Chapter 2.5, Problem 19PPSE

Solution

a.

To Graph:

Show the given data in a scatter plot. Draw a trend line.

Given:

Per capita revenue and expenditure for a recent year.

State	Per Capita Revenue	Per Capita Expenditure
Arizona	4144	3789
Georgia	3904	3834
Maryland	5109	4557
Mississippi	5292	4871
New Mexico	6205	5793
Nevada	4345	3723
New York	7081	6891
Texas	4030	3442
Utah	5439	4459

Concepts Used:

Graphing given data points using a graphing calculator.

Determining the trend line for a given data set.

Calculations:

Draw the data points in the given table as a scatter plot using a graphing calculator.

  

Fit the points to a trend line:

  

Conclusion:

The scatter plot graph of given points is made. The points have a trend line given by the equation $y=1.0116x-524.4869$ . Here, $x$ represents the per capita revenue and $y$ represents the per capita expenditure.

b.

To State:

The expected per capita expenditure for a state that collected revenue of $3000$ per capita.

  $\text{2508}\text{.9931}$ is the expected per capita expenditure for a state that collected revenue of $3000$ per capita.

Given:

Per capita revenue and expenditure for a recent year.

State	Per Capita Revenue	Per Capita Expenditure
Arizona	4144	3789
Georgia	3904	3834
Maryland	5109	4557
Mississippi	5292	4871
New Mexico	6205	5793
Nevada	4345	3723
New York	7081	6891
Texas	4030	3442
Utah	5439	4459

Known from previous part:

The given data points have a trend line given by the equation $y=1.0116x-524.4869$ . Here, $x$ represents the per capita revenue and $y$ represents the per capita expenditure.

Concepts Used:

Estimating using a model.

Calculations:

The per capita revenue is given to be $3000$ . Thus, in our model $y=1.0116x-524.4869$ , the value of $x$ is given to be $3000$ . Substitute $x=3000$ in the equation $y=1.0116x-524.4869$ and solve for $y$ :

  $\begin{array}{l}y=1.0116x-524.4869\\ y=1.0116\times 3000-524.4869\\ y=\text{2508}\text{.9931}\end{array}$

Conclusion:

  $\text{2508}\text{.9931}$ is the expected per capita expenditure for a state that collected revenue of $3000$ per capita.

c.

To State:

The expected per capita taxes collectedfor the state Ohio that spent $5142$ per capita.

  $\text{5601}\text{.5094}$ is the expected per capita taxes collected for the state Ohio that spent $5142$ per capita.

Given:

Per capita revenue and expenditure for a recent year.

State	Per Capita Revenue	Per Capita Expenditure
Arizona	4144	3789
Georgia	3904	3834
Maryland	5109	4557
Mississippi	5292	4871
New Mexico	6205	5793
Nevada	4345	3723
New York	7081	6891
Texas	4030	3442
Utah	5439	4459

Known from previous part:

The given data points have a trend line given by the equation $y=1.0116x-524.4869$ . Here, $x$ represents the per capita revenue and $y$ represents the per capita expenditure.

Concepts Used:

Estimating using a model.

Calculations:

The per capita expenditure is given to be $5142$ . Thus, in our model $y=1.0116x-524.4869$ , the value of $y$ is given to be $5142$ . Substitute $y=5142$ in the equation $y=1.0116x-524.4869$ and solve for $x$ :

  $\begin{array}{c}y=1.0116x-524.4869\\ 5142=1.0116\times x-524.4869\\ 5142+524.4869=1.0116x\\ \text{5666}\text{.4869}=1.0116x\\ \frac{\text{5666}\text{.4869}}{1.0116}=x\\ \text{5601}\text{.5094}=x\end{array}$

Conclusion:

  $\text{5601}\text{.5094}$ is the expected per capita taxes collected for the state Ohio that spent $5142$ per capita.

d.

To State:

Whether the given situation fits the linear trend line model for the given data table.

The model is satisfactorily accurate for the given situation.

Given:

Per capita revenue and expenditure for a recent year.

State	Per Capita Revenue	Per Capita Expenditure
Arizona	4144	3789
Georgia	3904	3834
Maryland	5109	4557
Mississippi	5292	4871
New Mexico	6205	5793
Nevada	4345	3723
New York	7081	6891
Texas	4030	3442
Utah	5439	4459

The situation:

New Jersey collected $5825$ per capita in taxes and spent $5348$ per capita.

Known from previous part:

The given data points have a trend line given by the equation $y=1.0116x-524.4869$ . Here, $x$ represents the per capita revenue and $y$ represents the per capita expenditure.

Concepts Used:

Verifying a model for a given point.

Calculations:

The per capita expenditure is given to be $5348$ and per capita revenue is given to be $5825$ . Thus, in our model $y=1.0116x-524.4869$ , the value of $y$ is given to be $5348$ and the value of $x$ is given to be $5825$ . Substitute $y=5348$ and $x=5825$ in the equation $y=1.0116x-524.4869$ and verify the resulting relation:

  $\begin{array}{c}y=1.0116x-524.4869\\ 5348=1.0116\times 5825-524.4869\\ 5348+524.4869=\text{5892}\text{.57}\\ \text{5872}\text{.4869}=\text{5892}\text{.57}\end{array}$

Conclusion:

The equality $\text{5872}\text{.4869}=\text{5892}\text{.57}$ is not exactly true but it is approximately true because the difference between the predicted and actual value is about $20$ , which is about 250 times smaller than the actual value of per capita spending.

Thus the model is satisfactorily accurate for the given situation.