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The following data resulted from an experiment performed for the purpose of regression analysis. The input variable, $x,$ was set at five different levels, and observations were made at each level.a. Draw a scatter diagram.b. Draw the regression line by eye.c. Place a star, $\star,$ at each level approximately where the mean of the observed $y$ values is located. Does your regression line look like the line of best fit for these five mean values?d. Calculate the equation of the regression line.e. Find the standard deviation of $y$ about the regression line.f. Construct a $95 \%$ confidence interval for the true value of $\boldsymbol{\beta}_{1}$.g. Construct a $95 \%$ confidence interval for the mean value of $y$ at $x=3.0$ and at $x=3.5$.h. Construct a $95 \%$ prediction interval for an individual value of $y$ at $x=3.0$ and at $x=3.5$.

a $\&$ c.d. $\Sigma x=37.5, \Sigma y=52.7, \Sigma x^{2}=104.75$$\Sigma x y=98.75, \Sigma y^{2}=159.49 ; \hat{y}=3.79-0.415 x$e. $s_{e}=0.196697$f. -0.496 to -0.334g. 2.43 to 2.672.19 to 2.49h. 2.12 to 2.981.90 to 2.78

Intro Stats / AP Statistics

Chapter 13

Linear Correlation and Regression Analysis

Section 6

Understanding the Relationship between Correlation and Regression

Linear Regression and Correlation

Ladymaica T.

