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 Jensen’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:

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