Q. 7.11 - Elementary Statistics

Question

The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat, One possible starting lineup for that team is as follows:

Part (a): Find the population mean height of the five players.

Part (b): For samples of size 2, construct a table similar to Table 7.2 on page 293. Use the letter in parentheses after each player's name to represent each player.

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, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be 1 inch or less; that is, determine the probability that $\overline{x}$ will be within 1 inch of $\mu$. Interpret your result in terms of percentages.

Step-by-Step Solution

Answer

Part (a): The mean population of the five players is $78.6$ inches.

Part (b): Constructing the table of samples of size 2 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 $0$.

Part (e): The probability that $\overline{x}$ is within 1 inch of $\mu$ is $0.4$.

Part (a) Step 1. Given information

Consider the given question,

Part (a) Step 2. Find the mean for five people.

The mean for five people is given below,

$$\begin{gathered} \mu =\frac{{\displaystyle \sum _{i=1}^n}{x}_i}{N} \\ =\frac{83+76+80+74+80}{5} \\ =\frac{393}{5} \\ =78.6 \end{gathered}$$

Thus, the population mean height for five players is $78.6$inches.

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

The samples of size 2 and the corresponding means are obtained,

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

We need to find the $P\left(\overline{x}=\mu \right)$.

From the table obtained in part (b), it is clear that none of the sample means is equal to the population mean.

Also, the number of samples of size 2 is 10.

$$\begin{gathered} P\left(\overline{x}=\mu \right)=\frac{0}{10} \\ =0 \end{gathered}$$

Part (e) Step 1. Find the probability that x will 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 4 of the sample means within 1 inch of the population mean.

$$\begin{gathered} P\left(\mu -1\le \overline{x}\le \mu +1\right)\mathit{=}P\mathit{(}\mathit{78}\mathit{.}\mathit{6}\mathit{-}\mathit{1}\mathit{\le }\overline{x}\mathit{\le }\mathit{78}\mathit{.}\mathit{6}\mathit{+}\mathit{1}\mathit{)} \\ =P\mathit{(}\mathit{77}\mathit{.}\mathit{6}\mathit{\le }\overline{x}\mathit{\le }\mathit{79}\mathit{.}\mathit{6}\mathit{)} \\ =\frac{4}{10} \\ =0.4 \end{gathered}$$