January 23, 2022

What is the value of 21.3481

Temple University

Missouri State University

Piedmont College

Lectures

0:00

08:34

Consider the following pai…

02:53

Regression was performed o…

all right. I put the data in here. I had to write down all the X values in a in a row and all the why values in a row? Um, Now I'm going to insert a scatter plot. So highlight the data. Insert a scatter plot chart. Sometimes it picks, scatter plot, and I don't know why it doesn't sometimes, but not always. Um, there it is. All right, Now I want my X axis to be acts, and I want to remove that now. These did have names. What were the names off them? No, they were just X and y, okay. They didn't have names trying to move this. Okay, now I want Teoh Inserted equation. I mean, ah, insert a trend line, and I'm gonna show the What am I doing here? Ensure the equation and the R squared? What did I do by accident number format was not They think What did I dio art. I'm just gonna undo what I did because I don't I don't like these numbers on there. Okay, let's do this again. I'm gonna show the r squared value and use the equation. Okay, Draw the regression line by law, by I. Well, it's a little late for that. Um, I probably would have drawn something at least very close to that. Um, place a star at each level about where the mean is located. Does it look like this is the best line for the means? Well, I would say the mean is right here, so that's a little bit, but no would say the mean is yet right there. Let's see the mean is a little lower. Say that means a little lower. I would say that means a little higher, but does it look like it's the best for the means? Yeah. Yep. Um, calculate the equation of the regression line. I have that right there. Find the standard deviation of Why about the regression line? Okay. Someone is trying to move this. So the distance of y from the regression line. Uh, well, first of all, I need to get my prediction. That would be the regression line. So, um, equals negative. 0.415 kinds. X plus three point. Have a nice All right, so there's a prediction. Now I need the distance of wife from that prediction. So, um, this is just gonna be. Why? Minus why carry equals? Why? Minus why, Carrot. All right, bring these all down here. Um, I can't Never mind. I just need to bring it down to the bottom of the data here. Okay, So I need to find the standard deviation of why about the regression line. So that would be this. So I'm just gonna dio standard deviation, like right here S t d e v of all of these things. Now he's got a sample standard deviation. I don't know. So I'm gonna do a little investigation. Here s T d E v estimates a standard deviation based on a sample, but it says I should use STD evey point s instead. So I'm actually going to change this. It shouldn't really have any effect. Women s 190512 Same thing. Gives me the same thing. Okay. On it. Now construct a 95% confidence interval for the true value of beta one. All right, So true value of the slope. So the way that I have done this in the past is that, um I put in B zero on B one over here, so b zero is 3.97 What? It said 3.79 be one is negative. 10.415 I think I'm reading these. Rome 3.79 and 0.415 You get 3.79 point 415 All right, Good. Now I want to make sure that this calculated the are squali our value correctly. I just put this in. Um, it says here that are squared is ah, 0.881 equals the square root of 0.81 93 a set. Yep. It looks good. Okay, so I have my spreadsheet here, which I created, um, calculating the sum of squares of axe sum of squares of why? Sum of squares of X y and so that it can calculate our That's not really important right now. Um, not really in the context of this question. So, um, what I am doing, though, is I'm can't calculating the standard error of the estimate. I'm using the method of equation 13.8. Um, where I take the some of the Y squared values minus B zero times the sum of the Y values minus B one times the sum of the X times y values and I divide that by end minus two. Give me that. Put the very variance in here for convenience. That's just the square of the standard error of estimate. Then I calculated the estimated standard error of regression. And how do you do that is You take the square root of the variants and you divide that by the quantity. Ah, the some of the X squared. Which is he here, minus of some of the axes. Then you square that and you divide by an which is how many there are right here. 18. I also use this function to figure out the company confidence coefficient. This this represents 95% confidence and ah, this is and minus two, which is the number of degrees of freedom e the air here is just this times this So then I take b one plus e and be one minus e. Elsa calculated t star. But that's not in this either. Okay, so now we need a 95% confidence interval for the mean value of why at axe equals three and X equals 3.5. And we also need a prediction interval. Okay, well, at X equals three and X equals 3.5. We can use the equation and the prediction. Um, but you know what? I'm just going to put it. Where am I going to put this? I don't need this anymore. Delete Put X Values here of 33.5. Okay. Prediction is going to be no e cools B zero F four plus, you won four in Vex. Okay, so there is the prediction. Just make sure that prediction makes sense. The re comma 2.55 32.55 Yep. So the 95% confidence value for the interval for the individual value, which is h um, So I'm going to write. Ah, individual here is just going to be this. All right, So, going to get back to my sheet here. I'm gonna move this over because I do want to get air of estimate here. Um, and I'm looking at the equation that says e equals t of end minus two, comma off over two. Well, that's just going to be often. It's coefficient equals the confidence coefficient times the standard air, which is going to be up here. Times the square root, Uh, one plus one over n plus X sub zero minus the mean squared Well, x of zero is here, minus the mean. Oh, I forgot to put parentheses around. That minus the mean I don't have to mean calculated, but I'm going to put it here except zero minus the mean squared over sum of squares are acts which I do have calculated here. Let's just make sure that my order of operations correct one plus one over end. Plus it's mine. Is this squared over that? Okay, that should work. That gives me a standard air. Except I don't have the means yet. I said I would put it right here. And so that's what I'm gonna dio. There we go. 2.8 The mean of the X values that looks about right. Okay. Looks like a reasonable air. And so now I just have to dio this minus this. And this looks this Plus this. All right, here. Um, I should have put a lot of ah fours in, um, the only thing that does not stay in its place is a 90 so f 44 f four F four F four. There we go. So now I should be able to just drag that right down. Ah, all right. Now, um, I need to do this for, um the mean value. So let me just copy this again. Give me the exact same praying yet. So now I'm gonna changes to me. All right, So now I see the equation, so it doesn't look much different and, ah, it really shouldn't look much different. So we just have to change. He here. Uh, it's going to be square root of one over an So we get rid of this one. Plus looks right. Leading clinics things one over an plus X minus X zero squared. That's the same thing, X minus X zero minus X squared over SS X squared. OK, so all we did was we not didn't put a one in here, and we still divide by you assess X. So I just want to verify that the only difference was that I didn't put a one in there. Yep. All right. So drag that down. Okay, so now I'm thinking about Does this make sense? The prediction has less air down here and that makes sense. Um, be 95 minus C 95. Okay, that makes that seems right. Be 95 plus D 95. That's not right. It's supposed to be. See 95. Okay, that must be my problem. Copy that up here. Okay, That makes that makes more sense. So 2.5 plus 1/2 is about three to any other miners have is about one. All right. Got it. So this is the means and these air the individuals, Um, and I'm gonna write predictions. I wanted an ass in the end. All right, so thanks.

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