401
.1635;'
.0681
.0823
.0985
.1170
.1379
-0.4
-0.3
-0.2
-0.1
-0.0
.3446
.3821
.4207
.4602
.5000
.3409
.3783
.4168
.4562
.4960
.3372
.3745
.4129
.4522
.4920
.3264
.3632
.4013
.4404
.4801
.3228
.3594
.3974
.4364
.4761
.3192
.3557
.3936
.4325
.4721
.3156
.3520
.3897
.4286
.4681
.3121
.3483
.3859
.4247
.4641
~~f: ·;:i";+i0~}_~1r~If~I'~'('i"";:$~
..
;~~~~il;~i'i~l~£.· 7'~0;'~JI~;c$ ··'·>'~~~,~,nj·, '~jllli:~~ii;
"i1762.'1736'
J6lf,.
.~~.~;;; .;i;·.• ··~;;~lit$~l~!f;·I~i'if~~'%';;:,·iik'{"~:t~iE.
.3336
.3707
.4090
.4483
.4880
.3300
.3669
.4052
.4443
.4840
-60­
Probability
Table entry for z is the
probability lying below z.
z
Table A
z
(Continuecl)
.00
.05
.06
.07
.08
.09
.5199
.5596
.5987
.6368
.6736
.5239
.5636
.6026
.6406
.6772
.5279
.5675
.6064
.6443
.6808
.5319
.5714
.6103
.6480
.6844
.5359
.5753
.6141
.6517
.6879
. 8051
80788106.8133
.8554
.8770
.8962
.9131
.9279
.8577
.8790
.8980
.9147
.9292
.8599
.8810
.8997
.9162
.9306
.8621
.8830
.9015
.9177
.9319
.9808
.9850
.9884
.9911
.9932
.9949 .'.
.9812
.9854
.9887
.9913
.9934
:9951 .
.9817
.9857
.9890
.9916
.9936
,9952
·i;;f;';~·/o!~~E>~~!'~;:r;!~rW.:;r, 'f;~:" '-:;~r[i
;;8~:f1:;,.·;s<·;&~4(J,;,ix;~~3,~~., ..:~389
1.0
J.I
1.2
1.3
1.4
Ji
.9788
.9834
.9871
.9901
.9925
:'.9943; ..
2.0
2.1
2.2
2.3
2.4
.9793
.9838
.9875
.9904
.9927
:9945
';S;-~!~~ii3.", .
.9798
.9803
.9842
.9846
.9878
.9881
.9906
.9909
,99299931
.9Q46> ..9948
9/8'," :~;~gt;~j.~'S:~~~t::f;q2jJ~~~~g~»
3.0
3.1
3.2
3.3
3.4
.9987
.9990
.9993
.9995
.9997
.9987
.9991
.9993
.9995
.9997
.9987
.9991
.9994
.9995
.9997
. 1:1)!~~~i:}
. ",',-,,
.9988
.9991
.9994
,9996
.9997
.9988
.9992
.9994
.9996
.9997
.9989
.9992
.9994
.9996
.9997
-61­
.9989
.9992
.9994
.9996
.9997
.9989
.9992
.9995
.9996
.9997
..·.,9Q6},
,'9~64
.9990
.9993
.9995
.9996
.9997
.9990
.9993
.9995
.9997
.9998
.
Table entry for p and
C is the point 1* with
probability plying
above it and
probability C lying
between -t * and 1*.
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
1.000
.816
.765
.741
1.376
1.061
.978
.941
.IS
.697
.695
- -.694
.692
.691
.873
.870
.868
.866
1.074
1.363
1.356
1.350
1.345
I.341
.686
.686
.685
.685
.684
.859
.858
.858
.857
.856
1.063
1.061
1.060
1.059
1.058
1.323
1.321
1.319
1.318
1.316
.681
.679
.679
.678
.677
.675
.674
.851
.849
.848
.846
.845
.842
.841
1.050
1.047
1.045
1.043
1.042
1.037
1.036
50%
60%
70%
.05
.025
.02
.01
.005
6.314
2.920
2.353
2.132
2.015
12.71
4.303
3.182
2.776
2.571
15.89
4.849
3.482
2.999
2.757
31.82
6.965
4.541
3.747
3.365
63.66
9.925
5.841
4.604
4.032
(,...' ?C#I'~:
I~i~Jjisfjl~
.0025
127.3
14.09
7.453
5.598
4.773
.001
318.3
22.33
10.21
7.173
5.893
00
636.6
31.60
12.92
8.610
6.869
; ;'i; ;lf!~,~.'_•:.;,',:',.._~i•. :.\•',. '.•_~.·.4:,.· ,~ ;~1.~ t,.'.: , .,'<.~!' ~ ,;,
..·,:,:
...•
.-.:,_.)• .•,. •..•·.,_•:.•.
••.
:••.•
·.8.• '''''.,.• .•. • .,:•;.'.,,. .
,,:,·.··;:.:·.·'.4:·
... ..' _
....;;.' ....
-,i,.·
..•
' ' '.
' '.
