Question

1. This question takes you through the design-based approach to sorting out sampling
distributions, their means and variances, and the estimation of these quantities when
you sample with and without replacement.
Consider a population of 5 individuals. The variable is their annual income in 1000s
of dollars per year, and you want to estimate the population mean income based on
a random sample of size two. The values of income are as follows:
Individual Income
A 80
B 18
C 24
D 52
E 24

(a) Make a list of all the samples of size 2 possible without replacement. Find the
sample mean and sample variance of each of these samples using the formula

yS = 1/n sigma yi

s^2 = 1/(n-1) sigma(yi - y(average))^2

(b) Find the sampling distributions of the sample mean and sample standard devi-
ation based on (a).
(c) Find the expected values of the sampling distributions of by using

E[W] = sigma wP[W = w]

(d) Find the variance of the sampling distribution of by using
V [W] =sigma (w - E[W])2P[W = w]
(e) Compare the results from (c) and (d) with the population mean, the population
variance as calculated from
S^2 = 1/(N-1) sigma (yi - yU)^2
and the variance of the sampling distribution of as calculated from
V [y] = S2/n(1 -n/N)

(f) For sampling with replacement, nd the sampling distribution of the sample
mean by listing all the samples that can occur, nding the sample mean values
for each sample, and nding the probability that each value occurs. Be careful:
in one kind of sampling, the order in which the individuals are sampled can't
be ignored.

Answer 1