Chapter 7, Survey Sampling
Solution 2
Denoting the proportion of sample greator than 3 by $y_i$ for the $i^{th}$ sample.
Sampling Distribution of the quantity $y$ of samples of size 2:
| Sample | $y_i$ | $p_i$ |
|---|---|---|
| $(1,2)$ | $\frac 0 2$ | $\frac 2 {10}$ |
| $(1,4)$ | $\frac 1 2$ | $\frac 1 {10}$ |
| $(1,8)$ | $\frac 1 2$ | $\frac 1 {10}$ |
| $(2,2)$ | $\frac 0 2$ | $\frac 1 {10}$ |
| $(2,4)$ | $\frac 1 2$ | $\frac 2 {10}$ |
| $(2,8)$ | $\frac 1 2$ | $\frac 2 {10}$ |
| $(4,8)$ | $\frac 2 2$ | $\frac 1 {10}$ |
Mean of the quantity, $y$, $\mathrm{E} y = \sum_{i=1}^{i=N} p_i y_i$, which is the dot product of last two columns $= \frac 4 {10} = 0.4$.
$\mathrm{Var} (y) = \mathrm{E} { \left( y_i - {\mathrm{E} y} \right) }^2$.
For clarity, using the following table for computation:
| Sample | $y_i$ | ${ \left( y_i - {\mathrm{E} y} \right) }^2$ | $p_i$ |
|---|---|---|---|
| $(1,2)$ | $\frac 0 2$ | ${ ( \frac 0 2 - 0.4 ) }^2 = 0.16$ | $\frac 2 {10}$ |
| $(1,4)$ | $\frac 1 2$ | ${ ( \frac 1 2 - 0.4 ) }^2 = 0.01$ | $\frac 1 {10}$ |
| $(1,8)$ | $\frac 1 2$ | ${ ( \frac 1 2 - 0.4 ) }^2 = 0.01$ | $\frac 1 {10}$ |
| $(2,2)$ | $\frac 0 2$ | ${ ( \frac 0 2 - 0.4 ) }^2 = 0.16$ | $\frac 1 {10}$ |
| $(2,4)$ | $\frac 1 2$ | ${ ( \frac 1 2 - 0.4 ) }^2 = 0.01$ | $\frac 2 {10}$ |
| $(2,8)$ | $\frac 1 2$ | ${ ( \frac 1 2 - 0.4 ) }^2 = 0.01$ | $\frac 2 {10}$ |
| $(4,8)$ | $\frac 2 2$ | ${ ( \frac 2 2 - 0.4 ) }^2 = 0.36$ | $\frac 1 {10}$ |
Variance is the dot product of last two columns, $\mathrm{Var} (y) = 0.09$.
$$\tag*{$\blacksquare$} $$