FORMULAS FOR STATISTICS 1

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "FORMULAS FOR STATISTICS 1"

ŌΣίμων Κυπραίος
5 χρόνια πριν
Προβολές:

1 FORMULAS FOR STATISTICS 1 X = 1 n Sample statistics X i or x = 1 n x i (sample mean) S 2 = 1 n 1 s 2 = 1 n 1 (X i X) 2 = 1 n 1 (x i x) 2 = 1 n 1 Xi 2 n n 1 X 2 x 2 i n n 1 x 2 or (sample variance) E(X) = µ, var(x) = σ2 n, E(S 2 ) = σ 2 Sample statistics for normal distribution Z = X µ σ/ n T = X µ S/ n N(0, 1) t with n 1 degrees of freedom V = (n 1)S2 σ 2 = (X i X) 2 σ 2 χ 2 with n 1 degrees of freedom Z = (X 1 X 2 ) (µ 1 µ 2 ) σ 2 1 /n 1 + σ 2 2/n 2 N(0, 1) T = (X 1 X 2 ) (µ 1 µ 2 ) S p 1/n1 + 1/n 2 t with n 1 + n 2 2 degrees of freedom, where S 2 p = (n 1 1)S (n 2 1)S 2 2 n 1 + n 2 2 W = (X 1 X 2 ) (µ 1 µ 2 ) S 2 1 /n 1 + S 2 2/n 2 t with assuming σ 1 = σ 2 (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 n 1 and a 2 = s2 2 n 2 (Welch Satterthwaite approximation) F = S2 1/σ 2 1 S 2 2/σ 2 2 F with n 1 1 and n 2 1 degrees of freedom i

2 Point estimation Parameter θ Estimate ˆθ Estimator ˆΘ µ ˆµ = x X σ 2 σ 2 = s 2 S 2 m ˆm = q(0.5) Q(0.5) Interval estimation of expectation Z = X µ σ/ n : x ± z σ α/2 n T = X µ S/ n : x ± t s α/2 n (with n 1 degrees of freedom) Z = (X 1 X 2 ) (µ 1 µ 2 ) σ1 2 : (x 1 x 2 ) ± z α/2 + σ2 2 σ 2 1 /n 1 + σ2/n 2 2 n 1 n 2 T = (X 1 X 2 ) (µ 1 µ 2 ) 1 : (x 1 x 2 ) ± t α/2 s p + 1 S p 1/n1 + 1/n 2 n 1 n 2 (with n 1 + n 2 2 degrees of freedom) W = (X 1 X 2 ) (µ 1 µ 2 ) s 2 1 : (x 1 x 2 ) ± t α/2 + s2 2 (Welch Satterthwaite) S 2 1 /n 1 + S2/n 2 2 n 1 (with = (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 n 1 and a 2 = s2 2 n 2 ) n 2 Estimation of proportion for the binomial distribution ( ) n P(X = x) = p x (1 p) n x x E(X) = np, var(x) = np(1 p) ˆp = x n i=x ( ) n ˆp i i L(1 ˆp L ) n i = α 2, x i=0 ( ) n ˆp i i U(1 ˆp U ) n i = α 2 V = F = S2 1/σ 2 1 S 2 2/σ 2 2 (n 1)S2 σ 2 : : s 2 1 s 2 2 Interval estimation of variance (n 1)s 2 h 2,α/2 and 1 f 2,α/2 and s2 1 s f 1,α/2 (n 1)s2 h 1,α/2 (with n 1 degrees of freedom) (with n 1 1 and n 2 1 degrees of freedom) ii

