Chapter 23, Problem 39E

(a)

To determine

To find a $95\%$ confidence interval for the mean drive distance.

Answer to Problem 39E

The confidence interval is $(287.308,289.892)$ .

Explanation of Solution

It is given that:

  $\begin{array}{l}n=202\\ \mu =288.6\\ \sigma =9.31\end{array}$

The degrees of freedom is:

  $\begin{array}{l}df=n-1\\ =202-1\\ =201\end{array}$

Thus, the t -value is:

  $t_{\alpha /2}=1.972$

Therefore the confidence interval is:

  $\begin{array}{l}\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =288.6-1.972\times \frac{9.31}{\sqrt{202}}\\ =287.308\\ \overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =288.6+1.972\times \frac{9.31}{\sqrt{202}}\\ =289.892\end{array}$

Thus, the confidence interval is $(287.308,289.892)$ .

(b)

To determine

To discuss interpreting this interval raises some problem.

Explanation of Solution

It is given that:

  $\begin{array}{l}n=202\\ \mu =288.6\\ \sigma =9.31\end{array}$

And the confidence interval is $(287.308,289.892)$ . In this question they are studying how far do professional golfers drive a ball. The histogram of the average driving distances of the leading professionals golfers along with summary statistics is given in the question. From the histogram we can say that the graph is approximately normal. Thus, interpreting this interval raises some problem that these data are not random sample of golfers. The top professionals are unfortunately not representative and were not selected at random. We might consider the $2006$ data to represent the population of all professional golfers, past, present and future.

(c)

To determine

To find out is this a concern in interpreting the interval.

Answer to Problem 39E

No, this will be not a concern in interpreting the interval.

Explanation of Solution

It is given that:

  $\begin{array}{l}n=202\\ \mu =288.6\\ \sigma =9.31\end{array}$

And the confidence interval is $(287.308,289.892)$ . In this question they are studying how far do professional golfers drive a ball. The histogram of the average driving distances of the leading professionals golfers along with summary statistics is given in the question. From the histogram we can say that the graph is approximately normal. And the data are mean driving distance for each golfer. Since the data are means for each golfers then this will not be a concern in interpreting the interval because they are less variable than if we looked at all the separate drives.