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The data from exercise I follow.$$\frac{x_{i}|1 \quad 2 \quad 3 \quad 4 \quad 5}{y_{i}|3 \quad 7 \quad 5 \quad 11 \quad 14}$$The estimated regression equation for these data is $\hat{y}=.20+2.60 x$$$\begin{array}{l}{\text { a. Compute SSE, SST, and SSR using equations }(12.8),(12.9), \text { and }(12.10) .} \\ {\text { b. Compute the coefficient of determination } r^{2} \text { . Comment on the goodness of fit. }} \\ {\text { c. Compute the sample correlation coefficient. }}\end{array}$$

a. $S S T=80, S S R=67.6, S S E=12.4$b. $0.845,84.5 \%$ of the variability between $x$ and $y$ is explained by theregression equation.c. $r=0.9192$

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

University of North Carolina at Chapel Hill

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Oregon State University

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Lectures

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05:43

The data from exercise 2 f…

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Refer to the data presente…

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In exercise $1,$ the follo…

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For the following exercise…

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The data from exercise I f…

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The following data are the…

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Compute the standard error…

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in this exercise, we are given the data set shown here and we're also told that the estimated linear regression line is given by the equation shown at the top here. For part A. We are asked to find the sum of squares due to error, the total sum of squares and the sum of squares duty regression. The S. S. E. Is given by this formula. PsST is given by this formula and the SSR by this formula. No, I'm going to solve these directly on the Excel spreadsheet. So the first step would be to find the predicted values for why using our regression equation. So the regression equation estimates why As 0.2 Plus 2.6 times the X value. So we can do this for each X value and we get the estimated Y values. Call that white hat. Now for the squared errors let's first find for the sum of squared errors. Let's first find the squared errors so far. Each data parrots the why value, why I subtracting the predicted value and then squared. And then we do this reach data pair and then we want to find the some of them. And now for the SST we can first find the squared deviations. This is by well let's first find the the average y value. And so the square deviations are each y value minus the average value, which is eight and then squared. So we do this for each row and then we sum them. And then for the SSR we can calculate the I've called it the squares duty regression and this will be the predicted value minus the average y value and then squared do this for each role and then find the sum. Yeah, something's gone wrong here. So let's Put an eight manually and try filling that down again. Okay, that's better. So the values we got were 12.4, 80 And 67.6. And we can confirm that the S. S. E. Plus the SSR is equal to the SST. Now for part B we are asked to calculate the coefficient of determination that's R squared, this is equal to the S. S. R. Divided by the SST. And this comes out to zero 845 And we are asked to comment on how well this regression equation fits the data and what we can say is based on R squared. About 84.5% of the variability in why can be explained by our regression equation. Yeah. And then for part C we are asked to calculate the sample correlation coefficient that is our so it's simply the square root of R squared. And this comes out to .919.

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