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Match the description below with its mathematical expression_Statistic used to detect dependence in sequences of residualsChoose the correct answer below:OA Heteros...

Question

Match the description below with its mathematical expression_Statistic used to detect dependence in sequences of residualsChoose the correct answer below:OA Heteroscedasticity0 B: Scatterplot of Y on X0€ Timeplot of residuals 0 D. Leveraged 0 E. Random sample from a population 0 F Homoscedasticity G Durbin-Watson statistic0 H: Plot of residuals on X 0 1 Normal quantile plot of residuals 0 J. Outlier

Match the description below with its mathematical expression_ Statistic used to detect dependence in sequences of residuals Choose the correct answer below: OA Heteroscedasticity 0 B: Scatterplot of Y on X 0€ Timeplot of residuals 0 D. Leveraged 0 E. Random sample from a population 0 F Homoscedasticity G Durbin-Watson statistic 0 H: Plot of residuals on X 0 1 Normal quantile plot of residuals 0 J. Outlier



Answers

Which of the following is a necessary assumption for performing inference analysis on the slope of a least squares regression line? (A) There is no strong skew or outliers in the data. (B) A straight line can be drawn through the set of paired observations in the scatterplot. (C) The distribution of the residuals is approximately uniform. (D) The distribution of the residuals is approximately linear. (E) The distribution of the residuals is approximately normal.

So I'm using a. T. I. 84. And I put my data in list one and list too. And when I did I find the equation Regression equation to be 21.024 plus .1389 x. And then I grabbed a scatter plot of the data and the scatter plot of the data. The data ends up looking like a pattern kind of like this. And then the regression equation ends up looking like that. So we can see that there is curvature in that data and that regardless of what the correlation coefficient we can see that there's definitely a pattern. And then you are asked to have a residual plot. And so a couple things you can do you can go over here in a to list three and you can um go into your editor and you can find the residuals longhand by putting the regression equation up here so go to variables and statistics and eq. And it will put the equation in there. And the only thing I need to do is change the X. value into list one. And when I do I get my residual excuse me, my predicted value. And that comes out to be 26.3, 25.7 and etc those are the first couple. And then we could take list to minus list three to find my residuals and make sure you're up in this L for icon not to do that and list to minus list three. And when I do I get the residuals of negative 2.3 is my first one, negative 3.7 point 42 and so on and not. That will give you an idea of what the residual should look like. And then I can go back and do a stat plot And my step plot would have list one versus list for. And also I would clear out that Y equals because we don't need that anymore and I don't want that to when I hit zoom and number nine presumed staff, I end up getting a residual plot that looks like clips, we put the axis there we go. And the residual plot will always have that access in the center by the way, and the residual plot has a couple dots and it does this and like so. And anyway, we definitely see a pattern of a curb, a curved pattern on here that tells us that our linear model is really not suited to this, that there's probably a different curved model that would fit this data better.

So I put all of my data in list one and list too. And then hit the linear regression button. And I found that the equation ended up being 3.912 Plus 1.71133 X. And we got a correlation coefficient which it doesn't ask for that. But it's important to look at a .9895. And then I did a stat plot graphing list one versus list too and having wise of one B. This regression equation. And when I look at that graph that graph looks very very linear. And the data points are not all specifically on the line. But when that line is drawn through them it looks very very close and we could tell that by that correlation coefficient that it was quite possible that it was going to look quite linear now for part C. After I've calculated that value for the regression equation, I went to list three and doctor instead of going through and finding the observed well we have the observed value but finding what the expected value was a predicted value by plugging all these list one values back into the equation and then subtracting. We do have that feature under second and list on A. T. I. 84 then the residual at the bottom. And so it's giving me all these residuals. When I look at my uh my chart For my residuals or my staff at it, the first residual comes out to be about .4. The next one is about .24. The next one is negative .58. So that gives you an indication of what the residuals should be. And now I'm going to go back to my staff plot And I'm going to turn my step plot on to go list one vs list three. So list one versus list three. And I'm also going to get rid of my regression equation as wise of one And just the second hair, list one vs list three and hit my wife someone and clear that out. And then again hit zoom number nine zoom number nine. And when I get that residual plot, that residual plot will always have that X axis in the center and my residuals, I have a result. You down here, I have up here, I have done here, down here, there's one up here and down here and I'm really not seeing it's kind of oscillating around this line. So I don't see not a not a particular pattern. There's no curvature in the data. So there's no pattern, which means that the linear model fits quite well, is quite good. Oh, no. All right. So I would use that model as a good predictor.

