Question

Data are given below on the adjusted gross income x and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation = 4.68 + .16x, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52.5 thousand is $13.08 thousand.

Adjusted Gross Income($1000s)	Reasonable Amount of Itemized Deductions ($1000s)
22	9.6
27	9.6
32	10.1
48	11.1
65	13.5
85	17.7
120	25.5

In the questions that follow, enter the dollar amounts requested. For example, if the regression results provide a value of 11.74 thousand, enter 11,740 as the dollar amount in the box. Use Table 1 of Appendix B.

a)Develop a 95% confidence interval for the amount of total itemized deductions for all taxpayers with an adjusted gross income of $52,500 (to the nearest whole number).

$ ( ____ , _____ )

b)Develop a 95% prediction interval for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of $52,500 (to the nearest whole number).

$ ( ____ , _____ )

c)If the particular taxpayer referred to in part (b) claimed total itemized deductions of $20,400, would the IRS agent's request for an audit appear to be justified?

d)Use your answer to part (b) to give the IRS agent a guideline as to the amount of itemized deductions that would suggest an audit for a taxpayer with an adjusted gross income of $52,500. Any deductions exceeding the upper limit of $_______ could suggest an audit.

Answer 1

Incomex Deductions x2 SUMMARY OUTPUTi 9.6 9.6 10.1 11.1 13.5 17.7 25.5 27 32 48 65 85 729 1024 2304 4225 7225 14400 30391 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.977173955 0.954868938 0.945842726 1.371565386 7 Total ANOVA Mean of x- Significance F MS 0.000149348 Regression Residual Total 199.0083277 199.0083 105.7884 9.405958047 1.881192 208.4142857 i Coefficients Standard Error t Stat P-value r 95% 2.020325666 7.33317478 0.120994464 0.20162583 Lower 95% 4.676750224 0.161310146 Intercept X Variable 1 1.033394277 4.52562 0.006251 0.015683485 10.28535 0.000149

$y=Deductions;~x=Income\\ \hat{y}=4.6768+0.1613x\\ \bar{x}=46.5,~S_{xx}=\sum x_i^2-7\bar{x}^2=30391-7*46.5^2=15255.25\\ t_{0.025,5}=2.5706\\ At~x=52.5~then~\hat{y}=4.6768+0.1613*52.5=13.1451,\\~s=standard~error=1.3716\\ \\ (a)~95\%~C.I.~for~the~amount~of~total~itemized~deductions~for~all~tax~payers~with\\ adjusted~gross~income~of~\$52500~is\\ \left((\hat{y}-t_{0.025,5}\times s\times \sqrt{\frac{1}{7}+\frac{(52.5-\bar{x})^2}{S_{xx}}})*1000,~\hat{y}+t_{0.025,5}\times s\times \sqrt{\frac{1}{7}+\frac{(52.5-\bar{x})^2}{S_{xx}}})*1000 \right )\\ =(\$11802,~\$14489)\\ (b)~95\%~P.I.~for~the~amount~of~total~itemized~deductions~for~all~tax~payers~with\\ adjusted~gross~income~of~\$52500~is\\ \left((\hat{y}-t_{0.025,5}\times s\times \sqrt{1+\frac{1}{7}+\frac{(52.5-\bar{x})^2}{S_{xx}}})*1000,~\hat{y}+t_{0.025,5}\times s\times \sqrt{1+\frac{1}{7}+\frac{(52.5-\bar{x})^2}{S_{xx}}})*1000 \right )\\ =(\$9372,~\$16918)\\$

(c) Since total itemized deductions of $20,400 aboce the 95% P.I. so the IRS agent's would request for an audit appear to be justified.

(d) Use your answer to part (b) to give the IRS agent a guideline as to the amount of itemized deductions that would suggest an audit for a taxpayer with an adjusted gross income of $52,500. Any deductions exceeding the upper limit of $16918 could suggest an audit.