Chapter 7.1, Problem 19E

To determine

To find: the proportion of red cars in the sample and display the sampling distribution of the sample proportion on dot plot and is the sample proportion an unbiased estimator of the population proportion, explain.

Answer to Problem 19E

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 of size 2	Number of red cars	Proportion of red cars
1,2	1	$\widehat{p}=\frac{1}{2}=0.5$
1,3	1	$\widehat{p}=\frac{1}{2}=0.5$
1,4	1	$\widehat{p}=\frac{1}{2}=0.5$
2,3	0	$\widehat{p}=\frac{1}{2}=0.5$
2,4	1	$\widehat{p}=\frac{1}{2}=0.5$
3,4	2	$\widehat{p}=\frac{1}{2}=0.5$

Graph:

  

A statistic is an unbiased estimator if the centre of the sampling distribution is the addition of all sample proportions divided by the total number of samples.

  $\begin{array}{c}\overline{x}=\frac{\text{Sum sample proportions}}{\text{Number of sample}}\\ =\frac{0+0.5+0.5+0.5+0.5+1}{6}\\ =\frac{3}{6}\\ =\frac{1}{2}\end{array}$

The proportion is 0.5, because there are 2 red cars from the 4 cars in the sample and 0.5, since the mean of the sampling distribution is same to the population proportion, the sample proportion is an unbiased estimator of the population proportion.