### Chapter 7, Survey Sampling

#### Solution 47

From Corollary B we have:

$\, \Exp(\bar Y_R) - \mu_Y \approx \frac 1 n \Prn{1 - \frac {n-1} {N-1} } \frac 1 {\mu_x} \Prn{ r \sigma^2_x - \rho \sigma_x \sigma_y} \,$.

We just need to plugin the values in the above expression. All the values are given in Example-D on page-226 in the book.

Skipping the computation part :)

I think the point of this problem is to compare the bias and variance as sample size $\, n \,$ increases. Looking at the formulae we can see that the bias should decrease as sample size increases as well as the ${bias}^2$ is very small compared to the variance(because ${bias}^2$ contains $\, n^2 \,$ but variance contains $\, n \,$). Thus the take away point is that bias can be ignored if a sample size is large.

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