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The commercial division of a real estate firm conducted a study to determine the extent of the relationship between annual gross rents $(\$ 1000$ s) and the selling price ( $\$ 1000$ s) for apartment buildings. Data were collected on several properties sold, and Excel's Regression tool was used to develop an estimated regression equation. A portion of the regression output follows.$$\begin{array}{l}{\text { a. How many apartment buildings were in the sample? }} \\ {\text { b. Write the estimated regression equation. }} \\ {\text { c. Use the } t \text { test to determine whether the selling price is related to annual gross rents. }} \\ {\text { Use } a=.05 .}\end{array}$$$$\begin{array}{l}{\text { d. Use the } F \text { test to determine whether the selling price is related to annual gross rents. }} \\ {\text { Use } a=.05 \text { . }} \\ {\text { e. Predict the selling price of an apartment building with gross annual rents of } \$ 50,000 .}\end{array}$$

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Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

University of North Carolina at Chapel Hill

Piedmont College

Idaho State University

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with this nova table. The first thing we have to do is figure out how many apartment buildings were in the sample. So that is basically our end. So how would we find our end? Well, our end is used to calculate a few things. One of those being the degrees of freedom. Um, And when we calculate the degrees of freedom for the total or for the, um, residual, we have to use our end so we could do this two ways. Uh, we can either calculate. So the degrees of freedom for our total is equal to end minus one. The degrees of freedom for our, um, error is equal to end minus two. So we know that the degrees of freedom for our total is equal to eight. So, degrees of freedom in general, um or so take this out so that our end would equal nine. Or weaken. Say that because and is or sorry Degrees of former error is equal to seven r n can equal night. Either one of these possibilities work either way. Our and is equal to nine. Next were asked to come up with a estimated regression equation. So at the bottom of this table were given these values here, we're giving the coefficients, which is what we're going to focus on. So when we see the intercept, we should think of be sub zero. And when we think of this value over here, this is our model, basically. So this is our V sub one. So our estimated regression equation why hat is going to be equal to 20 plus 7.21 x. So that's the answer to Part B now in part C were asked to come up with a, um T tests determine whether the selling price is related to the annual gross rents. So for this, we should first come up with, ah, hypotheses. So are no. Hypothesis is going to be that there is a relationship. There is no relationship. So when there is no relationship, our our beta value is going to equal zero, and our alternative hypothesis is going to be that there is a relationship. And when there is a relationship, that means that our beta value is not equal to zero. Now we have to come up with a T test statistic. Luckily, this table provides it for us and we are interested in looking at beasts of one. So 5.29 will be our t te statistic. So, with a we have to come up with a P value for t equals 5.29 But the degrees of freedom of N minus two. And since we have nine and minus two is seven, um, seven degrees of freedom s O. The P value is going to equal 0.0 zero. Ah, 11 Okay, s. So now we're going to compare this to Alfa of 0.5 And, um, this is less than 0.5 So we reject the no. And now we're going to do a similar thing for an F test statistics. So the F te statistic is equal to the mean for the is equal to them. Yeah, it's the mean square of the residual over means square of the air. So we have the means square of the residual. No, we don't. So the means square of the residual is equal to the sum of squares of the residual divided by the degrees of freedom for the residual. We're sorry. We're taking the mean square of the regression, not the residual means. Square of the regression is equal to sum of squares of the regression divided by the degrees of freedom of the regression Um, which is equal to 41,587 0.3. Divided by the number of independent variables we have, which is one so mean square of, um, the regression is 41,587 0.3. And now we have to come up with, um, mean square for the air, which is the means square for the residual. So the mean square of the air is equal to the sum of squares of the air divided by n minus two. So we need to come with the sum of squares of the air. The sum of squares for the error is equal to the sum of squares of the total, minus the sum of squares of the residual. So we just have to take the difference of these two, and we will get a on sum of squares of the air equaling 10,000 396.3. No 0.8. Sorry. 0.8. Now, to find the main square of the air we're going to take 10,396 0.8, divided by seven to get a value of 1485.3. And now we need to find an F te statistic. Ah, okay, so it's the main square of the residual or means square of the regression divided by the means square of the air. So we have 41,587 0.3, divided by 1485.3, which is equal to 28 with an F. So we have to come up with a P value now be value at F equals 28. Um, with a degrees of ah, freedom. Ah, and minus two, I believe, um, which is seven. We get a ah P value less than 0.0 one. So this is small. So then, compared to our Alpha of 0.5 we're going to reject the null. We reject the no. And now we have to come up with, um, an estimate for ah, our Why value at, um, 50,000. So, um, because everything's in terms of thousands are why hat at 50 is equal to 20 plus 7.21 times 50 which is equal to 380,000 500

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