Chapter 8.6, Problem 33E

a.

To determine

To make:

A scatter plot of the data points $(x,y)$ . Draw the line that appears to best fit the data points.

Answer to Problem 33E

The line of best fit would look like:

  

Explanation of Solution

Given:

The table below shows the men’s winning timesin the Boston Marathon for every tenth year from 1900 to 2000. In the table, x represents the number of years since 1900, and y represents the corresponding winning time (to the nearest minute).

  

Calculation:

First of all, we will write our given data as ordered pairs $(x,y)$ .

  $\begin{array}{l}(0,160),(10,149),(20,150),(30,155),(40,148),(50,153),(60,141),\\ (70,131),(80,132),(90,128),(100,130)\end{array}$

Now we will plotour given values on coordinate plane and draw a line of best fit as shown below:

  

b.

To determine

To write:

An equation for the line of best fit.

Answer to Problem 33E

The equation for the line of the best fit would be $y=-0.3x+160$ .

Explanation of Solution

Given:

The table below shows the men’s winning times in the Boston Marathon for every tenth year from 1900 to 2000. In the table, x represents the number of years since 1900, and y represents the corresponding winning time (to the nearest minute).

  

Calculation:

First of all, we will find slope of line passing through point (0,160) and (100,130) as:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{130-160}{100-0}\\ m=\frac{-30}{100}\\ m=-0.3\end{array}$

Now we will use slope-intercept form to write our equation as:

  $\begin{array}{l}y=mx+b\\ y=-0.3x+160\end{array}$

Therefore, the equation for the line of the best fit would be $y=-0.3x+160$ .

c.

To determine

To predict:

The men’s winning time in the Boston Marathon for the year 2010.

Answer to Problem 33E

Men’s winning time in the Boston Marathon for the year 2010would be 127 minutes.

Explanation of Solution

Given:

The table below shows the men’s winning times in the Boston Marathon for every tenth year from 1900 to 2000. In the table, x represents the number of years since 1900, and y represents the corresponding winning time (to the nearest minute).

  

Calculation:

To predict men’s winning time in the Boston Marathon for the year 2010, we will substitute $x=110$ in equation $y=-0.3x+160$ and solve for y as:

  $\begin{array}{l}y=-0.3x+160\\ y=-0.3(110)+160\\ y=-33+160\\ y=127\end{array}$

Therefore, men’s winning time in the Boston Marathon for the year 2010 would be 127 minutes.

d.

To determine

Do you think your equation will accurately predict winning times far into the future? Explain your reasoning.

Answer to Problem 33E

The equation will not accurately predict winning times far into the future.

Explanation of Solution

Given:

The table below shows the men’s winning times in the Boston Marathon for every tenth year from 1900 to 2000. In the table, x represents the number of years since 1900, and y represents the corresponding winning time (to the nearest minute).

  

Calculation:

The line of best fit is for our given data. The slope of the line is negative, so as the x values will increase the value of y will approach zero. Our equation for the line of best fit can predict winning times close to year 2000.

Since we cannot expect all data points to fall on the line of best fit, therefore, the equation will not accurately predict winning times far into the future.