Coefficient of Determination & using technology

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Coefficient of Determination & using technology

The coefficient of determination is the measure of the variation of the dependent variable that is explained by the regression line and the independent variable. The symbol for the coefficient of determination is 𝑟 2 The easiest way to find the Coeff. Of Deter. is to square the correlation coefficient or… (next slide)

Coefficient of determination 𝑟 2 = 𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛= 𝑦 − 𝑦 2 or predicted value – mean of dependent variable squared 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛= 𝑦− 𝑦 2 or (actual value – mean dependent variable) squared Total variation = explained + unexplained variation 𝑢𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛= 𝑦− 𝑦 2 or (actual – predicted values minus mean of the dependent variable) squared.

Reusing our data from the linear regression Slides—find the explained and total variation

Summarizing—we calculated total variation = 92.8 Total variation = total unexplained + total explained 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛=14.4+78.4=92.8 (checks) Coefficient of Determ. = 𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑢𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝐶𝑜𝑒𝑓. 𝐷𝑒𝑡𝑟. = 78.4 92.8 = .8448 We had already calculated r =.9191 (recall r is coeff. of correlation) 𝑟 2 = .9191 2 = .8447—so it checks!

Technology to the rescue Technology to the rescue! Load the following into the l1 and l2 data banks on the tI Hit Stats, edit, and enter: x y 1 10 2 8 3 12 4 16 5 20

Using the ti to calculate linear regression Now enter Stats, calc, selection 8 (LinReg) L1,L2 and enter. You should see a (y-intercept), b (slope), r (coefficient of correlation) and 𝑟 2 (coefficient of determination)