Question

A statistic is said to be an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, it is said to be a biased estimator. An unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not.

Part (a): Is the sample mean an unbiased estimator of the population mean? Explain your answer.

Part (b): Is the sample median an unbiased estimator of the population mean? Explain your answer.

Short answer

Part (a): Yes, as mean of all possible sample means for a fixed sample size is equal to the population mean.

Part (b): Yes, sample median is an biased estimator of population median.

Mean of all possible sample median for sample size n, is not equal to the population median.

Step-by-step solution

Part (a) Step 1. Explain the answer.

Yes, sample mean is an unbiased estimator of the population mean. Since, mean of all possible sample means for a fixed sample size is equal to the population mean.

Part (b) Step 1. Explain the answer.

No, sample median is not an unbiased estimator of the population median.

Suppose the population consists of five players says A, B, C, D, Eand the variable under consideration is height of players in inches.

The table shows the height of the players,

Here, the total number of popuation observation is $N=5$, which is odd. The observations are in increasing order.

So, the population meanrole="math" localid="1652638307953" $=\frac{N+1}{2}$th observation

$$\begin{gathered} =\frac{5+1}{2}thobservation \\ =3rdobservation \\ =79 \end{gathered}$$

Therefore, the population mean is79inches.

Part (b) Step 2. Make a table considering samples of size 3 from the population.

Consider samples of size 3 from the population,

Number of possible samples of size 3 from the population of size 5 is 10.Then,

$$\begin{gathered} =\frac{78+78+78+79+79+81+79+79+81+81}{10} \\ =\frac{793}{10} \\ =79.3 \end{gathered}$$

Therefore, mean of all possible sample median for sample size 3, is not equal to the population median 79 inches. Thus, sample median is an biased estimator of population median.