91Ó°ÊÓ

We have been given these six people a population of interest.

The samples of size 3 and the corresponding means is given below,

Here, Bill Gates is represented by G, Warren Buffett is represented by B, Larry Ellison is represented by E, Charles Koch is represented by C, David Koch is represented by D and Chris Walton is represented by W.

On constructing the dot plot for the sampling distribution of the sample mean,

The population mean wealth for six people is $\mathrm{μ}=46.5$billion.

From the table in part (b), it is clear that none of the sample means is equal to the population mean. Also, the number of samples size 3 is 20.

Thus,

$$\begin{gathered} P(\stackrel{¯}{x}=\mathrm{μ})=\frac{0}{20} \\ =0 \end{gathered}$$

So, there is zero chance that the sample mean is equal to the population mean.

We need to find $P(\mathrm{μ}-3≤\stackrel{¯}{x}≤\mathrm{μ}+3)$

Here, $\mathrm{μ}=46.5$.

So from the table constructed in part b, it can be seen that there are $8$sample means in the range $\left[46.5-3,46.5+3\right]=\left[43.5,49.5\right]$.

Also, the number of samples size 3 is 20.

Thus,

$$\begin{gathered} P(\mathrm{μ}-3≤\stackrel{¯}{x}≤\mathrm{μ}+3)=\frac{8}{20} \\ =0.40 \end{gathered}$$

So, there is a probability of 40% that the mean wealth of the three people obtained will be within 3 billion of the population mean.