Q. 7.17 - Elementary Statistics

Question

Each years, Forbers magazine publishes a list of the richest people in the United States. As of September 16, 2013, the six richest Americans and their wealth (to the nearest billion dollars) are as shown in the following table. Consider these six people a population of interest.

Part (a): Calculate the mean wealth, $\mu$, of the six people.

Part (b): For samples of size 2, construct a table similar to Table 7.2 on page 293. (There are 15 possible samples of size 2.)

Part (c): Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.

Part (d): For a random sample of size 2, what is the chance that the sample mean will equal the population mean?

Part (e): For a random sample of size 2, determine the probability that the mean wealth of the two people obtained will be within 3 of the population mean. Interpret your result in terms of percentages. 

Step-by-Step Solution

Answer

Part (a): The population mean wealth for six people is $46.5$billion.

Part (b): On constructing the sample of size 2 for the given population is given below,

Part (c): The dot plot is given below,

Part (d): The chance that sample mean is equal to population mean is $0$.

Part (e): The probability that $\overline{x}$ is within $3$billion of $\mu$ is $0.2$.

Part (a) Step 1. Given information

Consider the given question,

Part (a) Step 2. Find the population mean wealth for six people.

The population mean wealth for six people,

$$\begin{gathered} \mu =\frac{{\displaystyle \sum _{i=1}^n}{x}_i}{N} \\ =\frac{72+59+41+36+36+35}{6} \\ =\frac{279}{6} \\ =46.5 \end{gathered}$$

Part (b) Step 1. Construct samples of size 2 of the given population.

The samples of size 2  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.

Part (c) Step 1. Construct the dot plot.

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

Part (d) Step 1. Find the chance that the sample mean will equal the population mean.

Consider the table in part (b), it is clear that none of the sample means is equal to the population mean. Also, number of samples size 2 is 15.

$$\begin{gathered} P\left(\overline{x}=\mu \right)=\frac{0}{15} \\ =0 \end{gathered}$$

Part (e) Step 1. Find the probability that x is within 3 billion of μ .

We need to find the $P\left(\mu -3\le \overline{x}\le \mu +3\right)$.

From the table obtained in part (b), it is clear that there are 3 sample means are within 3 billion of the population mean.

$$\begin{gathered} P\left(\mu -3\le \overline{x}\le \mu +3\right)\mathit{=}P\mathit{(}\mathit{46}\mathit{.}\mathit{5}\mathit{-}\mathit{3}\mathit{\le }\overline{x}\mathit{\le }\mathit{46}\mathit{.}\mathit{5}\mathit{+}\mathit{3}\mathit{)} \\ =P\mathit{(}\mathit{43}\mathit{.}\mathit{\le }\overline{x}\mathit{\le }\mathit{49}\mathit{.}\mathit{5}\mathit{)} \\ =\frac{3}{15} \\ =0.2 \end{gathered}$$

On interpreting, we can say that there is $20\%$ change that the mean wealth of the two people will be within 3 billion of the population mean.