### Chapter 7, Survey Sampling

#### Solution 1

Population mean:

Population varaince:

Samples of size 2:

Note that every sample containing appears twice except for . Since in a sample order does not matter and both ’s are taken from the population, so there is only one possible sample for .

Thus we have total samples. The probability of occurance of a sample is the fraction of times it appears in the total(10) samples. For eg: Prob. of choosing sample is .

Sampling Distribution of the mean of samples of size 2:

Sample | Mean( ) | Probability( ) |
---|---|---|

Mean of all the sample means,

.

Variance of all the sample means, .

For better visibility, using the following table for computation:

Sample | Mean( ) | Probability( ) | |
---|---|---|---|

Clearly, Variance is the dot product of last two columns, .

Comparing the values of and population mean, , both are equal to , in agreement with the Theorem mentioned in problem.

Comparing the values of and population variance, , as per theorem . Here, is the sample size and , total population.

Putting these values ion RHS, , which is same as the .

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