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