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AS91581 代写 统计学 Statistic

 
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


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