AS91581 代写 统计学 Statistic
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AS91581 代写
AS91581 代写 统计学 Statistic
STA001 Examination Formula Booklet
Descriptive Statistics
Sample mean ¯ x =
n
X
i=1
x i /n
Sample variance s 2 =
1
n − 1
"
n
X
i=1
(x i − ¯ x) 2
#
or
1
n − 1
"
n
X
i=1
x 2
i
− n¯ x 2
#
z-score z =
x − µ
σ
Probability
pr(A) + pr(A C ) = 1
pr(A ∪ B) = pr(A) + pr(B) − pr(A ∩ B)
pr(A|B) =
pr(A ∩ B)
pr(B)
when pr(B) 6= 0
pr(A ∩ B) = pr(A)pr(B) if the events A and B are independent.
Discrete Random Variables
If X is a discrete random variable then the expectation
E(X) = µ =
X
xpr(x)
and the variance
V ar(X) = σ 2 =
X (x − µ) 2
pr(x) or
X
x 2 pr(x) − µ 2
Combining random variables
For any constants a and b, and random variables X and Y
E(aX + b) = aE(X) + b, V ar(aX + b) = a 2 V ar(X)
E(aX + bY ) = aE(X) + bE(Y ).
If X and Y are independent random variables, then
V ar(aX + bY ) = a 2 V ar(X) + b 2 V ar(Y ).
Sampling distributions
If X 1 ,X 2 ,X 3 ,...,X n are an independent and identically distributed random sample with
mean µ and standard deviation σ < ∞ then
E( ¯ X) = µ ¯
X
= µ V ar( ¯ X) = σ 2 ¯
X
=
σ 2
n
.
Inferences based on a single sample
Test statistic for a population mean µ:
t =
¯ x − µ 0
s/ √ n
where H 0 : µ = µ 0
and t is on n − 1 degrees of freedom.
Test statistic for a population proportion p:
z =
ˆ p − p 0
σ ˆ p
=
ˆ p − p 0
q
p 0 (1−p 0 )
n
where H 0 : p = p 0
Confidence interval for a population mean µ:
¯ x ± t α/2
s
√ n ,
where t α/2 is on n − 1 degrees of freedom.
Large sample confidence interval for a population proportion
ˆ p ± z α/2
s
ˆ p(1 − ˆ p)
n
.
Inferences based on two samples
Test statistic for comparing two independent population variances
F =
larger sample variance
smaller sample variance ,
where F is on n 1 − 1 numerator degrees of freedom and n 2 − 1 denominator degrees of
freedom.
Large sample confidence interval for comparing two independent population means (also for
small samples assuming unequal variances) estimated using:
(¯ x 1 − ¯ x 2 ) ± t α/2
s
s 2
1
n 1
+
s 2
2
n 2
,
where t α/2 is on the smaller of (n 1 − 1),(n 2 − 1) degrees of freedom.
Large sample test statistic for comparing two independent population means (also for small
samples assuming unequal variances) estimated using
t =
(¯ x 1 − ¯ x 2 ) − D 0
s
s 2
1
n 1
+
s 2
2
n 2
,
where H 0 : µ 1 − µ 2 = D 0 , and where t α/2 is on the smaller of (n 1 − 1),(n 2 − 1) degrees of
freedom.
Small sample confidence interval for comparing two independent population means
(¯ x 1 − ¯ x 2 ) ± t α/2
s
s 2
p
?
1
n 1
+
1
n 2
?
,
assuming equal variances estimated using
s 2
p
=
(n 1 − 1)s 2
1 + (n 2 − 1)s
2
2
n 1 + n 2 − 2
,
where t α/2 is on n 1 + n 2 − 2 degrees of freedom.
Small sample test statistic for comparing two independent population means assuming equal
variances
t =
(¯ x 1 − ¯ x 2 ) − D 0
s
s 2
p
?
1
n 1
+
1
n 2
? ,
where H 0 : µ 1 − µ 2 = D 0 and t is on n 1 + n 2 − 2 degrees of freedom.
Confidence interval for the mean paired difference between two populations
¯ x D ± t α/2
s D
√ n
D
,
where t α/2 is on n D − 1 degrees of freedom.
Test statistic for comparing the mean paired difference between two populations
t =
¯ x D − D 0
s D / √ n D
,
where where H 0 : µ D = D 0 and t is on n D − 1 degrees of freedom.
Large sample confidence interval for comparing two independent population proportions
(ˆ p 1 − ˆ p 2 ) ± z α/2 σ ˆ p 1 −ˆ p 2 ≈ (ˆ p 1 − ˆ p 2 ) ± z α/2
s
ˆ p 1 (1 − ˆ p 1 )
n 1
+
ˆ p 2 (1 − ˆ p 2 )
n 2
.
