### Chapter 7, Survey Sampling

#### Solution 12

#### (a)

To prove that is unbiased in estimating , we need to show that .

To Prove that we shall first prove the following Lemmas:

**Lemma 1**

, where denotes the population mean.

*Proof*

Since this is a random sampling with replacement and every element having equal chance to be selected:

Thus

Thus

**Lemma 2**

*Proof*

Now, we will prove the main result

We have:

Cancelling from both sides, it follows .

#### (b)

No, it is not an unbiased estimate for sigma. By Jensenâ€™s Inequality:

Thus . It follows that is not always equal to , or is not an unbiased estimate of .

#### (c)

We need to show

We have:

#### (d)

We have:

#### (e)

To show , where , the sample proportion.

We have:

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