Question

3. Following data on adjusted income and reasonable itemized deduction amount of 1040 Form is published by IRS. (20 points)

Adjusted Gross Income ($1000) in 1040	Reasonable itemized reduction amount ($1000)
22	9.6
27	9.6
32	10.1
48	11.1
65	13.5
85	17.7
120	25.5

1. Which variable is the predictor (X) and which is the response (Y)?
2. Find the least-square regression line y=a+bx
3. Plot the data and your least-square line in one graph. Do you think that the regression line does a job to describe the relationship?
4. What is the estimated reasonable itemized reduction amount for all the taxpayers who earn $52,500 a year? If John makes $52,500 and claim $20,400 itemized reduction, will IRS audit his tax return?
5. Compute the standard error of estimate S_y,x for the least-square regression.

Answer 1

ANSWER::

3Q)

Que.a
Predictor X = adjusted income

Response Y = Reasonable itemized deduction amount

In order to solve this question I used R software.

R codes and output:

> x=c(22,27,32,48,65,85,120)
> y=c(9.6,9.6,10.1,11.1,13.5,17.7,25.5)
> fit=lm(y~x)
> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5 6 7
1.3744 0.5679 0.2613 -1.3196 -1.6619 -0.6881 1.4660

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.67675 1.03339 4.526 0.006251 **
x 0.16131 0.01568 10.285 0.000149 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.372 on 5 degrees of freedom
Multiple R-squared: 0.9549, Adjusted R-squared: 0.9458
F-statistic: 105.8 on 1 and 5 DF, p-value: 0.0001493

> plot(x,y)
> abline(fit)

Que. b

Least square equation,

Y = 4.67675 + 0.16131 X

Que.c

  

Almost all points lie close to regression line. Hence regression line does a good job to describe the relationship.

Que.d

Taxpayer who earn $52500 (X = 52.5), estimated reasonable itemized reduction amount is,

Y = 4.67675 + 0.16131 X

Y = 4.67675 + 0.16131 (52.5)

Y = 13.145525

Que.e

Standard error of estimate = 1.372

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