Q. 7.41 - Elementary Statistics

Question

NBA Champs 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.

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).

Step-by-Step Solution

Answer

Part a. The population mean height of the five players is $\mu =78.6$.

Part b. The mean of the variable $\overline{x}$ when the sample size is 2 is ${\mu }_{\overline{x}}=78.6$.

Part c. ${\mu }_{\overline{x}}=78.6$.

Part (a) Step 1. Given Information

We are given a data in the table as

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

Part (a) Step 2. Find the population mean

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

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

Part (b) Step 2. Find the mean when sample size is 2

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.

Sample	$\overline{x}$
83,76	$\frac{83+76}{2}=79.5$
83,80	$\frac{83+80}{2}=81.5$
83,74	$\frac{83+74}{2}=78.5$
83,80	$\frac{83+80}{2}=81.5$
76,80	$\frac{76+80}{2}=78$
76,74	$\frac{76+74}{2}=75$
76,80	$\frac{76+80}{2}=78$
80,74	$\frac{80+74}{2}=77$
80,80	$\frac{80+80}{2}=80$
74,80	$\frac{74+80}{2}=77$

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 ${\mu }_{\overline{x}}=78.6$.

Part (c) Step 1. Find the sample mean when sample size is 2

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

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