### 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. Itâ€™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.

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