### Chapter 7, Survey Sampling

#### Solution 11

#### (a)

In simple random sampling, a sample of a given size is chosen from all possible samples with same size from the population. Thus the total samples are .

#### (b)

We have sample mean equals to:

Thus the expected value of the sample mean is

In the given samples , we can see that every element of the population is present in equal number of samples(i.e. 2). (Note that in a sample, order does not matter and are same samples.)

Thus every member of a **random sample** has an equal chance to be any element or of the population. In other words, every element has an equal probability to become a part of a *random sample*. Thus probability for any member of sample and any element of the population, is . It follows that .

( **Note:** Initially, I made a rookie mistake here by computing the probability , thinking that each member appears twice in total 4 samples. This is wrong because this probability is for whether a **specific element** is a member of a sample or not but the probability that we require is the probability of selection of an element in a random sample. Another way to see that is wrong is that the sum of probabilities of choosing each of the members should equal 1 but in this case it becomes , which is clearly wrong!)

Thus

But this expression is the population mean .

It follows that . Thus the sample mean is unbiased.

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