Stat Formula Sheet

FORMULA SHEET (page 1 of 3) x = (xi)/n Q1 Position = (n+3)/4 St.Dev. = Var s2 = 1/(n-1)*[x2i - 1/n*(xi)2] CV = ( s /x )*100 Q2 Position = (n+1)/2 Q3 Position = (3n+1)/4 Range = Max - Min IQR = Q3 - Q1 Chebyshev's Rule: at least [100*(1-1/z2)]% where z = the number of st. dev. Cov(X,Y) = sxy = 1/(n-1)*[xiyi - 1/n*(xi)*(yi)] r = sxy/[sx*sy] ________________________________________________________________________________________ P(A) = 1 - P(Ac) P(A or B) = P(A) + P(B) - P(A and B) P(A and B) = P(B|A)*P(A) P(B|A) = P(A and B)/P(A) Two events are mutually exclusive if P(A and B) = 0 Two events are independent if P(B|A) = P(B) ________________________________________________________________________________________ E(X) = {xi*P(X=xi)} Var(X) = [{x2i *P(X=xi)}] - [E(X)]2 n! = n*(n-1) *(n-2)*....*1 nCx Binomial Distribution: nCx * = n!/[x!*(n-x)!] px * (1-p)(n-x) E(X)=n*p Var(X)=n*p*(1-p) ________________________________________________________________________________________ Normal Distribution: E(X)= Var(X)=2 Standard Normal Distribution: E(Z)=0 Var(Z)=1 Z=(x-)/ Sampling Distribution of x: E(x )= Var(x )=2/n Z=(x- )/(/n) Sampling Distribution of p: E(p )=p Var(p )=p(1-p)/n where p = x/n Z=(p-p )/[p(1-p)/n] ________________________________________________________________________________________ FORMULA SHEET (page 2 of 3) Inference on , Known  Zstat = (x - 0 ) / (/n ) C.I.: x  Z/2 (/n ) Inference on , Unknown  tstat = (x - 0 ) / ( s/n ) C.I.: x  t/2,n-1 ( s/n ) Inference on p, Large Sample Zstat = (p - p0 ) / [ p0(1-p0)/n ] C.I.: p  Z/2 [ p(1-p)/n ] ________________________________________________________________________________________ Inference on 1 - 2, Known s, Indep.Samples Zstat = [(x1 - x2 ) - (1 - 2 )0] / [( 21/n1) + ( 22/n2)] C.I.: (x1 - x2 )  Z/2[( 21/n1) + ( 22/n2)] Inference on 1 - 2, Unknown s, Indep.Samples tstat = [(x1 - x2 ) - (1 - 2 )0 ] / [(s21/n1) + (s22/n2)] C.I.: (x1 - x2 )  t/2,df[(s21/n1) + (s22/n2)] where df = [s21/ n1 + s22/ n2]2 / [(s21/ n1)2/(n1-1) + (s22/ n2)2/(n2-1)] round df down Inference on 1 - 2, Unknown s, Indep. Samples,  21= 22 Assumption tstat = [(x1 - x2 ) - (1 - 2 )0 ] / [(s2pool/n1) + (s2pool/n2)] where s2pool = [(n1-1)s21 + (n2-1)s22] / (n1+n2-2) 2 2 C.I.: (x1 - x2 )  t/2,n1+n2-2[(s pool/n1) + (s pool/n2)] Inference on 1 - 2, Unknown s, Dep.Samples (1 - 2=d) tstat = (d - d,0 ) / ( sd/n ) whered = (di)/n C.I.: d  t/2,n-1 ( sd/n ) s2d = {1/(n-1)}[d2i - (1/n)(di)2] Inference on p1 - p2, Large Indep.Samples, (p1 - p2 )0 = 0 Zstat = [(p1 - p2 ) - 0] / [{ppool(1-ppool)/n1} + {ppool(1-ppool)/n2}] C.I.: (p1 - p2 )  Z/2 [{p1(1-p1)/n1}+ {p2(1-p2)/n2}] whereppool = (x1 + x2) / (n1 + n2) ________________________________________________________________________________________ FORMULA SHEET (page 3 of 3) SSxx = x2i - 1/n(xi)2 SSyy = y2i - 1/n(yi)2 SSxy = xiyi - 1/n(xi)(yi) b1 = SSxy/SSxx b0 = y - b1x yhat = b0 + b1x dfReg = 1 SSR = b1*SSxy = (yi,hat - y)2 MSR = SSR/dfReg dfErr = n - 2 SSE = SST - SSR = (yi - yi,hat)2 MSE = SSE/dfErr = s2e dfTot = n - 1 SST = SSyy = (yi - y)2 r2 = SSR/SST corr = r2 GlobalFstat = MSR/MSE sb1 = [MSE/SSxx] C.I. for  1: b1  t/2,n-2*sb1 tstat = syhat = [MSE{1/n + (x0 - x)2/SSxx}] [b1- 0]/ sb1 C.I. for the mean of y given x=x0: yhat  t/2,n-2*syhat sind = [MSE{1 + 1/n + (x0 - x)2/SSxx}] = [MSE+(syhat)2] for the same x0 P.I. for an individual new y given x=x0: yhat  t/2,n-2*sind