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Problem 13 Medium Difficulty

To the Internal Revenue Service, the reasonableness of total itemized deductions depends on the taxpayer's adjusted gross income. Large deductions, which include charity and medical deductions, are more reasonable for taxpayers with large adjusted gross incomes. If a taxpayer claims larger than average itemized deductions for a given level of income, the chances of an IRS audit are increased. Data (in thousands of dollars) on adjusted gross income and the average or reasonable amount of total itemized deductions follow.
a. Develop a scatter diagram for these data with adjusted gross income as the independent variable.
b. Use the least squares method to develop the estimated regression equation.
c. Predict a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $\$ 52,500$ . If this taxpayer claimed itemized deductions of $\$ 20,400,$ would the IRS agent's request for an audit appear justified? Explain.

Answer

a. See scatter diagram
b. $\dot{y}=4.6768+0.1613 x$
c. $\$ 13,145.53,$ No

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Video Transcript

Hi there. This is a three part problem will start in part A by diagramming the data on this chart with adjusted gross income on the X access, access and itemized deductions on the Y axis. Next, we'll use this data to find the mean value of X here, the mean value of the wise. Then wolf take each instance of x x l by minus the mean value of X. Then we'll find in each instance of why minus the mean value of why, Well, multiply these two together and add them up. That will be our numerator and it for our beasts of one equation here. Then we'll take ah, the each instance of X minus the mean value of X square. Them Adam up. That will be our denominator. This will tell us be sub one down here. Um, we'll use the equation given for to find, be not or be subzero, and then that will enable us to find our estimated regression equation here part that will be part B. Then we'll move over here to part C. Using the estimated regression equation, we will determine whether or not for adjusted gross income of 5 52,500 Whether or not 5 20,400 is a reasonable itemized deduction as viewed by the i. R s. So starting in part a was diagram. Our data that we have the first part is 20 to 9.6. So it's call that here the next piece of data is 27 9.6. Call that here Next is 32 10.1. Well, say that's right about here. Next is 48 11.1, next 65 13.5, 85 17.7. And by the way, these air all, um in thousands. But we're just ah doing ah, chopping off the trailing zeros to simplify things and make it a little quicker. Last piece of data is 100 and 20,000 in 25.5. So we'll say that's right about here. You can see the general trend here is we're looking for a linear trend and it's roughly like this. So we know there's a positive correlation between adjusted gross income and the itemized deductions, uh, the equation that the regression equation is going to describe the line that goes here, and I'll just say that if you can use a, um, spreadsheet program on your computer, this will be much faster and easier. So consider doing that. Okay, so next part of this is the mean value of X. So we'll do. Is we'll take the X value for each of these points of data, add them up and divide by seven. Since there are seven points of data, so 20 to 27 plus 32 plus 48 plus 65 plus 85 plus 120 divided by seven is 57. Next, we'll take the mean value of the wise 9.69 point 6 10.1 and so forth. Add them together and divide by seven is 13.9 rounding to the nearest 10th. Uh, now we'll take each instance of ah, except by ah, and subtract out the mean value of X. So, in other words, the first instance of excess 22 subtract the mean value of X is 57. Um, s 0 20 to minus 57 is negative. 35. The next piece of data is 27 minus and mean value of X is 57. Eso UM 27 minus 57 is negative. 30. And we'll do that for each value of X will subtract the mean value of X. So 30 to minus 57 is negative. 25 Negative. Nine 65 minus 57 is eight 85 minus 57 is 28 and finally 120 minus 57 is 63 and we're gonna do the same thing for the wise. So the first value of why is 9.6. We'll subtract out 13.9 and leaves you with negative 4.3 9.6 minus the mean value of y 13.9 is um, negative. 4.3, 10.1 minus 13.9 There's negative. 3.8 11.1. Minus 13.9 is native 2.8, 13.5 minus 13.9 is native 0.4 17.7 minus 13.9 is 3.8 and finally the last value. 25.5 minus 13.9. 11.6, huh? Now Teoh, find the numerator. The next part of this will multiply each of these values. Eso this column times the first row of this column times the first row of this column and so forth down each row and then we'll add them up. And that will be our numerator. So negative 35 times negative. 4.3 is 149.5. We'll do this again for each value negative. 30 times negative 4.3 is 128.1, 94.3, 24.9 Negative, 2.97 107.2. Uh huh. And finally, 732 26. Okay, we add those up, we get 1000 233 0.7. So that is your numerator. 101,233 0.7 Now will do the denominator, which is taking these values, squaring them and then adding them up. So negative 35 times negative. 35 is 1225 negative. 30 squared is 900 625. Native. Nine squared is 81 eight square to 64 28 squared of 784 and 63 squared is 3116 again, We're gonna take this some of each of these numbers, So ah, summing up all these numbers we just did, uh is 7000 648. Okay, so we've essentially done most of be someone we know that we just no need to divide these numbers and that will give us be sub one. So 1233 0.7, divided by 7648. Okay, that's gonna be 0.1613 now for be subzero. We know that is the mean value of why, um, here, minus beasts of one being 0.1613 times the mean value of x 57. So when we do those operations, that comes out to four point 6768 So that is 13.9 minus 0.1613 times 57 gives you 4.6768 So now, getting be not and be one we confined our estimated regression equation, which is going to be be not or be subzero. 4.6768 and we're going to add B one, which is 0.16 13 Uh, ex, this is our estimated regression equation. So this is the answer to part B. Okay, now we're gonna use this estimated regression equation. So again, it's, um he's a use a different color here. Ah, 4.6768 plus 0.1613 And we're gonna use the number of given 52,500 and, ah, when do this operation? Um Ah. So we'll dio 0.1613 times 52,500 and keep it simple just to 52.5, and then we'll add in 4.6768 And that answer is, um, 13,000 145 0.53 And the question asks is, that's our estimated itemized deduction. Based on 52,500 adjusted gross income, we would expect to see about $13,145 as the the itemized deduction. The the question is whether or not 20,400 would be a reasonable, um, itemized deduction. That's what they actually gave we estimated 13.1. So they're quite a bit higher than actually 55% higher than our estimate of 13,100. So I'm going to say that 20,400 is, um, not in line with the ah, our estimate. And we would expect to see ah, potentially an IRS agent audit that item.

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