Question

Consider a finite population of size N in which the mean of the variable of interest Y is u. Suppose a sample of size n is taken from this population using simple random sampling with replacement. Suppose that the sample mean Y is used to estimate u.

(a) [3 marks] Calculate the probability of sample and probability of inclusion for this sam- pling protocol.

(b) (4 marks] Show that Y is unbiased for u.

(c) [4 marks] Calculate Var(Y). Make sure to provide complete details of your deriva- tion/calculation.

Hint: Define Z; to be the number of times that unit i apears in the sample, i = 1,2, ..., N. Find the distribution of Z; and use it parts (a) and (b). You may use Cov(Z;,Z;) = - Mz for i tj without proof.

Answer 1

Ans.

(A) In the random sampling method, an equal probability of selection is assigned to each unit of the population at first draw. It also implies an equal probability of selecting any unit from the available units at subsequent draws. Thus in SRS from a population of N units, the probability of drawing any unit at the first draw in 1/N , The probability of drawing any unit in the second draw from the available (N-1) units in 1/(N-1) and so on.

Let be the event that any specified unit is selected at the r-th draw.

Then, =Prob. that any specified unit is not selected in anyone of the previous (r-1) draw and then selected at r-th draw.

.

The selection probability remains same.

For sample,

The probability that a specified unit is included in the sample= , by addition theorem of probability.

B. Let us take indicator variable

So,

Y is unbiased for .

C.Using the given hint ,

SUBSTITUTING VALUES,