So this problem, the first thing we're asked to do is come up with an estimated regression equation. And this is the common regression equation. Ah, why hacked our estimated Why value is equal to beta subzero or be subzero, plus piece of one times X. So the first thing we're gonna do is come up with a beast of one which is equal to the sum of a product of the difference between each X value, Um and I mean for the x value. And why value to each individual? Why value and be? Why bar over the sum of each the sum of square differences for each individual X value end, um, expert. So in order to do this, we first have to come up with an ex bar and X bar is going to be equal to two plus three plus four plus five plus seven plus seven plus ah seven plus seven plus eight plus nine divided by +123456789 divided by nine values which is equal to ah, 5.78 And now we need a y bar and why bar is going to be equal to four plus five plus four plus six plus four plus six plus nine plus five plus 11 divided by nine, which is equal to six. Okay, so now that we have these, we can come up with our beta sub one be sub one value. So this is going to be equal to, um, the beast. One is going to be equal to the different the some of the product of each individual X value. So, um, I'm just going to give you the values that we get after re compute all the multiplication addition divided by, ah, the sum of difference between each individual x value and the export of 45.6. So we get a piece of one value of 0.64 Okay. Um, So now how did we get these values? So the sum of yeah, product of the differences is gonna be equal. Thio two minus ah. 5.78 times four minus six plus three minus 5.78 times five minus six plus four minus 5.78 times four minus six and so on and so forth. And we have to do this with all the X's and y's in our data set and this value down here a blues and red is equal to, um, to minus 5.78 squared plus three minus 5.78 squared and so on and so forth until we get to the end of our data set. And then we gonna be someone value of 0.64 And now we need to come up with aby subzero value, which is equal to why hat minus, um, piece of one times X bar. Sorry. Why? Bar not expert. Why bar not a white hat? Um, so we figured out that our Y bar is equal to six minus, and we just found our beasts of one value to be 0.64 times our ex bar of 5.78 So we gotta be sub 00 of, um, 2.32 So now the answer to part A is equal to high hat is equal to 2.32 plus zero point 64 X. Now, using this, we have to come up with a plot of the residuals. So what are the residuals? The residuals are the difference between the actual Why values and the estimated y values. And how do we get these estimated? Why values these estimated why values come from he formula for White Hat s 02 0.32 plus your 0.64 x. And for each x value that we have, we're going to get a predicted value. So at wise, equal to or at X is equal to two. We get a predictive value of 3.60 at why is or X is equal to three. Get a predictive value of 4.23 And we do this, Um, until we get to the end of our ex list. So this keeps going on and on and on s so we have a ah, the actual y value Or why I and then our ah predicted value, which is why hat so by. So we're going to take the difference between these and square them. Do you get our residuals? And after we do that, we get a plot that looks something like this. So what does this plot seem to tell us? Um, so it seems like, um, as X increases exit over here as X increases there seems to be more variance between, um the values in our plot or the residuals. Um, so ah, this doesn't mean that there is a constant variant. Um, so that means that the assumption about the error term is satisfied.

So here's our scatter plot and You've never find two points on it and make a line through it. So just for simplicity sake I'm gonna pick these two points to the origin and 14. That's gonna give me the creeps me out. His X. Equals what's it to mix? No four X. There we go. That's what it is as we're looking at that is not the line of best fit. We could we could do better. And that's what the the least squares regression line is going to give us. Uh huh. So the way we can use our software for that is using this notation. Yeah. Why one until the and X. One plus B. There's are ideal in an ideal B. I'm gonna give it a function name called G. You can see that looks a bit better than the red F. Function but I want to give you the sum of the squared residuals for each of them. So it's going to do that. So what we're gonna do is take the given y value and subtract out the function value about the X. Give an excellent you square this and that's what's that gonna be 16 plus 25 plus 9 50. Great. Now if we can find the sum of squared residuals for the regression line that's why I want to buy this G. Explained square that stuff all this stuff so we could sum that out but I'm gonna use the software to help me do a equals This function. This expression which is going to give us a five element list To some from 1 to 5 A. in which is 6.7 which is way less than the some squares an estimate line. Seven squares of the residuals based on the estimate. So this is the smallest is released. Some of these great residuals that we would ever get. No matter what we did, turn off those points so we can better see it and there again coming on the fit. Found the line in part B versus the at least squares regression line found in the other part. Well, obviously much better visually. And also by that some of these residuals. Second


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