Question

Data given below are 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 y=4.68+0.16x, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52,500 is $13,080.

Adjusted Gross Reasonable Amount of Itemized
Income ($1000s) Deductions ($1000s)
22 9.6
27 9.6
32 10.1
48 11.1
65 13.5
85 17.7
120 25.5
Use the estimated regression coefficients rounded to 2 decimals in your calculations.
а. Develop a 95% confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of $52,500 (to 2 decimals).
$ thousand to $ thousand
b. Develop a 95% prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of $52,500 (to 2 decimals).
$ thousand to $ thousand
d. Use your answer to part (b) to give the IRS agent a guideline as to the amount of total itemized deductions a taxpayer with an adjusted gross income of $52,500 should claim before an audit is recommended (to the nearest whole number).
Any deductions exceeding the $ upper limit could suggest an audit.

Answer 1

The confidence interval for the mean amount of total itemized deductions cannot be calculated due to a standard error of 0. Similarly, the prediction interval estimate cannot be calculated. Therefore, no guideline on the amount of total itemized deductions can be provided.

Step-by-step explanation:

To develop a 95% confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of $52,500, we can use the estimated regression equation: y = 4.68 + 0.16x.

Substituting x = 52.5 (adjusted gross income in thousands of dollars) into the equation, we get: y = 4.68 + 0.16 * 52.5 = 13.38. So, the estimated mean amount of total itemized deductions for a taxpayer with an adjusted gross income of $52,500 is $13,380.

To construct the confidence interval, we need to calculate the margin of error, which is the critical value times the standard error of the estimate. Assuming a normal distribution, we can calculate the critical value at 95% confidence level as approximately 1.96. The standard error of the estimate can be calculated as the square root of the sum of the squared residuals divided by the degrees of freedom. Degrees of freedom can be calculated as the number of observations minus the number of variables in the regression equation minus 1 (7 - 2 - 1 = 4).

Using the given data, we can calculate the residuals: 9.6-9.6, 9.6-10.1, 11.1-11.1, 13.5-13.5, 17.7-17.7, 25.5-25.5.

So, the sum of the squared residuals is 0, and the standard error of the estimate is 0.

Since the standard error of the estimate is 0, the margin of error will also be 0, and hence the confidence interval cannot be calculated. The point estimate of $13,080 should be taken as an estimate of the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of $52,500.

To develop a 95% prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of $52,500, we can use the same estimated regression equation. Assuming a normal distribution, the prediction interval can be calculated as the point estimate plus or minus the critical value times the standard error of the prediction. The critical value at 95% confidence level is approximately 1.96. But since the standard error of the estimate is 0, the prediction interval cannot be calculated.

Therefore, there is no prediction interval estimate available in this case.

Based on the analysis in part (b), we cannot provide a guideline to the IRS agent regarding the amount of total itemized deductions a taxpayer with an adjusted gross income of $52,500 should claim.