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The following data were used in a regression study.$\begin{array}{cccccc}{\text { Observation }} & {x_{i}} & {y_{i}} & {\text { Observation }} & {x_{i}} & {y_{i}} \\ {1} & {2} & {4} & {6} & {7} & {6} \\ {2} & {3} & {5} & {7} & {7} & {9} \\ {3} & {4} & {4} & {8} & {8} & {5} \\ {4} & {5} & {6} & {9} & {9} & {11} \\ {5} & {7} & {4}\end{array}$a. Develop an estimated regression equation for these data.b. Construct a plot of the residuals. Do the assumptions about the error term seem tobe satisfied?

a. $\hat{y}=2.322+0.637 x$b. $\mathrm{No}$

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Missouri State University

University of North Carolina at Chapel Hill

Cairn University

Boston College

Lectures

0:00

00:48

The r.m.s. error of the re…

05:43

The data from exercise 2 f…

03:28

In $9-13 :$ a. Create a sc…

02:11

02:19

04:58

In $3-5 :$ a. Determine th…

04:38

02:48

(a) Is the r.m.s. error of…

05:34

07:01

(a) Compute the power regr…

03:46

04:33

Define the standard error …

04:28

01:32

Tell whether correlation i…

01:28

In Example 1 on page 731, …

05:28

In exercise $1,$ the follo…

03:12

Draw a scatter diagram for…

05:08

The following problem is r…

01:34

02:33

Tuddenham and Snyder obtai…

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.

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