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The data from exercise 3 follow.$$\frac{x_{i}}{y_{i}} \left| \begin{array}{ccccc}{2} & {6} & {9} & {13} & {20} \\ \ {} {7} & {18} & {9} & {26} & {23}\end{array}\right.$$$$\begin{array}{l}{\text { a. What is the value of the standard error of the estimate? }} \\ {\text { b. Test for a significant relationship by using the } t \text { test. Use } \alpha=.05 \text { . }} \\ {\text { c. Use the } F \text { test to test for asignificant relationship. Use } \alpha=.05.\text { What is your conclusion? }}\end{array} $$

a. 6.5141b. Fail to reject the null hypothesis $H_{0}$ or not significantc. Fail to reject the null hypothesis $H_{0}$ or not significant

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Missouri State University

Oregon State University

Idaho State University

Boston College

Lectures

0:00

11:56

The data from exercise 2 f…

08:57

The data from exercise I f…

07:48

Refer to the data presente…

here's a solution to the problem number 25 we're supposed to find the standard error of the estimate which is the square root of M. S. E. Which is the square of SSS divided by N -2 and of course Sse is the sum of the residuals squared. So let's go to our graphing calculator. Go to stat and edit and go and pipe in your data values in L. One and L. Two 269 13 and 20 is L. One and then 7 18 9 26 and 23. That's your those are your Y values. Okay so we need the regression equation. So if you had a stat and then cow and then let's go down to that eighth option where it says Lynn rig a plus bx and then go ahead and calculate And there's your regression line. So it's pretty nice. So it's 7.6 plus .9 X. So we'll need that. So let's go and write that down. So that's why hat equals 7.6 plus 0.9. Okay so now we need to get the white hats. So if we go back to stat and edit and we're gonna use this third column here to get the predicted wise now so if we go to use our regression line we type In 7.6 plus .9 Times L one and those are predicted. And then we can go ahead and find the residual squared if we just take parentheses L two minus L. Three and then we square that weird. And we're gonna go ahead and add these up. So if you go to stat and then couch and then one bar stats. And we're gonna look at L4 And calculate. So the summation of that, that's the summation of the residual squared is 127.3. Okay, so that's what we need. So the yes is equal to um the square root of the sse which we just found to be 1 27. Um whatever it was .3, so 127.3 and then divided by N -2. And it is the sample size and there were five pairs of Numbers there says divided by three. So you type that into your calculator. You should get about 6.5 141. So that's the first answer there. That's the Standard error of the estimate. six 5141. Alright, so part B. That was part A right? That's a part A and part B uh says to find the T. Value or to test the hypothesis using the T. Test. So we need to find the T value and we're going to be one over S sub B one. All right, so be one we know is .9. That's the actual estimate. But the standard error of the one. We don't know what it is. So we need to find that. So the s sub B one is equal to s divided by the square root of the summation of X. I minus X. bar squared. I already found s that was the 6.5 141. But I need to get the summation of X I minus X bar squared. So I go back to the calculator, go back to stat and then we'll go ahead and find the X bar first. Remember the X values were in L. One. Okay, so the x bar is 10. So let's write that down. The x bar is tense, only that. And then now we can go to stat and then edit and let's use this 5th column now and we'll use parentheses no one minus the 10. So this is the X I minus X bar and then we square it and we're going to add these together, which means we can just go to stat coke one of our stats and we'll just change it to five now. And that summation is 1 90. So that is this denominator here. So we're going to do s which is the 6.5141 divided by the square to 1 90. That's the summation of X I minus X bar squared. And that should give you about one point. Actually, I'm sorry, that was not one point so respect actually zero point 4726. So that's this number here, .4726. And we want to replay that in, You should get about That's your 1.90 for so that's your T value. So we can use that to get the probabilities. This is actually a two tail test with a hypothesis. And so that's 1.9044. So we're going to find this probability and then we're just going to double it to get this probability over here. So we do T c d E F. Using that 1.9044. And so we'll go back to the calculator. You can also use the table to if you want. But second distribution and then it's the T C D E. F. The lower bound is that 1.90 4 4? The upper bound is just some big number. Yeah, Big Enough. I do. NT 99. And then the degrees of freedom here is going to be three. We paste because it's that that's just one tail. So you got to multiply that by two to get the two tails. So it's about .153. Alright, so power .153. So this is the p value if you didn't know that that's the p value and that is greater than the alpha. So remember alpha, alpha equals five. So it's great Annapolis that means we fail to reject. H not so we don't have a significant slope here. And then the last portion is basically doing the same thing. We're just finding the F. Test the F value is MSR over MSC. Alright. And MSR is just the SSR over MSC and the S. S. R. Is equal to the summation of why I had minus Y. Bar square. Okay, so a little bit more work now. So I need the why I had now I already know what those are. So let's actually we're probably gonna have to now we have one more. So the Y. I. Hats are the L. Three's and the white bar, I don't know what it is. Um it's the average of the L. two. So let's go to stat and then cow one of our staff For his second l. two And that is 16.6. Right, so why bars 16.6. Okay, so we need that. So now let's go back to stat and then To use L. six now. So this is going to be um parentheses L three. Those are the predicted y values minus the Y. Bar, which is 16.6. And then we're going to square that and then we're gonna add those together. So we'll go to someone more and less time stat Cal one of our stats and we'll do L six. That's 1 53.9. That is 1 53.9. So we have our SSR so it's 1 53.9 divided by the MSC and the MSC we found, well, okay, so we found the square root of MSC. So the MSC is s squared which is six point 5141 Squared, which is 42 .4335. Okay, so let's plug that in. So 42 .4335. So 153.9 divided by 42.335 will give you 3.6- seven. That's the F. Value. And then we need to find that probability. So the F test, there's always a right tail test will go to F c d e F. All right. The lower bound will be the three point 6-7 and then the upper bound will be some big number. All right. The degrees of freedom for enumerators, always one degrees of freedom for the denominators in minus two. So that would be three And notice that's the same thing. So .153 and that's what it should be. So, the p value 1.153, which is also greater than alpha. So again, we fail to reject each night.

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