Question

1. (Sampling means)

(a) Explain as thoroughly as you can why the mean of the sampling distribution of means is equal to the population mean. (

b) Explain as thoroughly as you can why the variance of the sampling distribution of means gets smaller with larger sample sizes.

Answer 1

Assume that the population mean is M and variance B². Then a random sample (X₁,X₂,..,X_n) of size n is independent and identically distributed observations from the population. Then E(X_i)=M and Var(X_i)=B² and cov(X_i,X_j)=0 for every i not equal to j.

(a) Then sample mean T=( X₁+X₂+..+X_n)/n.

Then by sum law of expectation, E(T)=(E( X₁)+E(X₂)+..+E(X_n))/n=nE(X₁)/n= E(X₁) =M, as the observations are iid. Hence T is unbiased for population mean M.

(b) Since the observations are independent, Var(T)= [Var( X₁)+Var(X₂)+..+Var(X_n)]/n². As the observations are iid variances are all equal and hence Var(T)=n Var( X₁)/n²=Var(X₁)/n.

Since, as n increases, (1/n) decreases and as Var(X₁) is fixed for the population,   Var(X₁)/n also decreases with increased n. Thus Var(T) decreases as sample size (=n) increases.