The IRS wants to develop a method for detecting whether or not individuals have overstated their deductions for charitable contributions on their tax returns. To assist in this effort, the IRS supplied data found in the table below listing the adjusted gross income $($AGI$)$ and charitable contributions for $11$ taxpayers whose returns were audited and found to be correct.

AGI $($in $$1,000$s$)$ Charitable Contributions

$$54$ $$4,300$

$$57$ $$4,700$

$$64$ $$6,429$

$$68$ $$8,117$

$$75$ $$7,500$

$$77$ $$8,700$

$$82$ $$12,290$

$$87$ $$10,506$

$$91$ $$11,720$

$$99$ $$11,990$

$$104$ $$14,575$

$($a$)$

Prepare a scatter plot of the data. Does there appear to be a linear relationship between these variables?

The relationship looks approximately linear. Charitable contributions decrease as adjusted gross income increases.

The relationship is highly nonlinear. Charitable contributions increase and then decrease as adjusted gross income increases.

The relationship looks approximately linear. Charitable contributions increase as adjusted gross income increases.

The relationship is highly nonlinear. Charitable contributions decrease and then increase as adjusted gross income increases.

The relationship looks approximately linear. Charitable contributions are relatively constant as adjusted gross income increases.

$($b$)$

Develop a simple linear regression model that can be used to predict the level of charitable contributions $($in dollars$)$ from a return's AGI $($in thousands of dollars$).$ What is the estimated regression equation? $($Round your numerical values to four decimal places. Use X$1$ for AGI $($in thousands of dollars$)$ and Y hat for charitable contributions in dollars.$)$

Y hat $=$

$($c$)$

Interpret the value of R$2.($Enter your answer as a percent. Round your answer to two decimal places.$)$

The value of R$2$ suggests that

$\%$ of

$---$Select$---$

charitable contributions

$---$Select$---$

adjusted gross income.

$($d$)$

How might the IRS use the regression results to identify returns with unusually high charitable contributions?

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions between the lower the upper limits would be considered suspect.

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions above the upper limit would be considered suspect.

The IRS could predict the charitable contributions at each level of adjusted gross income. Charitable contributions that are equal to the predicted value would be considered suspect.

The IRS could predict the charitable contributions at each level of adjusted gross income. Charitable contributions that are not equal to the predicted value would be considered suspect.

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions below the lower limit would be considered suspect.