00:03
Hi there.
00:04
This is a three -part problem.
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We'll start in part a by diagramming the data on this chart with adjusted gross income on the x -axis and itemized deductions on the y -axis.
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Next, we'll use this data to find the mean value of x here, the mean value of the y's.
00:26
Then we'll take each instance of x, x sub i, minus the mean value of x.
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Then we'll find in each instance of y minus the mean value of y.
00:35
We'll multiply these two together and add them up.
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That'll be our numerator for our b sub 1 equation here.
00:45
Then we'll take each instance of x minus the mean value of x, square them, add them up.
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That'll be our denominator.
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This will tell us b sub 1 down here.
00:58
We'll use the equation given to find b not or b sub zero, and then that'll enable us to find our estimated regression equation here.
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That'll be part b.
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Then we'll move over here to part c using the estimated regression equation.
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We will determine whether or not for adjusted gross income of 52 ,500, whether or not 20 ,400, is a reasonable itemized deduction as viewed by the irs.
01:32
So starting in part a, let's diagram our data that we have.
01:37
The first part is 22, 9 .6.
01:41
So let's call that here.
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The next piece of data is 27, 9 .6.
01:48
Let's call that here.
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Next is 32, 10 .1.
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We'll say that's right about here.
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Next is 48, 11 .1.
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Next, 65, 13 .5, 85, 17 .7.
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By the way, these are all in thousands, but we're just doing a, chopping off the trailing zeros to simplify things and make it a little quicker.
02:34
Last piece of data is 120 ,000 and 25 .5.
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So we'll say that's right about here.
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You can see the general trend here is we're looking for a linear trend and it's roughly like this.
02:48
So we know there's a positive correlation between adjusted gross income and the itemized deductions.
02:55
The equation, the regression equation is going to describe the line that goes here.
03:01
And i'll just say that if you can use a spreadsheet program on your computer, this will be much faster.
03:10
Year and easier.
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So consider doing that.
03:13
Okay.
03:14
So the next part of this is the mean value of x.
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So what 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.
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So 22, 27 plus 32, 27 plus 48, plus 65, plus 85, plus 120 divided by seven is 57.
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Next, we'll take the mean value of the y's, 9 .6, 9 .6, 10 .1, and so forth, add them together and divide by 7 is 13 .9, rounding to the nearest 10th.
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Now we'll take each instance of x sub i and subtract out the mean value of x.
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So in other words, the first instance of x is 22, subtract the mean value of x is 57.
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So 22 minus 57 is negative 35.
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The next piece of data is 27 minus the mean value of x is 57.
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So 27 minus 57 is negative 30.
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And we'll do that for each value of x.
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We'll subtract the mean value of x.
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So 32 minus 57 is negative 25.
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Negative 9.
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65 minus 57 is 8.
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85 minus 57 is 28.
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And finally, 120 minus 57 is 63.
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And we're going to do the same thing for the ys.
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So the first value of y is 9 .6...