# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 3

The best way to solve it is to think of random variable as a function which has a probability distribution. A random variable can take any values as per its distribution function.

a. No. It’s a constant value for a population.

b. No. It is also fixed for a population.

c. No. This is also a constant.

d. Yes. It is not fixed and can acquire a different value as we change the sample. It is a function of the sample, if sample changes, then sample mean changes.

e. No. Sample mean takes on different values based on sample and but variance of sample mean is the average of the squares of the differences between a sample mean and expected value of sample mean, - which is a constant. This can be verified by looking at its formulae for Simple Random Sampling ${\sigma}^2_{\bar X} = \frac { {\sigma}^2 } n \left( \frac {n-1} {N-1} \right)$. As we can check this value does not change irrespective of the sample!

f. Yes. Again this is a function of the sample. sample changes, largest-value changes.

g. No. It is also fixed for a population.

h. Yes. It also depends on the sample. Change of the sample also changes the estimated variance. This can be checked by looking at its formulae for Simple random sampling: $s_{\bar X} = \frac { s^2 } n \left( 1 - \frac n N \right)$, where $s^2 = \frac 1 {n-1} \sum_{i=1} n { \left( X_i - {\bar X} \right) }^2$. The $s^2$ depends on the given sample.

$$\tag*{\blacksquare}$$