Q. 7.54 - Elementary Statistics

Question

7.54 Unbiased and Biased Estimators. 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.
a. Is the sample mean an unbiased estimator of the population mean? Explain your answer.
b. Is the sample median an unbiased estimator of the population median? (Hint: Refer to Example $7.2$ on pages $292-293$. Consider samples of size $2.$)

Step-by-Step Solution

Answer

(a) Yes, the sample mean is an unbiased estimator of population mean.
Because the population mean is equal to the mean of all possible sample means for a specific sample size.

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

Part (a) Step 1: Given information

To determine the sample mean is an unbiased estimator of the population mean.

Part (b) Step 2: Explanation

Yes, sample mean is a reliable predictor of population mean.
Because the population mean is equal to the mean of all possible sample means for a specific sample size.

Part (b) Step 3: Given information

To explain the sample median is an unbiased estimator of the population median.

Part (b) Step 4: Explanation

Assume the population consists of five players,$A, B, C, D,$and $E,$ and the variable in question is the players' height in inches.
The heights of the players are listed in the table below:
Player as $A B C D E$
Height (in inches) $76 78 79 81 86$
The total number of population observations in this case is$N=5$, which is odd.
The observations are listed in ascending order of importance.
Then, the population median$=\frac{N+1}{2}-th$ observation

$=\frac{5+1}{2}-th$ observation

$=3-rd$ observation
$=79$
Hence, the population median is $79$inches.

Part (b) Step 5: Explanation

Consider the population's sample size number $3$.

The number of size $3$ samples that can be taken from a population of size $5$ is $10$.

Sample	Observations (Height in inches)	Sample median$=\frac{n+1}{2}-{\text{ th obs }}^n$ $=\frac{3+1}{2}-{\text{ th obs }}^{\mathrm{n}}{\text{=2nd obs}}^n$
A,B,C	76,78,79	78
A,B,D	76,78.81	78
A,B,E	76,78,86	78
A,C,D	76,79,81	79
A,C,E	76,79,86	79
A,D,E	76,81,86	81
B,C,D	78,79,81	79
B,C,E	78,79,86	79
B,D,E	76,81,86	81
C,D,E	79,81,86	81

As a result, for samples of size$3$, the mean of all possible sample medians
$\frac{78+78+78+79+79+81+79+79+81+81}{10}$
$=\frac{793}{10}$
$=79.3$
As a result, the average of all possible sample medians for size $3$ samples does not equal the population median of $79$ inches.
As a result, the sample median is a biased estimator of the population median.