Question

More sample proportions List all $4$possible SRSs of size n=$3$, calculate the proportion of red cars in the sample, and display the sampling distribution of the sample proportion on a dot plot with the same scale as the dot plot in Exercise $19$. How does the variability of this sampling distribution compare with the variability of the sampling distribution from Exercise $19$? What does this indicate about increasing the sample size?

From exercise$19$:

Car Number	Color	Age
$1$	Red	$1$
$2$	White	$5$
$3$	Silver	$8$
$4$	Red	$20$

Short answer

Required dot plot is

Dot plots with sample sizes of n=$3$have less variability than dot plots with sample sizes of n=$2$.

As the sample size grows, the sampling variability reduces.

Step-by-step solution

Given Information

We are given following data:

Car Number	Color	Age
$1$	Red	$1$
$2$	White	$5$
$3$	Silver	$8$
$4$	Red	$20$

We need to calculate the minimum age for each sample, and draw it's dot plots.

We need to explain how variability of this sampling distribution compare with the variability of the sampling distribution from Exercise$19$

Explanation

All possible samples of size $3$then contain any three cars all different of population of $4$cars.

Car Number	Color	Age
$1$	Red	$1$
$2$	White	$5$
$3$	Silver	$8$
$4$	Red	$20$

The sample proportion of red cars is calculated by dividing the number of red cars by the sample size.

Sample of size$3$	Numbers of red cars	Proportion of red cars
$1,2,3$	$1$	$\stackrel{\wedge }{p}=\frac{1}{3}$
$1,2,4$	$2$	$\stackrel{\wedge }{p}=\frac{2}{3}$
$1,3,4$	$2$	$\stackrel{\wedge }{p}=\frac{2}{3}$
$2,3,4$	$1$	$\stackrel{\wedge }{p}=\frac{1}{3}$

From above data our Dot plot will be:

In exercise $19$dot plots varies in range of $0$to $1$, whereas in this problem dot plot ranges from $0.33$to $0.667$

As a result, dot plots with sample sizes of n= $3$have less variability than dot plots with sample sizes of n= $2$.

This also means that as the sample size grows, the sampling variability reduces.