Question

Let p1,p2p1,p2 denote the probability that a randomly selected male and female, respectively, has allergy to nuts. Let n1,n2n1,n2 be the sample size of a random sample for male and female, respectively. Assume two samples are indepedent. Let X1,X2 be the number of male and female who have allergy to nuts in the random sample, respectively.

(1)(3pts) For parameters p1,p2,p1,p2, and p1−p2p1−p2, find one unbiased estimator for each of them. And show why they are unbiased.

(2)(3pts) Derive the formula for the standard error of those estimators in (1). Note that V(X−Y)=V(X)+V(Y)V(X−Y)=V(X)+V(Y) for two independent rv's X,YX,Y.

(3)(3pts) For given samples, let n1=100,n2=150,x1=5,x2=9.n1=100,n2=150,x1=5,x2=9. Compute the the value of those estimators in (1).

(4)(3pts) For given samples, let n1=100,n2=150,x1=5,x2=9.n1=100,n2=150,x1=5,x2=9. Compute the estimated standard errors of those estimators in (2).

Answer 1