Let'sTo help with the address the problem problem step shown by in step using your the image data, I'll and regression address equation the provided:

Key questions Information step by step. Below are:

• Regression Equation: (\hat{ my solutions:

###y Given} Regression Equation =The regression4 equation. is:

68$$\hat + 0.16{yx}\ = )

• Adjusted4.68 Gross + Income 0.16x$$

Where: (-AGI) for x = 52.5 \( x \ (in thousands), ): Adjusted Gross Income(\ in (hat{ $y} =1,000 \ 13)

• $.08$ \ (hat{ypredicted} item \ized): deductions Pred inicted reasonable thousands). amount- of For total item confidenceized interval and prediction deductions interval in $, standard \$ deviation and1,000$

variance data are derived from the### Step-by-Step Solution:

regression dataset.

####---

a####) a .95 **%Develop Confidence a 95% Interval for confidence the Mean The formula for interval a for confidence the mean itemized interval for deductions for the mean is (x:

=\ [ \52.5hat{):y**

} \Formulapm t for the Confidence \cd Interval: ot\ SE[ _{\text\hat{{mean}} y}]

\Wherepm t:

• (_{\alpha /2hat{} \y}cdot = s _{\13hat{.y08 }} ) (pred] ictedWhere value: for- \( x =s_{\ hat52{y.5}} =, s i_e.e \sqrt.,{\ frac \$52{1,500}{) n} +- ( \frac t {): Critical( value fromx - \ (bar t{x}) ^)-distribution for2}{\ 95%sum confidence (x_i level -- \ bar SE{x_{\})text^{2mean}}: Standard error of the mean}}\

),

• $s_e$: Standard error#### of the b) 95% Prediction Interval The estimate, formula- (t_{\alpha/2}\ for): a (t\ prediction)-value for interval the is:

[ \hat {95y%} confidence \ levelpm with t ( \n -cd ot2 SE_{\text){ degreespred of}} freedom.

b.]

Where:

• $SE_{\text **{predDevelop a}}$: Standard error for95% prediction interval for a specific taxpayer prediction (,( whichx accounts for = more52. variability than the5)): mean.

c**

)Formula Audit Just forification Compare the claimed Prediction Interval: deductions (\[ (hat{y} \ pm20 t.{\4alpha /))2 to} the \ predictioncd intervalot s upper limit to{pred} determine if\ it's] unusuallyWhere large.

: -#### ( ds)_{ Guidpredeline} = s for Auditing_e \Usesqrt the{1 upper + limit \ of thefrac prediction interval as a{1}{n threshold} for + suggesting \frac an{ audit(.

x ----

I \ willbar calculate{x these}) values^ if2 you}{ provide\ the necessary statistics (esum.g (.,x_i - \bar{x standard})^2 error, degrees of freedom). Let me know if you have}}).

this---

data#### c. Evaluate the IRS justification or for $y need further clarification =! 20.4$:

Compare the taxpayer's reported deductions ($y = 20.4$) against the prediction interval. If it falls outside, the deduction is "larger than anticipated."

d. Suggest audit guideline:

The upper limit of the prediction interval from part (b) serves as a threshold for audits.

To Proceed:

I'll calculate the exact intervals using your uploaded data. Would you like to proceed with precise computational values?