Q.7.22 - Elementary Statistics

Question

America's Riches. Each year, Forbes magazine publishes a list of the richest people in the United States. As of September l6, 2013, the six richest Americans and their wealth (to the neatest billion dollars) are as shown in the following table. Consider these six people a population of interest. 

(a) For sample size of $6$ construct a table similar to table 7.2 on page293  what is the  relationship between the only possible sample here and the population?

(b) For a random sample of size $6$ determine the probability that themean wealth of the two people obtained will be within $3$(i.e,$3$ billion) of the population mean. interpret your result in terms of percentages.

Step-by-Step Solution

Answer

(a) 

The  relationship between the only possible sample here and the population is that both are equal.

(b) There is $100\%$ chance that the mean wealth of the six people will be within $3$ billion of the population mean.

Part (a) Step 1: Given Information

Given in the question that,

 we have to construct a table for sample size $6$

Part (a) Step2 : E xplanation

The sample size of $6$ and the corresponding means are obtained as shown in the below table :

The  relationship between the only possible sample here and the population is that both are equal.

Part (b) Step 1: Given Information

Given in the question that,

we have to deterrmine the probability that $\overline{x}$ is within $3$ billion of$\mu$.

Part(b) Step 2: Explanation

We have to obtain $P(\mu -3\le \overline{x}\le \mu +3)$

$P(\mu -3\le \overline{x}\le \mu +3)=P(46.5-3\le \overline{x}\le 46.5+3)$

                                  $=P(43.5\le \overline{x}\le 49.5)$ 

                                  $$\begin{gathered} =\frac{1}{1} \\ =1 \end{gathered}$$

     Therefore, the probability that $\overline{x}$ is within $3$ billion of$\mu is 1$ .

Interpretation:

There is$100\%$ chance that the mean wealth of the six people will be within $3$billion of the population mean.