Chapter 2, Problem 12PS

a.

To determine

Use the regression feature of a graphing utility to find a quadratic model for the data.

Answer to Problem 12PS

  

Explanation of Solution

Calculation:

Consider the following table;

  $\begin{array}{cc}Age,x& Near,po\mathrm{int},y\\ 16& 3.0\\ 32& 4.7\\ 44& 9.8\\ 50& 19.7\\ 60& 39.4\end{array}$

Use graphing utility Ti-83 to create the scatter plot.

Go to STAT and select EDIT

Then enter the data in the L1 as year and L2 as sales in that list. The list look like below:

Now, go to the STAT PLOT and choose Plot1.......Off.

  

Now, ENTER and turn ON Plot-1

  

Select the first type of Plot and turn off the other types.

  

Now, go to ZOOM option and choose ZOOMSTAT:

  

Put ENTER and you will get the plot as below:

  

Use regression option and find the quadratic regression for the data:

Then again go to the STAT option and go to CALC option.

  

Then go to the QuadReg option, thyen go to VARS option and then Function- $Y_1$ :

  

Then put ENTER. You will get the equation as below:

  

Hence, the quadratic equation is: $y=0.031x^2-1.586x+21.018$

Now, graph the model in same viewing model. So, go the $Y=$ option and put the function as below:

  

Now, do not change the WINDOW as the window is already adjusted to the scattered plot. So,

You will get the WINDOW as:

  

Now, go to GRAPH option and you will get the plot as:

  

Hence, now you can see that it almost goes with the scattered plot. So, it fit the data well.

b.

To determine

Use the graphing utility to plot the data and graph the model in the same viewing window.

Answer to Problem 12PS

  

Explanation of Solution

Calculation:

Consider the following table;

  $\begin{array}{cc}Age,x& Near,po\mathrm{int},y\\ 16& 3.0\\ 32& 4.7\\ 44& 9.8\\ 50& 19.7\\ 60& 39.4\end{array}$

Find the rational model of the data. So, the rational model will be as below:

  $\frac{1}{y}=ax+b$

Here $a$ and $b$ are constants.

Now, consider $y=3$ at $x=16$

So, the reciprocal of $y$ is:

  $\begin{array}{l}\frac{1}{y}=\frac{1}{3}\\ =0.333\end{array}$

So, the new table is:

  

Use graphing utility Ti-83 to create the scatter plot.

Go to STAT and select EDIT.

  

Then enter the data in the L1 as year and L2 as sales in that list. The list will look like below:

  

Now, go to the STAT PLOT and choose Plot1...Off.

  

Now, ENTER and turn ON Plot-1

  

Select the first type of plot and turn off the other types.

  

Now, go to the Zoom option and choose ZOOMSTAT

  

Put ENTER and you will get the plot as below:

  

Now, use regression option and find the linear regression for the data:

Then again go to the STAT option and go to CALC option.

  

Then go to the LinReg(ax+b)option, then go VARS option and then Function- $Y_1$ :

  

Then put ENTER. You will get the equation as below:

  

So, the rational equation is : $\frac{1}{y}=-0.0074x+0.4443$

Now, graph the model in the same viewing model. So, go to the Y= option and pit the function as below:

  

Now, do not change the WINDOW as the window is already adjusted to the scattered plot. So, you will get the WINDOW as:

  

Now, go to GRAPH option and you will get the plot as:

  

Hence, now you can see that it almost goes with the scattered plot. So, it fit the data well.

c.

To determine

Verify that the model fit the data.

Answer to Problem 12PS

The data match the quadratic model.

Explanation of Solution

Calculation:

Consider the quadratic model

  $y=0.031x^2-1.586x+21.018$

Go to the TABLE option of this and get the table as:

  

Now, from the table get the values as

  

Hence, here at lower age, the data matches the rational model at the higher age, the data match the quadratic model.

d.

To determine

Use both models to estimate the near point for a person $25$ years old.

Answer to Problem 12PS

  $y=0.8$ , $y=3.86$

Explanation of Solution

Calculation:

Consider the quadratic model:

  $y=0.031x^2-1.586x+21.018$

Find the near point of the person who is $25$ year old. So, put the function at TI-83 as below:

  

Now, go to QUIT option. Then go to VARS option and then Function- $Y_1$ and put the value $25$ :

  

Put ENTER and you will get:

  

Hence, according model, the near point is at $y=0.8$

Consider the rational model

  $\frac{1}{y}=-0.0074x+0.4443$

Find the near point of the person who is $25$ year old. So, put the function at Ti-83 as below:

  

Now, go to QUIT option. Then go to VARS option and then Function- $Y_1$ and put the value $25$ :

  

Put ENTER and you will get:

  

So, you get:

  $\begin{array}{l}\frac{1}{y}=0.2593\\ y=3.86\end{array}$

Hence, according to rational model, the near point is at $y=3.86$

To determine

Predict the near point for a person $70$ years old.

Answer to Problem 12PS

Only Quadratic model will fit at the age of $70$ .

Explanation of Solution

Calculation:

Consider the quadratic model:

  $y=0.031x^2-1.586x+21.018$

Find the near point of the person who is $70$ year old. So, put the function at Ti-83 as below:

  

Now, go to QUIT option. Then go to VARS option and then Function- $Y_1$ and put the value $70$ :

  

Put ENTER and you will get

  

Hence, according to quadratic model, the near point is at $y=61.96$

Consider the rational model

  $\frac{1}{y}=0.0074x+0.4443$

Find the near point of the person who is $70$ year old. So, put the function at TI-83 as below:

  

Now, go to QUIT option. Then go to VARS option and then Function- $Y_1$ and put the value $70$ :

  

Put ENTER and you will get

  

This is absurd.

Hence, only quadratic will fit at the age of $70$ .