"Old Vehicles In Exercises 31–34, use the figure shown at the ...

Question

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Table showing the average age of vehicles on U.S. roads from 2014 to 2021, with years and corresponding ages in years.

Regression Line Find and draw the regression line."

Accepted Answer

Hi everyone. Welcome back. Let's take a look at our next problem. Says the table below shows the average vehicle age in years. For each of 8 different years. So we have a table with the year and then the corresponding average vehicle age in years for 8 values. We'll look at those in a little bit. Using this data, find the least squares regression equation of the form Y hat equals MX plus B, where X is the year and Y hat is the predicted vehicle H. We've got 4 answer choices. A, Y hat equals 0.0673 X minus 175.02. By hat equals 0.0629 X minus 119.37. C Y hat equals 0.0893 X minus 168.45, or D, Y hat equals 0.1013X minus 153.26. So we want this uh linear regression equation. What do we have and what do we need? Well, we're going to need to find M and B. So, let's think of our equations there. So, our slope equation is M equal to. The sum Of and then in parentheses X sub i. Minus X bar close parent and that difference multiplied by. In parentheses Y I minus Y bar close parentheses. And then that whole thing in the numerator, then divided by, and then our denominator, the sum of in parentheses Xi minus X bar, and then that difference, so we have closed parentheses squared. So that's a big one. Let's think about which of those values we have, what do we need to calculate? And then we'll also think briefly about our Y intercept, so B, and we'll recall that B is equal to. And we just rearrange our equation, and if we do it for the mean values, which we can calculate from our values, B is equal to Y minus MX bar. So looking at this, we know we will need to calculate our mean values of X and Y, so we can do that. And that will lead us to the other values we need for our slope equation. So, let's go ahead and get our mean values. So we have Our X bar equals, so sum up all of the X values. So that's our years. So we have the years 2014 through 2021. So, we'll sum those all up, and then divide them by eight, the number of years. And that will give us a value for X bar of 2017. 0.5 Or I should say not the average age, but the average year in which they were made. It's a little bit confusing. And then we'll look for Y bar, or the mean of the Y values. So, our Y values are 11.4. So, we're just summing these all up at 11.5, 11.6, 11.7. Another 11.7. 11.8, 11.9, and 12.1. And then divide that sum by 8 as well. That's going to give us a value. Actually approximately equal to 11.71. So now we have our M value we can compute, but first we will find it helpful to make a table because we have to do these sums and we have to subtract each individual X value, and then subtract X bar, do the same for all the Y values, then we have to square X individual minus X bar. And multiply the difference in the X's and the difference in the Y. Anyway, it's easiest to make a table for each of these. Expressions and then we can then read the values we need from our table. So, I'm not going to write that all out, you know, and have you watch me write it out. I'll paste the table up here. If you want to compute that on your own or check it over, you can pause the video and do so. So you would make a table similar to this one. We have In the first column, the individual X values, so Xi, from 2014 through 2021. Then, in column number 2, the individual Y values. So, Y sub I, all the values we just listed, so the average vehicle age in years. And then, we can make a column 3, that's the values of X I minus X bar, since we've calculated X bar. So, those values are negative 3.5, -2.5, -1.5, 0.5, 0.5, 1.5, 2.5, 3.5. In the next column four, we have the difference of Yi minus Y bar. So those values are 0.31, 21, 11, 0.001, 01, sorry, 0.09, 0.19 and 0.39. So then, since we have to, looking down at our M equation, we're going to have to sum up the product of the difference between the X values and X bar, and the difference between the Y values and Y bar. Our fifth column will be in parentheses X by minus X bar. And that multiplied by parentheses Y sub I minus Y bar. So we calculate those values we get. 1.085, 0.525, 0.165, 0.005, negative 0.005, 0.135, 0.475, and 1.365. And then we go ahead and put down at the bottom. Underneath that column, we can sum up all those values, because we will need the sum of all of those, and that sum is 3.75. And then finally, in our last column, X by minus X bar, and that expression squared, because we'll need that in the denominator of our M equation. So, those values are 12.25, 6.25, 2.25, 0.25, 0.25, 2.25, 6.25, 12.25. And then again, we will need the sum of those values in our denominator. So, if we calculate that sum, it's 42.0. So that uh is a bit tedious, but the table helps cause you can because you can just read those sums off of it. So let's fill in the values in our M equation. So our numerator and our M equation. Is that 12345, 5th column. So, the sum of that fifth column of the product of the difference in X and X bar, the difference in Y and Y bar, and that sum was 3.75 from that table. We divide by, and our denominator, remember, was the sum of the squares of X by minus X bar, that difference. And we added that up in the last column, and our sum was 42.0. 3.75 divided by 42.0. And that will give us an M value. Rounded a bit of approximately 0.0893. So that is our a value, we'll go ahead and highlight that in blue. So we can find it easily. And then let's go over to our B equation. So, B is equal to Y bar. So, we have Y bar, that was 11.71. And then minus M, so that we said was 0.0893. Multiplied by ASA, which is 2017.5. And when we go ahead and calculate those values, we're going to get B approximately equal to -168.45. Highlight those, that value as well. And now we have what we need to write out the equation, that regression equation for this data. So we know we need. Our form Y equals MX plus B. So, when we look over at our answer choices, the M value is 0.0893, only choice C has that correct M value. So then check our B value, and indeed it's 0.0893 X minus 168.45 our value of B. So, that is going to be our correct linear regression equation. Choice C, Y equals 0.0893 X minus 168.45. See you in the next video.

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Regression Line Find and draw the regression line."

Key Concepts

Regression Line

Dependent and Independent Variables

Correlation

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