Hello guys!

I am having problems with the following problem (from Cochran's "Sampling Techniques", 3rd edition, p.49, question 2.19):

Suppose in a population of N units, the value of $\displaystyle {Y}_{1}$ is very small while that of $\displaystyle {Y}_{n}$ is very large as compared to other values of the variable of interest. Show that for simple random sample of size n the following estimator is unbiased for estimating $\displaystyle \bar{Y} $.

$\displaystyle {\hat{\bar{Y}}}_{S} $=

{$\displaystyle \bar{y} +c$ if the sample contains $\displaystyle {Y}_{1}$ but not $\displaystyle {Y}_{n}$

{$\displaystyle \bar{y} -c$ if the sample contains $\displaystyle {Y}_{n}$but not $\displaystyle {Y}_{1}$

{$\displaystyle \bar{y}$ for all other samples

where c is a fixed constant.

Prove that:

$\displaystyle Var({\hat{\bar{Y}}}_{S}) = (1-\frac{n}{N} )(\frac{S^{2}}{n }-\frac{2c}{(N-1) }({Y}_{n}-{Y}_{1}-nc) ) $

Okay, firstly, to show that the estimator is unbiased, is the following method correct?

Remembering that $\displaystyle E(\bar{y})=\bar{Y}$,

$\displaystyle Bias({\bar{Y}}_{S})=E(\bar{y} +c)-\bar{Y}=0 $

$\displaystyle Bias({\bar{Y}}_{S})=E(\bar{y} -c)-\bar{Y}=0 $

$\displaystyle Bias({\bar{Y}}_{S})=E(\bar{y})-\bar{Y}=0 $

But then I get stuck on how to prove how to prove the variance of the estimator. I believe that the following formula for the variance of the sample mean from a simple random sample without replacement:

$\displaystyle V({{\bar{y}})=(\frac{1}{n})(1- \frac{n}{N})({S}^{2}) $

might come in handy but I am really stuck as I do not even know where to begin to find this required variance. I am really stuck on this one and any hints to help me would be GREATLY appreciated.