3 Testing expectations z = x µ 0 σ/ n and H 0 : µ = µ 0 : H 1 Critical region P-probability µ > µ 0 z z α 1 Φ(z) µ < µ 0 z z α Φ(z) µ µ 0 z z α/2 2 min ( Φ(z), 1 Φ(z) ) Φ is the standard normal cumulative distribution function. t = x µ 0 s/ n and H 0 : µ = µ 0 : H 1 Critical region P-probability µ > µ 0 t t α 1 F (t) µ < µ 0 t t α F (t) µ µ 0 t t α/2 2 min ( F (t), 1 F (t) ) F is the cumulative t-distribution function with n 1 degrees of freedom. z = x 1 x 2 d 0 and H 0 : µ 1 µ 2 = d 0 : σ 2 1 /n 1 + σ2/n 2 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 z z α 1 Φ(z) µ 1 µ 2 < d 0 z z α Φ(z) µ 1 µ 2 d 0 z z α/2 2 min ( Φ(z), 1 Φ(z) ) Φ is the standard normal cumulative distribution function. t = x 1 x 2 d 0, where s 2 p = (n 1 1)s (n 2 1)s 2 2, and H 0 : µ 1 µ 2 = d 0 : s p 1/n1 + 1/n 2 n 1 + n 2 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 t t α 1 F (t) µ 1 µ 2 < d 0 t t α F (t) µ 1 µ 2 d 0 t t α/2 2 min ( F (t), 1 F (t) ) F is the cumulative t-distribution function with n 1 + n 2 2 degrees of freedom. t = x 1 x 2 d 0 and H 0 : µ 1 µ 2 = d 0 (Welch Satterthwaite) : s 2 1 /n 1 + s 2 2/n 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 t t α 1 F (t) µ 1 µ 2 < d 0 t t α F (t) µ 1 µ 2 d 0 t t α/2 2 min ( F (t), 1 F (t) ) F is approximatively the cumulative t-distribution function with (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 and a 2 = s2 2. n 1 n 2 iii

4 v = Testing variances (n 1)s2 σ 2 0 and H 0 : σ 2 = σ 2 0 : H 1 Critical region P-probability σ 2 > σ0 2 v h 2,α 1 F (v) σ 2 < σ0 2 v h 1,α F (v) σ 2 σ0 2 v h 1,α/2 or v h 2,α/2 2 min ( F (v), 1 F (v) ) F is the cumulative χ 2 -distribution function with n 1 degrees of freedom. f = 1 k s 2 1 s 2 2 and H 0 : σ 2 1 = kσ 2 2 : H 1 Critical region P-probability σ1 2 > kσ2 2 f f 2,α 1 G(f) σ1 2 < kσ2 2 f f 1,α G(f) σ1 2 kσ2 2 f f 1,α/2 tai f f 2,α/2 2 min ( G(f), 1 G(f) ) G is the cumulative F-distribution function with n 1 1 and n 2 1 degrees of freedom. Goodness-of-fit test H o : P(T 1 ) = p 1,..., P(T k ) = p k H = k (F i np i ) 2 np i χ 2 with k 1 degrees of freedom Independence test. Contingence tables P(T 1 ) = p 1,..., P(T k ) = p k and P(S 1 ) = q 1,..., P(S l ) = q l P(T i S j ) = p i,j (i = 1,..., k and j = 1,..., l) H 0 : p i,j = p i q j (i = 1,..., k and j = 1,..., l) H = k l (F i,j F i G j /n) 2 j=1 F i G j /n χ 2 with (k 1)(l 1) degrees of freedom Contingence table: S 1 S 2 S l Σ T 1 f 1,1 f 1,2 f 1,l f 1 T 2 f 2,1 f 2,2 f 2,l f T k f k,1 f k,2 f k,l f k Σ g 1 g 2 g l n iv

5 Maximum likelihood estimation L(θ 1,..., θ m ; x 1,..., x n ) = f(x 1 ; θ 1,..., θ m ) f(x n ; θ 1,..., θ m ) l(θ 1,..., θ m ; x 1,..., x n ) = ln f(x 1 ; θ 1,..., θ m ) + + ln f(x n ; θ 1,..., θ m ) (ˆθ 1,..., ˆθ m ) = argmax L(θ 1,..., θ m ; x 1,..., x n ) or θ 1,...,θ m (ˆθ 1,..., ˆθ m ) = argmax θ 1,...,θ m l(θ 1,..., θ m ; x 1,..., x n ) Model: Data: Linear regression y = β 0 + β 1 x β k x k + ɛ x 1 x 2 x k y x 1,1 x 1,2 x 1,k y 1 x 2,1 x 2,2 x 2,k y x n,1 x n,2 x n,k y n 1 x 1,1 x 1,2 x 1,k y 1 ɛ 1 β 0 1 x 2,1 x 2,2 x 2,k X = , y = y 2., ɛ = ɛ 2., β = β 1. 1 x n,1 x n,2 x n,k y n Data model: y = Xβ + ɛ. ɛ n β k ˆβ = b = (X T X) 1 X T y = β + (X T X) 1 X T ɛ C = (c ij ) = (X T X) 1 H = XCX T (hat matrix) P = I n H E(b i ) = β i, var(b i ) = c ii σ 2 and cov(b i, b j ) = c ij σ 2 σ 2 = ŷ = Xb e = y ŷ = Py = Pɛ SSE = e 2 = e 2 i = (y i ŷ i ) 2 SSE n k 1 = MSE, MSE = RMSE v