Large sample test statistic for comparing two independent population proportions
z =
(ˆ p 1 − ˆ p 2 ) − D 0
σ ˆ p 1 −ˆ p 2
≈
(ˆ p 1 − ˆ p 2 ) − D 0
s
ˆ p(1 − ˆ p)
?
1
n 1
+
1
n 2
? ,
where H 0 : p 1 − p 2 = D 0 and ˆ p =
ˆ p 1 n 1 + ˆ p 2 n 2
n 1 + n 2
.
Categorical Data
χ 2 =
P
[n i −E(n i )] 2
E(n i )
One-way table
where n i = count for cell i
E(n i ) = np i,0
p i,0 = hypothesized value of p i under H 0
and χ 2 is on k − 1 degrees of freedom
χ 2 =
P [n ij − ˆ
E(n ij )] 2
ˆ
E(n ij )
For testing association in a two-way table
where n ij = count for cell in row i column j
ˆ
E(n ij ) = r i c j /n
r i = total for row i
c j = total for column j
n = total sample size
and χ 2 is on (r − 1)(c − 1) degrees of freedom
POSITIVE z Scores
T ABLE A-2 (continued) Cumulative Area from the LEFT
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
3.50 .9999
and up
NOTE: For values of z above 3.49, use 0.9999 for the area.
*Use these common values that result from interpolation:
z score Area
1.645 0.9500
2.575 0.9950
Common Critical Values
Confidence Critical
Level Value
0.90 1.645
0.95 1.96
0.99 2.575
0 z
*
*
T ABLE A-3 t Distribution: Critical t Values
Area in One Tail
0.005 0.01 0.025 0.05 0.10
Degrees of Area in Two Tails
Freedom 0.01 0.02 0.05 0.10 0.20
1 63.657 31.821 12.706 6.314 3.078
2 9.925 6.965 4.303 2.920 1.886
3 5.841 4.541 3.182 2.353 1.638
4 4.604 3.747 2.776 2.132 1.533
5 4.032 3.365 2.571 2.015 1.476
6 3.707 3.143 2.447 1.943 1.440
7 3.499 2.998 2.365 1.895 1.415
8 3.355 2.896 2.306 1.860 1.397
9 3.250 2.821 2.262 1.833 1.383
10 3.169 2.764 2.228 1.812 1.372
11 3.106 2.718 2.201 1.796 1.363
12 3.055 2.681 2.179 1.782 1.356
13 3.012 2.650 2.160 1.771 1.350
14 2.977 2.624 2.145 1.761 1.345
15 2.947 2.602 2.131 1.753 1.341
16 2.921 2.583 2.120 1.746 1.337
17 2.898 2.567 2.110 1.740 1.333
18 2.878 2.552 2.101 1.734 1.330
19 2.861 2.539 2.093 1.729 1.328
20 2.845 2.528 2.086 1.725 1.325
21 2.831 2.518 2.080 1.721 1.323
22 2.819 2.508 2.074 1.717 1.321
23 2.807 2.500 2.069 1.714 1.319
24 2.797 2.492 2.064 1.711 1.318
25 2.787 2.485 2.060 1.708 1.316
26 2.779 2.479 2.056 1.706 1.315
27 2.771 2.473 2.052 1.703 1.314
28 2.763 2.467 2.048 1.701 1.313
29 2.756 2.462 2.045 1.699 1.311
30 2.750 2.457 2.042 1.697 1.310
31 2.744 2.453 2.040 1.696 1.309
32 2.738 2.449 2.037 1.694 1.309
33 2.733 2.445 2.035 1.692 1.308
34 2.728 2.441 2.032 1.691 1.307
35 2.724 2.438 2.030 1.690 1.306
36 2.719 2.434 2.028 1.688 1.306
37 2.715 2.431 2.026 1.687 1.305
38 2.712 2.429 2.024 1.686 1.304
39 2.708 2.426 2.023 1.685 1.304
40 2.704 2.423 2.021 1.684 1.303
45 2.690 2.412 2.014 1.679 1.301
50 2.678 2.403 2.009 1.676 1.299
60 2.660 2.390 2.000 1.671 1.296
70 2.648 2.381 1.994 1.667 1.294
80 2.639 2.374 1.990 1.664 1.292
90 2.632 2.368 1.987 1.662 1.291
100 2.626 2.364 1.984 1.660 1.290
200 2.601 2.345 1.972 1.653 1.286
300 2.592 2.339 1.968 1.650 1.284
400 2.588 2.336 1.966 1.649 1.284
500 2.586 2.334 1.965 1.648 1.283
1000 2.581 2.330 1.962 1.646 1.282
2000 2.578 2.328 1.961 1.646 1.282
Large 2.576 2.326 1.960 1.645 1.282
AS91581 代写 统计学 Statistic
