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Exercise 3 Three variables N, D, and Y all have zero sample means and unit sample variances. A fourth variable is C = N + D. In the regression of € OH Y, the ...

Question

Exercise 3 Three variables N, D, and Y all have zero sample means and unit sample variances. A fourth variable is C = N + D. In the regression of € OH Y, the slope is 0.8. In the regression of € OH N,the slope is 0.5. In the regression of D On Y the slope is 0.4_ What is the error Sum of squares in the regression of C On D? There are 21 observations.

Exercise 3 Three variables N, D, and Y all have zero sample means and unit sample variances. A fourth variable is C = N + D. In the regression of € OH Y, the slope is 0.8. In the regression of € OH N,the slope is 0.5. In the regression of D On Y the slope is 0.4_ What is the error Sum of squares in the regression of C On D? There are 21 observations.



Answers

For the data in Exercise 9 of Section 10.2 "The Linear Correlation Coefficient" a. Compute the least squares regression line. b. Can you compute the sum of the squared errors SSE using the definition $\Sigma\left(y-y^{\wedge}\right)_{2}$ ? Explain. c. Compute the sum of the squared errors SSE using the formula $S S E=S S_{y y}-\beta^{\wedge} 1 S S x y$.

The following is a solution number five. And what This one were asked to compute the least squares regression line from section 10.2, number five. And so that data, if you remember, I'm going to use a T. I 84. But if you go to stat and then edit, you can punch in your data value. So in L one I put my ex values 11345 and then an L two I put my Y values to 1534 So if I want to find my least squares regression line again using technology, you can just go to stat and then air over to Kaltag and then we're gonna go to this fourth option here, lin Wreg A X plus B. So click four. The X list will be L one, the wireless will be L two and then we're going to calculate And the slope is .6-5. And then the Y intercept is 1.25. So let's go and write that down. So the way we right the least squares regression line, we write it as y hat equals 0625 x Plus 1.25. Okay, so the second part is to compute the sum of the squared estimates. Um using the summation or some of the standard areas, using the summation of y minus y hat quantity squared. So that's the actual Y value minus the predicted y value. We can use this equation for that quantity squared. So I'm going to do the calculator again because it cuts down a lot of the work. So if you go back to stat and then edit on this third column, we're going to make that into our predicted Y values and the way you do that, you just click up where the L three is highlighted and we're just going to type in our equation so remember that this is equal to y. Hat. So we can just type in 30.625 which was the slope and then it was times X. Now I'm not gonna use X. Because I actually want data value. So I'm gonna do L. One so second L. One And then plus that 1.25. Okay so you can do this manually but you would have to do it five different times. This is basically just plugging in one for X. And then one for X. M. Three for X. And so on. So whenever you click enter you should get five data values here. So these your Y. Hats In L. four. So golden arrow over in L. four we're going to do the y minus Y hat. So remember that L. Two is the why and why hat is L. Three? So in L. Four I'm gonna say second. L two minus second. L. Three. Okay so that's my Y. And Y. Hat. So this is the column that's the y minus Y hat. Now you could square that but if you just go back to stat and then air over to Kaltag and do one of our stats And just choose the 4th list. So L four is your list. And what you're gonna do is you're gonna do this some of X squared. Because that's that why minus y hat squared? Is the is the excess the column? So it's equal to five? Okay so the S. S. E. using that formula is equal to five. Okay now we're supposed to compute the S. S. E. Using this formula S. S. Y Y minus the predicted that beta one hat. S. S. Xy. So here we've got to find some things. Now a lot of this stuff, you know, just some nation of X. So you just add your X values. You can probably do that mentally. You should get 14. And if you some your why values should get 15. You can also get that in one bar stats the summation of Y squared. If you if you square those anatomy together you should get 55 And then if you multiply your exes and wise together and add all those together, you should get 50. And then there are five days values. So it equals five. So remember S. S. Y. Y is equal to the summation of Y squared. So that's 55 -1 over end. So 1/5 Times the summation of Y, which is 15 quantity squared. And you plug that in the calculator and you get 10. The S. S. Of X. Y. is equal to the summation of XY. Which remember was 50 -1 over N Times the summation of X, which is 14 times the summation of Y, which is 15. And so there you should get eight. And then the beta one. Hat we actually already found that was the slope. So .625, is that estimate? Okay? So now we all we need to do is just plug that stuff in. So S. S. E. Is equal to S. S. Y. Y., which is 10. Mine is I'm using this formula or minus 0.6 to five times eight. And whenever you plug in that, you should get the same answer which is five. So that's just a nice verification. We can do the residual squared, you have those together. Or you can use this formula here.

