Q. 3.199 - Elementary Statistics

Question

Heights of Basketball Players. In Section 3.2, we analyzed the heights of the starting five players on each of two men's college basketball teams. The heights, in inches, of the players on Team II are $67, 72, 76, 76, and 84.$ Regarding the five players as a population. solve the following problems.

a. Compute the population mean height, $\mu$

b. Compute the population standard deviation of the heights, $\sigma$

Step-by-Step Solution

Answer

Part (a):

The population mean height is $75$ inches.

Part (b):

The population standard deviation of the heught is $5.6$ inches. 

Step 1. Given information is:

The heights, in inches, of the players on Team II are $67, 72, 76, 76, and 84.$

Part (a) Step 1. Calculating Mean Height

As we know, 

$$\begin{gathered} \mu =\frac{\sum _{}{x}_i}{N} \\ =\frac{67+72+76+76+84}{5} \\ =\frac{375}{5}=75 \end{gathered}$$

Thus, the population mean height is $75$ inches.

Part (b) Step 1. Calculating Standard Deviation

The formula for population standard deviation is:

$\mathrm{\sigma }=\sqrt{\frac{\sum _{}{\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}^2}{\mathrm{N}}}$

Obtain $\sum _{}{\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}^2$

x	$\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)$	${\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}^2$
67	-8	64
72	-3	9
76	1	1
76	1	1
84	9	81
		$\sum _{}{\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}^2 = 156$

$$\begin{gathered} \mathrm{\sigma } = \sqrt{\frac{156}{5}} \\ = 5.6 \end{gathered}$$

Thus, the population standard deviation of the heught is $5.6$ inches.