',·.... , ·
,I' ,
2.201
2.179
2.160
2.145
2.I31
2.328
2.303
2.282
2.264
2.249
2.718
2.681
2.650
2.624
2.602
3.106
3.055
3.012
2.977
2.947
3.497
3.428
3.372
3.326
3.286
4.025
3.930
3.852
3.787
3.733
1.721
1.717
1.714
1.711
1.708
2.080
2.074
2.069
2.064
2.060
2.189
2.183
2.177
2.172
2.167
2.518
2.508
2.500
2.492
2.485
2.831
2.819
2.807
2.797
2.787
3.135
3.119
3.104
3.091
3.078
3.527
3.505
3.485
3.467
3.450
1.303
1.299
1.296
1.292
1.290
1.282
1.282
1.684
1.676
1.671
1.664
1.660
1.646
1.645
2.021
2.009
2.000
1.990
1.984
1.'962
1.960
2.123
2.109
2.099
2.088
2.081
2.056
2.054
2.423
2.403
2.390
2.374
2.364
2.330
2.326
2.704
2.678
2.660
2.639
2.626
2.581
2.576
2.971
2.937
2.915
2.887
2.871
2.813
2.807
3.307
3.261
3.232
3.195
3.174
3.098
3.091
3.551
3.496
3.460
3.416
3.390
3.300
3.291
80%
90%
\
95%
96%
98%
99%
99.5%
99.8%
99.9%
11:!··.;i~! ~~I~~i&~i'ii!tJ:it~ir~i,~~~!jt)~, Ii .···JJI!.... .!~!.,.
40
50
60
8.0
100
1000
.0005
'.C}:~~~;-:.'.{:1\ ': .\ : i j j , : { ) - , " t r
Confidence level C
-62­
4.437
4.318
4.221
4.140
4.073
!HI
3.819
3.792
3.768
3.745
3.725
" ·,q·~~~;/;S['f/;W:ft:gz!l;
,.,-,-~ -,~.'!:.'
Probability p
Table entry for p is the point
(X 2 ) with probability plying
above it.
Cx2 )
Table C
df
1
2
3
4
5
X 2 critical values
.25
.20
.15
1.32
2.77
4.11
5.39
6.63
1.64
3.22
4.64
5.99
7.29
2.07
3.79
5.32
6.74
8.12
2.71
4.61
6.25
7.78
9.24
14.63
15.81
16.98
18.15
19.31
15.77
16.99
18.20
19.41
20.60
17.28
18.55
19.81
21.06
22.31
]]
12
13
14
15
:>r6~'
'jit;~f~z.;
22,76
.025
.02
.01
.005
.0025
.001
.0005
5.02
7.38
9.35
11.14
12.83
5.41
7.82
9.84
11.67
13.39
6.63
9.21
11.34
13.28
15.09
7.88
10.60
12.84
14.86
16.75
9.14
11.98
14.32
16.42
18.39
10.83
13.82
16.27
18.47
20.51
12.12
15.20
17.73
20.00
22.]]
22.62
24.05
25.47
26.87
28.26
24.72
26.22
27.69
29.14
30.58
26.76
28.30
29.82
31.32
32.80
28.73
30.32
31.88
33.43
34.95
31.26
32.91
34.53
36.12
37.70
:~M
e•••_.••.•.•.• • •:".,
•• .••.• .•,.'
.•.•,.• •'••
• .',•• • ••
24·
·.. '.••.
:·•..•,:..·.,·
.•.•:.•·
-·:·.·.·..·..•·•·.••
:·.•
•.·'·.·.:.:
••·,,·
•.•.
47<;.·.·.·.:,c
••
:••
..·.. •, .•.;...••..,...•.•.••
-,•.: .
v
27.66
28.82
29.98
31.13
32.28
21
22
23
24
25
40
50
60
80
100
.05
3.84
5.99
7.81
9.49
11.07
.• ~;:~;
;i:~~ . .~;:~~ > ·J~·:~~·;:·~~:~~~3;9l?;·
1
. 2416 •... 25 991& 87
3153
. 3235'
34·.8. 1 ,.. .·• :•.. ~:· 4..•3.3'. 7. 8•:0. . . •:.,'•,.0.5'.1.,:··•.0.6•~...·. .~• :. •. •:. :.. . •. .,•:.•'.::.,•:. . .,,•. . ~9~4 : .~ ~.•·.4·.·•.,•.•..•2:.~:•.·:.: .·3.38·. ·. .· .· .2•.'.:.'::.,",.•.:•,.•:.:•.'•.:.;,,.'•.;',:•. .:•. •. . .:• .•,4. . ·. ~9·5 . · .cP7 A.' ·
44:r
·1~~!~i"\\·.~ti~~l{~:·:. ··'·il~~,~!i,.~':~.~~~~:r.i;ji:~~r~~f~f;~~..jt:9 . :' ... '. . .~. ....1
~~:~i .~~:~~
17
.10
45.62
56.33
66.98
88.13
109.1
47.27
58.16
68.97
90.41
111.7
49.24
60.35
71.34
93.11
114.7
29.62
30.81
32.01
33.20
34.38
32.67
33.92
35.17
36.42
37.65
35.48
36.78
38.08
39.36
40.65
;'. \!:,,:,-.'..t;;~·~}::~i':;" '"
59.34
71.42
83.30
106.6
129.6
-63­
36.34
37.66
38.97
40.27
41.57
,-'"
60.44
72.61
84.58
108.1
131.1
38.93
40.29
41.64
42.98
44.31
- .•
43.78
45.20
46.62
48.03
49.44
46.80
48.27
49.73
51.18
52.62
49.01
50.51
52.00
53.48
54.95
69.70
82.66
95.34
120.1
144.3
73.40
86.66
99.61
124.8
149.4
76.09
89.56
102.7
128.3
153.2
>
63.69
76.15
88.38
112.3
135.8
66.77
79.49
91.95
116.3
140.2