Chapter 1, Problem 65RE

Solution

a.

To Sketch: The scatter plot for the given data.

Given:

The average U.S. exports of the product for the years $2001-2011$

Year	Exports $\left(1000bls/day\right)$
$2001$	$971$
$2002$	$984$
$2003$	$1027$
$2004$	$1048$
$2005$	$1165$
$2006$	$1317$
$2007$	$1433$
$2008$	$1802$
$2009$	$2024$
$2010$	$2924$
$2011$	$1033$

Graph:

Let $x$ as the number of years and $y$ as the export numbers, graph the scatterplot of the provided data using graphing calculator as follows:

Use the instruction to generate a scatter diagram for a set of data:

Press the data column and $\boxed{\text{Enter}}$ the  $x$ and  $y$ data in the text box. 

And Press the $\boxed{\text{Submit Data}}$ button to perform the computation.

To clear the graph and enter a new data set, press $\boxed{\text{Reset}}$ .

The scatter plot for the given data with years in $x$ axis and number of products in $y$ axis has been represented below.

  

b.

To Model: Superimpose the parabolic curve on the scatter plot using quadratic regression with a quadratic function. Analyse whether it is good fit or not.

The quadratic function is $y=25.34x^2-122.22x+1117.38$ . The fit is very good since the graph passes through all the points.

Given:

The average U.S. exports of the product for the years $2001-2011$

Year	Exports $\left(1000bls/day\right)$
$2001$	$971$
$2002$	$984$
$2003$	$1027$
$2004$	$1048$
$2005$	$1165$
$2006$	$1317$
$2007$	$1433$
$2008$	$1802$
$2009$	$2024$
$2010$	$2924$
$2011$	$1033$

Calculation:

Compute the Quadratic regression equation for the provided data using Ti-83 calculator as follow:

Press $\boxed{\text{Stat}}$ key, scroll to $\boxed{\text{Calc}}$ and press $\boxed{4}$ key.

Press $\boxed{\text{Enter}}$ key.

The screenshot is as follows:

  

Thus, the quadratic function is $y=25.34x^2-122.22x+1117.38$

Superimpose the parabolic curve on the scatter plot:

Enter the model $y=25.34x^2-122.22x+1117.38$ to $\text{Y}\text{1}$ and graph it to superimpose with the data points.

  

Thus, the fit is very good since the graph usually passes through the points.

c.

To Determine: The number thousands of barrels of oil the country will export per day in $2016$ based on the quadratic model.

Approximately $4986$ thousands of barrels of oil the country will export per day in the year $2016$ .

Given:

The average U.S. exports of the product for the years $2001-2011$

Year	Exports $\left(1000bls/day\right)$
$2001$	$971$
$2002$	$984$
$2003$	$1027$
$2004$	$1048$
$2005$	$1165$
$2006$	$1317$
$2007$	$1433$
$2008$	$1802$
$2009$	$2024$
$2010$	$2924$
$2011$	$1033$

Calculation:

Let, the number of years be $x$ :

To find the number of gallons of oil will be produced per day in $2016$ , the number of years should be required.

So, subtract the initial year $2001$ from the goal year $2016$ :

  $\begin{array}{l}x=2016-2001\\ x=15\end{array}$

Now, substitute $x=15$ in the quadratic equation from part $\left(b\right)$

  $y=25.34x^2-122.22x+1117.38$ :

  $\begin{array}{l}y=25.34{\left(15\right)}^2-122.22\left(15\right)+1117.38\\ y=5701.5-1833.3+1117.38\\ y=3868.2+1117.38\\ y=4985.58\end{array}$

Round to the approximate values:

  $\begin{array}{l}y=4985.58\\ y\approx 4986\end{array}$

Thus, approximately $4986$ thousands of barrels of oil the country will export per day in the year $2016$ .