Data are given below on the adjusted gross income x and the amount of itemized d

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 yˆ = 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.

22

In the questions that follow, enter the dollar amounts requested. For example, if the regression results provide a value of 11.74 thousand, enter 11740 as the dollar amount in the box.

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?
Yes or No?

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 $52,5000.
Any deductions exceeding the upper limit of $ ? could suggest an audit.

AGI ($1000') Itemized Deductions ($1000s)

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9.6 27 9.6 32 10.1 48 11.1 65 13.5 85 17.7 120 22.5

Explanation / Answer

We shall analyse this in the open source statistical package R , the snippet is as follows

agi <-c(22,27,32,48,65,85,120)
deductions <- c(9.6,9.6,10.1,11.1,13.5,17.7,22.5)

## fit the regression equation

fit <- lm(deductions ~ agi)

summary(fit)

## new data for prediction

newdat <- data.frame(agi = 52.5)

## predict

predict(fit,newdata=newdat,interval = "confidence")

predict(fit,newdata=newdat,interval = "predict")

The results are

> predict(fit,newdata=newdat,interval = "confidence")
fit lwr upr
1 12.82817 11.98999 13.66634
> predict(fit,newdata=newdat,interval = "predict")
fit lwr upr
1 12.82817 10.4764 15.17994

c)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?

fit lwr upr
1 12.82817 10.4764 15.17994

as the prediction interval does not include 20.4(outside the upper side) , hence IRS agent's request for an audit is justified

Any deductions exceeding the upper limit of 13.66634 could suggest an audit. This is based on the confidence interval.

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