6 SST = (y i y) 2, SSR = (ŷ i y) 2 SST = SSE + SSR MST = SST n 1, MSR = SSR k ANOVA-table: Source of variation Degrees of freedom Sums of squares Mean squares F Regression Residual Total variation k n k 1 n 1 SSR SSE SST MSR σ 2 = MSE (MST) F = MSR MSE H 0 : β 1 = = β k = 0 : F = MSR MSE F with k and n k 1 degrees of freedom H 0 : β i = β 0,i ; T i = b i β 0,i RMSE c ii t with n k 1 degrees of freedom R 2 = SSR SST = 1 SSE SST Radj 2 = 1 MSE MST = 1 n 1 n k 1 Categorical regressors: z i z i,1 z i,2 z i,mi 1 A i, A i, A i,mi A i,mi SSE SST, y = β 0 + β 1 x β k x k + l (β i,1 z i,1 + + β i,mi 1z i,mi 1) + ɛ Logistical regression P(y = A) = e β 0 β 1 x 1 β k x k L(β 0,..., β k ) = L 1 (β 0,..., β k ) L n (β 0,..., β k ), where 1 p i = 1 + e β 0 β 1 x i,1 β k x i,k, if y i = A L i (β 0,..., β k ) = 1 p i = e β 0 β 1 x i,1 β k x i,k 1 + e β 0 β 1 x i,1 β k x i,k, if y i = B vi

7 (b 0, b 1,..., b k ) = argmax β 0,β 1...,β k L(β 0, β 1,..., β k ) ˆp 0 = e b 0 b 1 x 0,1 b k x 0,k. H = 12 n(n + 1) k j=1 Kruskal Wallis-test H 0 : µ 1 = = µ k W 2 j n j 3(n + 1) χ 2 with k 1 degrees of freedom Spearman rank correlation coefficient Samples: x 1,1,...,x 1,n and x 2,1,...,x 2,n Sequence numbers: r 1,1,...,r 1,n and r 2,1,...,r 2,n r S = r S = 1 (r 1,i r)(r 2,i r) (r 1,i r) 2 6 n(n 2 1) (r 2,i r) 2, where r = d 2 i, where d i = r 1,i r 2,i n (Assuming there are no duplicate numbers in the samples!) vii

8 Standard normal distribution Quantiles of the standard normal distribution z viii

9 χ 2 -distribution Quantiles of the chi-square distribution (left tail) d.o.f Quantile (left tail) E This table was produced using APL programs written by William Knight. ix

10 t-distribution Quantiles of the t-distribution (right tail) (both tails) D e g r e e s o f f r e e d o m (both tails) (right tail) This table was calculated by APL programs written by William Knight. x

11 F-distribution Right tail quantiles f 2, for F-distribution for = 0.05, 0.025, 0.01 and degrees of freedom v 1 and v 2. The inverses are the left tail quantiles f 1, for the same :s, and degrees of freedom v 2 and v 1. v 2 v 1 v 2 xi

12 Signed-rank test Critical Values of the Wilcoxon Signed Ranks Test n Two-Tailed Test One-Tailed Test! =.05! =.01! =.05! = xii

13 Mann Whitney-testi Critical Values of the Wilcoxon Ranked-Sums Test (Two-Tailed Testing) n! m Note: n is the number of scores in the group with the smallest sum of ranks; m is the number of scores in the other group. xiii

14 Two-sided tolerance interval table The table gives the coefficient k for a two-sided tolerance interval. k: γ = 0.1 γ = 0.05 γ = 0.01 n α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = xiv

15 One-sided tolerance interval table The table gives the coefficient k for a one-sided tolerance interval. k: γ = 0.1 γ = 0.05 γ = 0.01 n α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = xv

Σχετικά έγγραφα

APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679

APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared