Chapter 1.6, Problem 6E

a.

To determine

To graph: the data on the scatter plot.

Answer to Problem 6E

  

Explanation of Solution

Given information:

The table of values given is as follows:

Number of years of football coaching (x)	Number of wins (y)
33	347
40	324
29	270
33	229
25	222
23	209
23	201
22	193
18	170
18	168
16	?

Calculation:

The graph for the given data is shown below.

  

b.

To determine

To find: best fit line using two ordered pairs.

Answer to Problem 6E

  $y=7.09x+40.38$

Explanation of Solution

Given information:

The table of values given is as follows:

Number of years of football coaching (x)	Number of wins (y)
33	347
40	324
29	270
33	229
25	222
23	209
23	201
22	193
18	170
18	168
16	?

Calculation:

Let’s select any two ordered pair like $\left(18,168\right)$ and $\left(40,324\right)$ . Plug in the values in the slope equation and solve m as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{324-168}{40-18}\\ =\frac{156}{22}\\ =7.09\end{array}$

Now, plug the value of m and any one point coordinate into the slope intercept equation and evaluate b as follows:

  $\begin{array}{c}y=mx+b\\ 168=\left(7.09\right)\left(18\right)+b\\ 168=127.62+b\\ b=168-127.62\\ =40.38\end{array}$

Now, plug the values of m and b in the slope intercept formula as follows:

  $\begin{array}{l}y=mx+b\\ y=7.09x+40.38\end{array}$

Thus, the line equation is $y=7.09x+40.38$ .

c.

To determine

To find: the equation for regression line and correlation value using the graphing calculator.

Answer to Problem 6E

  $\begin{array}{c} y=33.378552971577+7.57278208441x\\ r=0.88275526058625\end{array}$

Explanation of Solution

Given information:

The table of values given is as follows:

Number of years of football coaching (x)	Number of wins (y)
33	347
40	324
29	270
33	229
25	222
23	209
23	201
22	193
18	170
18	168
16	?

Calculation:

The graph for the above equation is as follows:

  

Using the graphing calculator, the regression line equation is as follows:

  $y=33.378552971577+7.57278208441x$

Using the graphing calculator, the correlation coefficient comes out to be, $r=0.88275526058625$

d.

To determine

To find: the missing value if regression line equation has strong or moderate relationship, and explains whether the prediction is reliable.

Answer to Problem 6E

  $\approx 155$

Prediction is reliable.

Explanation of Solution

Given information:

The table of values given is as follows:

Number of years of football coaching (x)	Number of wins (y)
33	347
40	324
29	270
33	229
25	222
23	209
23	201
22	193
18	170
18	168
16	?

  $\begin{array}{c} y=33.378552971577+7.57278208441x\\ r=0.88275526058625\end{array}$

Calculation:

The equation of regression line shows strong relationship because $0.75\le r\le 1$ .

Plugging $x=16$ into the regression formula and evaluate y as follows:

  $\begin{array}{c} y=33.378552971577+7.57278208441x\\ y=33.378552971577+7.57278208441\left(16\right)\\ =33.378552971577+\text{121}\text{.16451335056}\\ =154.543066322137\\ \approx 155\end{array}$

The skill of the coaching is only a partial factor for this situation. Assuming that the rate of "bad coaches with long careers" was consistent throughout the years, the effect of the bad coaching would be covered by the data. According to the regression line, it represents number of wins of an average coach in a given length of time.

Assume if a coaching is done for one season, the chances of losing 100% game in a session are very rare. In that respect, winning 155 games over 16 years shows that a team won nine games in a year. Mostly teams play about 16 games per season. According to the bell-curve prediction, the average team would be the ones that won eight games and lost eight games per season. So, the predicted value is reliable.