Data given below are on the adjusted gross income z and the...

Data given below are on the adjusted gross income z and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y} = 4.68 + 0.16z$, 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. Click on the datafile to reference the data. Excel File: data14-37.xlsx Adjusted Gross Income ($1000s) 22 27 32 48 65 85 120 Reasonable Amount of Itemized Deductions ($1000s) 9.6 9.6 10.1 11.1 13.5 17.7 25.5 Use the estimated regression coefficients rounded to 2 decimals in your calculations. Round all intermediate calculations to 2 decimals. a. 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 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 ✔✔✔, it is larger ✔✔ than anticipated. 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.

Transcript

00:01 All right, so we're looking at income, adjusted gross income in the thousands.

00:06 So this column is, and then the reasonable amount of itemized deductions.

00:10 And this is based on a linear model.

00:13 Y hat is 4 .68 plus 0 .16x.

00:20 All right, so there's our model.

00:23 And this, that's what we have.

00:29 And we want to use this regression coefficient here.

00:34 Or excuse me, this regression equation to make a 95 % confidence interval for the mean amount of total atomized deductions and the 95 % prediction interval.

00:46 Let's give the formulas for those.

00:47 So the confidence interval, and this is for 95 % of course, is going to be the predicted amount, y hat, plus minus t with the alpha.

01:04 Well, the alpha is 0 .05 over 2.

01:09 And degrees of freedom is going to be given is n minus 1.

01:13 Or n minus 2, excuse me.

01:14 So it's going to be, there are 7 here.

01:15 1, 2, 3, 4, 5, 6, 7.

01:18 So it's going to be 5 because degrees of freedom for regression is given as n minus 2.

01:24 All right.

01:25 Then we have the t value.

01:26 And then we'd multiply it by this large square root term.

01:30 So this is for the confidence interval.

01:32 It's the mean square error times 1 over n.

01:36 And that's going to be 7 plus x minus x bar squared divided by the sum of the x is all the x is minus the means quantity squared that's the confidence interval the prediction interval is very similar you take your predicted value plus minus the t scores the same t value and we're going to see some familiar faces here it's the mean square error just like before and we're going to have the one over n just like before so one over seven plus the same x value minus the mean squared divided by the sum of the x minus x bar values squared but we have an extra one plus and this is because we're we're looking for one person an individual this is for an individual for the prediction interval and conference interval this is for the mean, like if you're looking for the average person.

02:53 Ok, so let's go ahead and answer these questions.

02:57 So for part a, we want to make the confidence interval.

03:01 For part b, you want to make the prediction interval.

03:04 So let's see.

03:05 The t value is easy enough to find.

03:07 Let me just show you everything.

03:08 And we'll go through it.

03:15 There we go.

03:18 All right, here we go.

03:19 This is all the stuff we need to calculate this by hand, which is a good exercise to go through.

03:24 So we need the mean.

03:26 See the x bar, we need the mean.