Mathematical Statistics and Data Analysis - Solutions

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, $\mathrm{Var} (y) = 0.09$.

$$\tag*{\blacksquare}$$