Chapter 5, Problem 15CR

The following data represent the percent of all drivers by age who have been stopped by the policy for any reason within the past year. The median age represents the midpoint of the upper and lower limit for the age range.

$\begin{array}{ccc}\text{Age}\text{Range}& \text{Median}\text{Age,}\text{x}& \text{Percentage}\text{Stopped, y}\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Using a graphing utility, draw a scatter plot of the data treating median age, $x$, as the independent variable.

Determine a model that best describes the relation between median age and percent stopped. You may choose from among linear, quadratic, cubic, exponential, logarithmic, and logistic models.

Provide a justification for the model that you selected in part (b).

To determine

To graph: The scattered plot of given data using graphing utility by treating median age $x$ as the independent variable. The table is

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Explanation of Solution

Given information:

The table

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Graph:

Step-1: Plug the median price, $x$ values in $L_1$ list and percent stopped, $y$ values in the $L_2$ list of the graphing calculator.

Step-2: Go to $\left[2nd\right]\gg \left[\text{STAT PLOT}\right]$ make $\text{Plot}1$ ON.

Step-3: Click on $\left[\text{Zoom}\right]\gg \text{ZoomStat}$

Therefore, the scattered plot of the given data is

Interpretation:

The scattered plot is showing behavior like graph of the quadratic function. Therefore, the relation between median price, $x$ values and percent stopped, $y$ would be given by the quadratic function polynomial function.

To determine

The best fit model that describes the relation between values of the median price, $x$ and percent stopped, $y$. The table is

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Answer to Problem 15CR

Solution:

The best fit function for data is $y=0.0022398179x^2-0.4721408517x+26.04517861$.

Explanation of Solution

Given information:

The table

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Explanation:

To find the equation of the best fit model for this data using graphing calculator press

$\left[\text{STAT}\right]\gg \text{CALC }\gg \text{QuadReg }\gg \text{ Calculate}$.

Therefore, the best fit function for the given data is

$y=0.0022398179x^2-0.4721408517x+26.04517861$

To determine

The justification for the model for the given data. The table is

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Answer to Problem 15CR

Solution:

Justification: As the graph of the quadratic function shows similar behavior like the scattered plot, that is why, the quadratic function is the best fit function for this given data values.

Explanation of Solution

Given information:

The data is

$\begin{array}{ccc}\text{Age Range}& \text{Median Age, }x& \text{Percent stopped, }y\\ 16-19& 17.5& 18.2\\ 20-29& 24.5& 16.8\\ 30-39& 34.5& 11.3\\ 40-49& 44.5& 9.4\\ 50-59& 54.5& 7.7\\ \ge 60& 69.5& 3.8\end{array}$

Explanation:

Graph the scattered plot and the best fit function $y=0.0022398179x^2-0.4721408517x+26.04517861$ on the same $xy$ coordinate system.

In the graphing calculator, press $\left[Y=\right]$, then type the function $y=0.0022398179x^2-0.4721408517x+26.04517861$ under $Y_1$.

Then, click on $\left[GRAPH\right]$ button

That gives the graph,

As the graph of the quadratic function shows similar behavior like the scattered plot, that is why, the quadratic function is the best fit function for this given data values.