Q. 7.18 - Elementary Statistics

Question

Repeat parts (b)-(e) of Exercise 7.17 for samples of size 1.

Step-by-Step Solution

Answer

Part (b): On constructing the sample of size 1 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 .

Part (e): The probability that $\overline{x}$ is within $3$ billion of $\mu$ is $0$.

There is $0\%$ change that the mean wealth of one person will be within 3 billion of the population mean.

Part (b) Step 1. Given information.

Consider the given question,

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

The samples of size 1 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.

The population mean wealth for six people is 46.5 billion.

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.

$$\begin{gathered} P\left(\overline{x}=\mu \right)=\frac{0}{6} \\ =0 \end{gathered}$$

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 none sample means is equal to the population mean.

$$\begin{gathered} P\left(\mu -3\le \overline{x}\le \mu +3\right)\mathit{=}P\mathit{(}\mathit{46}\mathit{.}\mathit{5}\mathit{-}\mathit{3}\mathit{\le }\overline{x}\mathit{\le }\mathit{46}\mathit{.}\mathit{5}\mathit{+}\mathit{3}\mathit{)} \\ =P\mathit{(}\mathit{43}\mathit{.}\mathit{\le }\overline{x}\mathit{\le }\mathit{49}\mathit{.}\mathit{5}\mathit{)} \\ =\frac{0}{15} \\ =0 \end{gathered}$$

On interpreting, we can say that there is $0\%$ change that the mean wealth of one person will be within 3 billion of the population mean.