Question

Repeat parts (b)-(e) of Exercise 7.11 for samples of size5.

Short answer

1. Part (b): Constructing the table of samples of size 5 of 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 $1$.

Part (e): The probability that $\overline{x}$ is within 1 inch of $\mu$is $1$.

On interpreting, there is $100\%$ change that the mean height of the four players selected will be within 1 inch of the population mean.

Step-by-step solution

Part (b) Step 1. Given information

Consider the given question,

Part (b) Step 2. Construct samples of size 5 of the given population.

The samples of size 5 and the corresponding means are obtained,

Here, Chrish Bosh by B, Dwyane Wade by W, LeBron James by J, Mario Chalmers by C and Udonis Haslem H.

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 previous question,

The population mean height for five players is $7.86$inches.

From table obtained in part (b), it is clear that none of the sample means are equal to the population mean. Also, number of samples of size 5is 1.

$$\begin{gathered} P\left(\overline{x}=\mu \right)=\frac{1}{1} \\ =1 \end{gathered}$$

Part (e) Step 1. Find the probability that xwill be within 1 inch of μ.

We need to find the $P\left(\mu -1\le \overline{x}\le \mu +1\right)$.

From the table obtained in part (b), it is clear that none of the sample means are within 1 inch of the population mean.

$$\begin{gathered} P\left(\mu -1\le \overline{x}\le \mu +1\right)\mathit{=}P\mathit{(}\mathit{78}\mathit{.}\mathit{6}\mathit{-}\mathit{1}\mathit{\le }\overline{x}\mathit{\le }\mathit{78}\mathit{.}\mathit{6}\mathit{+}\mathit{1}\mathit{)} \\ =P\mathit{(}\mathit{77}\mathit{.}\mathit{6}\mathit{\le }\overline{x}\mathit{\le }\mathit{79}\mathit{.}\mathit{6}\mathit{)} \\ =\frac{1}{1} \\ =1 \end{gathered}$$

On interpreting, we can say that there is $100\%$change that the mean height of the four players selected will be within 1 inch of the population mean.