Chapter 1.CR, Problem 95CR

To determine

a)

To graph:

The scatter plot for the data of amount of beef produced on private farms for the years from 2007 through 2014.

To determine

b)

To find:

The equation of the least squares regression line that fits the data of the amount of beef produced on private farms for the years from 2007 through 2014 and to graph the model and the scatter plot in the same graph and to interpret how closely the model represents the data.

Solution:

The least squares regression equation is $B=-5.02t+135.6$ and the graph of the model with the scatter plot on the same graph is:

The model represents a very close relationship with the data.

Explanation:

Definition:

Regression is a statistical method used to model the relationship between two sets of variables. The least squares equation is defined as,

$Y=a+bX$

Here, X is the independent variable, Y is the dependent variable, a is the y-intercept and b is the slope of the line.

Calculation:

Consider the provided information of the amount B (in millions of pounds) of beef produced on private farms for years from 2007 through 2017.

$\left(2007,102.7\right)$	$\left(2010,84.2\right)$	$\left(2013,70.4\right)$
$\left(2008,95.9\right)$	$\left(2011,75.0\right)$	$\left(2014,67.9\right)$
$\left(2009,90.2\right)$	$\left(2012,76.3\right)$

Assume that t represents the year with $t=7$ representing the year 2007.

Excel can be used to determine the least squares regression equation. The steps followed to construct the scatter plot are:

Step 1: Open Excel worksheet. Enter the provided data of years and amount of beef.

Step 2: In the menu bar, select Data➔Data Analysis.

Step 3: In the dialog that appears, select Regression. Click ‘Ok’.

Step 4: In the window that appears, select the Input Y-Range and Input X-Range and click Ok.

The least squares regression equation is obtained as,

$B=-5.02t+135.6$

Now, graph the model with the scatter plot in the same graph. The steps followed in Excel are:

Step 1: Open the obtained scatter plot in part (a).

Step 2: Right click on the graph and select Add Trendline from the dropdown list.

Step 3: Again Right click on the trend line added and select Format Trendline.

Step 4: In the window that appears select “Display Equation on Chart”. Click Ok.

The graph is obtained as:

The obtained graph shows the scatter plot and the regression model on the same graph. The graph shows that the data points are very close to the regression line. This indicates that there is a close relationship between the data and the model.