Question

Consider a random sample (WOR) of two households from a population of households having monthly income (in Rs.) as follows:
Household 1 2 3 4 5
Income 1000 1200 900 1500 1300
Enumerate all possible samples (WOR) of size 2 and show that the sample mean gives an unbiased estimate of population mean.

Answer 1

Answer to Question #84129 - Math - Statistics and Probability

Question: Consider a random sample (WOR) of two households from a population of households having monthly income (in Rs.) as follows:

Enumerate all possible samples (WOR) of size 2 and show that the sample mean gives an unbiased estimate of population mean.

Solution: The given population has total 5 households, and here we shall consider a random sample "without replacement". Therefore, there are total $\binom{5}{2} = 10$ possible outcomes for a sample of size 2.

In the following table, we represent these 10 outcomes and compute sample mean for each case:

True mean of the population is,

$$\mu = \frac {1 0 0 0 + 1 2 0 0 + 9 0 0 + 1 5 0 0 + 1 3 0 0}{5} = \frac {5 9 0 0}{5} = 1 1 8 0$$

As each of the 10 possible outcomes for the sample of size 2 is equally likely, each of them will occur with probability $\frac{1}{10}$. Therefore, the expectation is sample mean is,

$$\begin{array}{l} E(\bar{X}) \\ = \frac{1}{10} \times (1100 + 950 + 1250 + 1150 + 1050 + 1350 + 1250 + 1200 + 1100 + 1400) \\ = \frac{1}{10} \times 11800 \\ = 1180 = \mu \end{array}$$

Hence, we have, $E(\bar{X}) = \mu$

This shows that, sample mean is unbiased estimate of population mean.

Answer

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