### Chapter 7, Survey Sampling

#### Solution 37

#### (a)

We need . Since , it follows that . Since and are unbiased estimates of the population mean, , it follows . Thus . Thus to make , we need .

#### (b)

We have . Since and are independent it follows that .

Thus we have to minimize the function, say f, under the constraint, say g, . We can use Lagranges multuplier here:

, which gives and . We also have . Thus by solving these equations we get:

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