The following is a solution to number 10 and were asked to compute the least squares regression line for number 10. For the summary stats of number 10 and section 10.2. And we need to find, so we're not giving a data set here. So a lot of this work is already done for us to find the slow to find beta one hat you take the ss of XY, which already found in a couple sections ago. It's negative 3 38.5 And you divide that by SS of XX, which again, we already found up to 62.1. So we plugged that in, I'm going to go and round to the nearest 1000. That should be negative one point ah 291. There's our slope negative 1.291. To find the Y intercept beta zero hat we take the mean of the wise minus beta one hat times the mean of the X's and the way you find the mean of the Y. It's just like you normally would find the mean. You add up the Wise and you divide by the sample size. Well we're not giving the Y values but were given the summation of why? Which The summation of? Why was 55 And the sample size was 10. So that's 5.5. And then the summation of X was negative three, Divided by 10 is negative .3. Okay, so then when we plug those in you get 5.5 -2.291 times negative three. And you should get about five 1127. So there's our y intercept 5.1127. So we put it all together and form a regression line. So why hat equals negative 1.291 X Plus 5.1127. Okay, so that's a regression line. Given summary stats the next question and ask us can you use the summation of the residual squared to get your S. S. E. And you cannot because we don't know any X's or wise. So to find the white hat, you need to know each individual X. And to find the Y. While you need to know why. So because we don't we're just giving the summation of X. Were just given the summation of why. We can't answer what S. S. E. Is using the residual squared method but we can use uh this other formula. So S. S. Y. Y. Which we found already to be 614 0.5 minus beta one hat. Which again we already found to be negative one point 291 times S. S. F. X. Y. Which was this negative? 3 38 5. So we can do it this way. So this is your SSE And whenever you plug that in, you should get 177 point 4965.

The following is a solution to number six. And were asked to compute the least squares regression line from section 10.2 number six. Using that data. So I'm gonna use the T. I. 84. If you go to stat and then edit, you can see in L. One. I put the X. Values 13558 And then in L. Two I put the Y. Values five negative 22 negative one and negative three. So now if we go back to stat and then air over the couch, we're gonna do the linear regression A. X plus B. So go to option four And your excellence should be a one in your wireless should be L2. And then we're going to calculate that. And here's our regression line. So why equals negative 0.897? X plus 4.147 So let's go and write that down. I should say why hat actually since it's a prediction so why hat is negative zero 897. X. Plus 4.147. Okay so that's my least squares regression line. Then I'm supposed to find the S. S. E. Using the summation of residual squared. So it's why minus Y. Hat. Which is the actual Y. Value minus the predicted Y. Value squared. That's called the residual squared. And we're gonna add those together. So the summation of that and I'm also going to use a calculator for this. So if you go back to edit we're gonna use this third column and we're gonna we're gonna apply the formula or the equation that we just used or that we just found. So in L. Three. So air over to L. Three, make sure L. Three is highlighted. You can type in negative point 897 and then times X. Which in this case our exes going L. One. So I'm gonna say times L one And then plus the 4.147 whenever you do that, that should give you the Y hats. So these are the Y hats, these are the actual wise. So if I go back over To L four and arrow up so L four is highlighted. Now I need to find why minus Y hat, which in this case would be L two minus L three, so second to minus second three and that will give me my residuals. Now I need to square them. Now you could make that squared in L. five if you want. But we can actually just go to stat and then couch one, bar stat and just change that to L. Four and we can find the summation of of the squares there. So some nation of X squared. So this this column is X. Now it is or we're calling it actually. So some nation of X square, that's the y minus Y hat squared is 20.912 So let's go and write that down 20.912 mm. Now we're going to compute SSC using the formula S S Y minus beta one hat times ss X. Y. So we need some nation of X. So you just add the excess together. This is the original data set. So you add the excess together, should get 22. Yeah, the wise together should just get one. You add the y squares together, you should get 43 then you multiply the exes and the wise and then you add those together, you get negative 20 and there are a total of five data values. So the way you find SS, Y, Y is you take the summation of Y Square which is 43 -1 over in which is 1/5 times the summation of Y, which is one quantity squared. So whenever you do that you should get 42.8. Then you find S. X. Y. Which is the summation of X. Y. which is negative 20 minus 1/5 times the summation of X, Which is 22 Times The Summation of Y, which is one. And whenever you plug that in you should get negative 24.4 And then your beta one hat We already found was about negative 0.897. So now we just have to plug it into our formula. This formula is up here in black SS Why? Why? Which is 42.8? I'll do SSC equals -10 Hat which is negative .897 times negative 24 4. Which is that S. S. X. Y. And you plug that in the calculator and you should get About the same answer 20.912. So if your answer is slightly different, it probably just means you round it differently. I looked around at the very end and that's typically what you should do. But you know if you rounded to four decimal places or whatever, that's fine, but you should get about 20.9 either way.

The following is a solution # three. And were asked to compute the least squares regression line from section 10.2. Using the data from section 10.2, number three. And I went ahead and put that in my tea I 84 calculator. But you can use any sort of technology you want. But if you have a T. I. T. For you go to stat and edit and again that's to recall L one is your ex value. So 1346 and eight should be your ex data values and then L two should be a Y. Data values 413 negative one and zero. So if you go to stat and couch the way this book is set up, it's set up like number four. So it does it in standard form where does the slope first and then the Y intercept second. So the slope is A. And the Y intercept is B. As opposed to this eight Lynn rig A plus B X. O. Some books are written where the where the Y intercept is first and the slope a second. But this one's not so we're gonna go to four. Excellence should be L one wireless should be L two unless you put it in different columns. And then when you recalculate, uh, here's the slope negative. So don't forget that negative, their negative 0.6. If we round, let's go and say 610 And the Y intercept is 4.82 So let's go and write that down. So why hat is the notation there? So why hat Equals -610 x Plus 4.08 two. So this is the least squares regression line for that small data set in section 10.2, number three.


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