### Chapter 7, Survey Sampling

#### Solution 24

Since this is a simple random sampling, we can use few results from the book:

, where is the population mean.

, where is population variance.

.

#### (a)

For , to be unbiased estimate of population mean, must be true.

Thus for to be true, we must have .

#### (b)

Lets first find an expression for the variance of the estimate, i.e. :

Thus, to minimize , we have to minimize under the given condition constraint .

This can be solved using Lagranges multiplier where is the function we have to minimize under the constraint .

We have:

Let is the partial differentiation of w.r.t and similarly is partial differentiation of w.r.t. .

Then by lagranges multiplier we should have for . It follows that , or . Now putting the value of in the constraint , gives . Thus . Now since , we get .

Now to check that this value of minimizes the , we can do the second derivative test, which I am skipping :)

Thus is the value that minimizes .

$$\tag*{$\blacksquare$} $$