Question

NBA Champs Repeat parts (b) and (c) of Exercise 7.41 for samples of size 4. For part (b), use your answer to Exercise 7.14(b).

Short answer

For the samples of size 4, the mean of the variable$\overline{x}$is ${\mu }_{\overline{x}}=78.6$.

Step-by-step solution

Step 1. Given Information

We have to determine the sample mean for exercise 7.41 for samples of size 4. The given sample in exercise 7.41 as:

Step 2. Find the sample mean

In the table below, the samples of size 4 and their respective means are obtained:

Sample	Height	Mean Height ($\overline{x}$)
B,W,J,C	83,76,80,74	$\frac{83+76+80+74}{4}=78.25$
B,W,J,H	83,76,80,80	$\frac{83+76+80+80}{4}=79.75$
B,W,C,H	83,76,74,80	$\frac{83+76+74+80}{4}=78.25$
B,J,C,H	83,80,74,80	$\frac{83+80+74+80}{4}=79.25$
W,J,C,H	76,80,74,80	$\frac{76+80+74+80}{4}=77.5$

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

role="math" localid="1652632975136" $$\begin{gathered} {\mu }_{\overline{x}}=\frac{78.25+79.75+78.25+79.25+77.5}{5} \\ {\mu }_{\overline{x}}=\frac{393}{5} \\ {\mu }_{\overline{x}}=78.6 \end{gathered}$$

So when the sample size is 4, the variable $\overline{x}$has a mean ${\mu }_{\overline{x}}=78.6$.

Step 3. Find the sample mean using the population mean

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

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

So the mean of the sample mean of sample size 4 is

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