Question

x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate the standard error? (c) Show that the sample variance S1 defined by n 11 Ση11(Xi-X1)2 is an unbiased estimator of true 2 variance σ (d) Suppose that both populations have the same variance, that is, σ-σ: σ2 . Show that mi + n2-2 is an unbiased estimator of σ2

Answer 1

solution : oiven that unbiased estimat0918 Mer H2 are zi, 7, respecthere we know that unbiased estimators of 0,2,0,2 are S;?, se respectively cas estimatong Of Ho-H2 Ž -X2 is an unbiased central limit theoriem forom we know X, ND CH 102 X2 nd Che, E [, - X2J = E [X] - Elže] Hence. it is an unbiased estimator

(br. The standaard error of Xi - X2 mane Xn0 CH₂ , 021ni X2 und CM2 , 02² ingo X, - Xq wo CH - H2, of +02² Ines standand deviation consider fop imitol my > 0 . (el. The sample variance s² defined by CX7X, )2 Let X1, X2 ----xn be a grandom sample from population with mean = 41. Svarliance = 0,2 Ex; = H, Vexin-07,2

expe=c727H 13 → EX, EH, v Că, 2 =VC Exi) = 11n? EUCxí) =o2 in, E (22) = VCX, ) + E CXj2 Det = colin,)th, ELX, 2) = (0,2 niotho now Sample variance $id = im,-i Cx; -,52 C Ex, - n, x, ?, EGP = my ec Exp2-nities Eins (EEx; 2-n,E772) E n - s

ES? = I, C Cor+M? 2-n, (of in the = Ź, (2,0,2-0,2 f970 m equations and ESP = (no) infin ne in 2 (d). Both populations have the same variance sp2 - 09,2=022 202 CM, -1)5,+(12-13522 nitha-2 E [SP2] = Ecn,-1) 5²+(12-1) 52² nith2-2

- cn,-1) E {S2 + (ne- E [S22] nit12-2 since 0,2-0,2-02 = (1,-1262+(12-1)002 - = 02 nitna-2 It is an onblased estimatoon , -X- Thank you