Question

Question: Refer to Exercise 5.5, in which we found the sampling distribution of the sample median. Is the median an unbiased estimator of the population mean m?

Short answer

Yes, the median is an unbiased estimator of the population mean “m”.

Step-by-step solution

List of probabilities

The list of the probabilities found in Exercise 5.5 corresponding to the respective mean is shown below.

Mean	Probability
1	0.04
1.5	0.12
2	0.17
2.5	0.20
3	0.20
3.5	0.14
4	0.08
4.5	0.04
5	0.01

Determination of the biasedness of the median

The calculation of the meanandis shown below.

$$\begin{gathered} {\mathrm{\mu }}_{\mathrm{x}}=\mathrm{xp}\left(\mathrm{x}\right) \\ =\left\{1(0.2)+2(0.3)+3(0.2)+4(0.2)+5(0.1)\right\} \\ =2.7 \end{gathered}$$

$$\begin{gathered} \mathrm{E}\left(\overline{\mathrm{m}}\right)=\mathrm{E}\overline{\mathrm{m}}\mathrm{p}\left(\overline{\mathrm{m}}\right) \\ =\left(1\times 0.04\right)+\left(1.5\times 0.12\right)+\left(2\times 0.17\right)+\left(2.5\times 0.20\right)+\left(3\times 0.20\right) \\ +\left(3.5\times 0.14\right)+\left(4\times 0.08\right)+\left(4.5+0.04\right)+\left(5\times 0.01\right) \\ =\left(0.04+0.18+0.34+0.5+0.6+0.49+0.32+0.18+0.05\right) \\ =2.7 \end{gathered}$$

As the value oflocalid="1661429803872" $\mathbf{\mu }$andlocalid="1661429812869" $\mathbf{E}\mathbf{\left(}\overline{\mathbf{m}}\mathbf{\right)}$is 2.7 each, solocalid="1661429822357" $\overline{\mathbf{m}}$is an unbiased estimator oflocalid="1661429831113" $\mathbf{\mu }$.