Question

Consider a random sample (WOR) of two households from a population of
households having monthly income (in $) 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 on Question #55135 – Math – Statistics and Probability

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

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

Solution

Let $i$ be the number of household, $X_i$ is the corresponding income.

The sample mean of pair $i,j$ is given by

$$\frac{X_i + X_j}{2}.$$

All possible samples (WOR) of size 2:

Population mean is

$$\mu = \frac{1000 + 1200 + 900 + 1500 + 1300}{5} = 1180.$$

The sample mean is

$$\frac{\sum \bar{x}_t}{n} = \frac{1100 + 950 + 1250 + 1150 + 1050 + 1350 + 1250 + 1200 + 1100 + 1400}{10} = 1180.$$

Thus, formulae (1) and (2) show that

$$\frac{\sum \bar{x}_t}{n} = \mu.$$

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