Chapter 7, Survey Sampling

Solution 12


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.


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



Lemma 2


Now, we will prove the main result

We have:

Cancelling from both sides, it follows .


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 .


We need to show

We have:


We have:


To show , where , the sample proportion.

We have:

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