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R Squared

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r = -.944

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r = -.79

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y = -0.8141x + 9.1332y = -0.9402x + 43.721 if x = 15, y = ? y = -0.9402(15) + 43.721 y = 29.6195 if x = 6, y = ? y = -0.8141(6) + 9.1332 y = 4.2486 Which value for Y is a more accurate prediction for the given X value?

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To find how well the Line of Best Fit actually fits the data, we can find a number called R-Squared by using the following formula: 1- Sum of squared distances between the actual and predicted Y values Sum of squared distances between the actual Y values and their mean

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XY 340 1035 1130 1532 2219 2226 2324 2822 2818 356 Equation for Line of Best Fit:y =.94x + 43.7 Correlation = -.94 For example, here’s how to find the R Squared value for the data/graph below:

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XY Predicted Y Value Error Error Squared Distance between Y values and their mean Mean distances squared 340 1035 1130 1532 2219 2226 2324 2822 2818 356 Mean:Sum: Equation for Line of Best Fit:y =.94x + 43.7

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XY Predicted Y Value Error Error Squared Distance between Y values and their mean Mean distances squared 340 40.88.88.7714.8219.04 1035 34.30-.70.499.896.04 1130 33.363.3611.294.823.04 1532 29.60-2.405.766.846.24 2219 23.024.0216.16-6.238.44 2226 23.02-2.988.88.8.64 2324 22.08-1.923.69-1.21.44 2822 17.38-4.6221.34-3.210.24 2818 17.38-.62.38-7.251.84 356 10.804.823.04-19.2368.65 Mean:25.2Sum:91.81Sum:855.60 Equation for Line of Best Fit:y =.94x + 43.7

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1- Sum of squared distances between the actual and predicted Y values Sum of squared distances between the actual Y values and their mean To calculate “R Squared”… 1- 91.81 855.60 1- 0.11=.89

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XY 340 1035 1130 1532 2219 2226 2324 2822 2818 356 r = -.944 OK. Don’t kill me. Remember this was the data/graph we were finding “R Squared” for? The value we got for R Squared was.89 Here’s a short-cut. To find R Squared… …Square r r 2 = -.944 -.944 r 2 =.89

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R Squared To determine how well the regression line fits the data, we find a value called R-Squared (r 2 ) To find r 2, simply square the correlation The closer r 2 is +1, the better the line fits the data r 2 will always be a positive number

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r = -.944 r 2 =.89

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r = -.79 r 2 =.62

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9.2A- Linear Regression Regression Line = Line of best fit The line for which the sum of the squares of the residuals is a minimum Residuals (d) = distance.

9.2A- Linear Regression Regression Line = Line of best fit The line for which the sum of the squares of the residuals is a minimum Residuals (d) = distance.

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