Q.7.2 - 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 $4$ construct a table similar to table 7.2 on page293.(There are 15 possible sample of size $4$

(b) For a random sample of size $4$ 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) Sample size of $4$ and corresponding means are obtained.

(b) There is $26.67\%$ chance that the mean wealth of the four 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 with sample size of $4$

Part(a) Step 2: Explanation

The sample mean of size $4$:

$\overline{x}=\frac{{\sum }_{i=1}^4{x}_i}{4}$

Thus, the samples of size $4$ and the corresponding means are obtained as shown in the table below:

Thus Sample size of $4$and corresponding means are obtained. 

Part (b) Step 1: Given Information

Given in the question that,

we have to determine the probability that the mean wealth of the two people obtained will be within $3$(i.e,$3$billion) of the population mean.  

Part(b) Step 2: Explanation

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

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

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

                                    $$\begin{gathered} =\frac{4}{15} \\ =0.2667 \end{gathered}$$

Therefore, the probability that $\overline{x}$ is within $3$ billion of$\mu is 0.2667$   

Interpretation:

There is $26.67\%$ chance that the mean wealth of the four people will be within $3$ billion of the population mean.