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