"Finding the Coefficient of Determination and the Standard Err...

Question

"Finding the Coefficient of Determination and the Standard Error of Estimate In Exercises 11-20, use the data to (a) find the coefficient of determination r^2 and interpret the result,

12. [APPLET] Median and Mean Hourly Wages The table shows the median and mean hourly wages (in dollars) in 10 states in a recent year. The equation of the regression line is y = 1.208x + 1.495. (Source: U.S. Census Bureau)

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Accepted Answer

All right, hello, everyone. So, this question says, a real estate analysts collected the following data for house size in hundreds of square feet, and corresponding average sale price in thousands for 10 neighborhoods. The regression equation for predicting price from size is as follows. Find the coefficient of determination are squared and interpret your result. Here we have 4 different answer choices labeled A through D. All right, so first and foremost. Here we have two different samples to consider. We have not only the size of each house but also the prices. This means that we have to find both sample means. So that's X bar representing the mean of all of the sizes, and Y bar representing the means of all of the prices. Recall that the mean is equal to the sum of all data points divided by N, which is the number of data points. After you do this calculation, X bar is equal to 15.0. And Y is equal to 283.8. This now brings you to the next set of calculations. That you would then carry out. With the sample means that you just found. So let me put those on the screen here. All right. So, ultimately, the next step is to create a table for computing the correlation coefficient R, whose formula we'll discuss later on. But for now, let's talk about the information that you would have to find. So first You would take each X value listed and subtract it from X bar. And then you do the same thing for the Y values. You take each Y value and subtract it from Y bar. And then At this point, you have two differences, right, for both X and Y. The next step is to multiply. Those differences together. And in addition to oops. Multiplying those differences. By each other You would then square. Each difference So you would also calculate X X subtracted by X bar squared. And why subtracted by Y bars squared. So once this data has been collected. You would have to find the sum of 3 columns. You would find the sum of. X subtracted by X bar multiplied by Y subtracted by Y bar. You would find the sum of all. X subtracted by X squared and of all Y subtracted by Y squared. Now, the sums in the order mentioned are as follows. The first is 397.0. The second is 30. And the third Is 5,300.6. So now let's go back to our coefficient of determination. Now, the coefficient of determination is the square of the correlation coefficient. And the correlation coefficient are. Is equal to the sum. Of all values of X subtracted by X bar multiplied by y subtracted by y bar. Divided by The square root of. The sum of all values of X subtracted by X bar squared. Multiplied by the sum of all values of y subtracted by Y bar square. So plugging in the information found earlier, R is equal to 397. Divided by the square root of 30 Multiplied by 5300. 0.6 And this gives you approximately 0.99 by 5. So R2d ultimately is equal to 0.9955 squared. Which gives you 0.991. Therefore This means that about 99.1% of the variation in price is explained by the house size. Meaning your answer is option C in the multiple choice. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Coefficient of Determination (r²)

Standard Error of Estimate

Regression Line

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