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

a. Determine the population mean height, $\mu$, of the five players:

b. Consider samples of size 2 without replacement. Use your answer to Exercise 7.11(b) on page 295 and Definition 3.11 on page 140 to find the mean, ${\mu }_{\overline{x}}$, of the variable $\overline{x}$.

c. Find ${\mu }_{\overline{x}}$using only the result of part (a).

We are given a data in the table as

And we need to find the population mean of the data.

From the table, the population data is given as $83,76,80,74,80$.

So the population mean height for the five players is given as

$$\begin{gathered} \mu =\frac{83+76+80+74+80}{5} \\ \mu =\frac{393}{5} \\ \mu =78.6 \end{gathered}$$

For the population data: $83,76,80,74,80$.

The sample and sample mean for a sample of size $n=2$are shown in the table below.

The variable $\overline{x}$has the following mean

$$\begin{gathered} {\mu }_{\overline{x}}=\frac{79.5+81.5+78.5+81.5+78+75+78+77+80+77}{10} \\ {\mu }_{\overline{x}}=\frac{786}{10} \\ {\mu }_{\overline{x}}=78.6 \end{gathered}$$

So when the sample size is 2, the variable $\overline{x}$has a mean ${\mu }_{\overline{x}}=78.6$.

We know that mean of the sample mean is equal to the population mean irrespective of the sample size.

Here, the population mean is $\mu =78.6$.

So the mean of the sample mean of sample size 2 is

$$\begin{gathered} {\mu }_{\overline{x}}=\mu \\ =78.6 \end{gathered}$$