Chapter 7.1, Problem 20E

To determine

To find: the minimum age for each sample and display the sampling distribution of the sample minimum on a dot plot and is the sample minimum an unbiased estimator of the population minimum, explain.

Answer to Problem 20E

Not an unbiased estimator

Explanation of Solution

Given:

Car number	color	Age
1	Red	1
2	White	5
3	Sliver	8
4	Red	20

Calculation:

All possible sample of size 2 then contain any two cars of the population of 4 cars.

Sample of size 2

1,2

1,3

1,4

2,3

2,4

3,4

Sample minimum is the smallest age of the two cars

Sample of size 2	Sample minimum
1,2	$\min \left(1,5\right)=1$
1,3	$\min \left(1,8\right)=1$
1,4	$\min \left(1,20\right)=1$
2,3	$\min \left(5,8\right)=5$
2,4	$\min \left(5,20\right)=5$
3,4	$\min \left(5,20\right)=5$

Graph:

  

For every given data value places a dot above the corresponding number on the number line.

Unbiased estimator if the centre of the sampling distribution is the true value of the parameter,

  $\begin{array}{c}\overline{x}=\frac{\text{Sum sample minimum}}{\text{Number of sample}}\\ =\frac{1+1+1+5+5+8}{6}\\ =\frac{21}{6}\\ =\frac{7}{2}\\ =3.5\end{array}$

The population minimum is 1; the reason is that the smallest age among the 4 cars is 1.

Since the mean of the sampling distribution is not same to the population minimum, the sample minimum is not an unbiased estimator of the population minimum.