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Bio-Tech NOTES FOR LINEAR REGRESSION Introduction: Prediction? ________________________________________________________ ____________________________________________________________________________ Objective: ___________________________________________________________________ Bivariate data _____________________________________________ Warm-Up: 1. Write an equation of the line that has slope ¾ and y-intercept -2. ___________________ 2. Write an equation of the line that passes through the y-axis at 3 and has slope of -4. ___________________ 3. Write an equation of the line shown I the graph on the screen. __________________ 4. If a line passes through (0, 5) and (2, 9), then the slope is ______ and an equation of the line is __________. 5. Write an equation of the line which passes through (-3, 8) and has a slope of -2. ______________________. Equation of a line in “slope-intercept” form _____________________ or _____________________________ (statistics books) “Linear Regression” ____________________________________________________________ Summation Mean ∑𝑥 ______________________________________ 𝑥̅ ______________________________________________ Predicted value of y 𝑦̂ ____________________________________ All regression lines pass through the point ______________________ Table set-up (fill in the headings of the columns) x y Formula for the slope (b) of the regression line ____________________ Bio-Tech Notes p. 2 Example problem: Fill in each cell in the table. Calculate the means for x and y. Interesting fact: ∑(𝑥 − 𝑥̅ ) = 0 This means that you can check the 3rd and 4th column values by adding them and checking to see if the sum is equal to 0! X 3 7 10 15 18 Y 16 45 82 102 116 ̅ 𝒙−𝒙 (𝒙 − 𝒙 ̅)(𝒚 − 𝒚 ̅) ̅ 𝒚−𝒚 ∑ Calculate the slope of this line 𝑏= ∑(𝑥−𝑥̅ )(𝑦−𝑦̅) ∑(𝑥−𝑥̅ )2 (𝒙 − 𝒙 ̅ )𝟐 ∑ = Regression line must pass through the (mean, mean) ______________ Substitute into the equation and solve for ‘a’ Equation ___________________________ Use your equation to find the possible y-value when the x = 12. _________ Use your equation to find the possible value of x when y = 32. _________ Use your equation to predict y when x = 20. ______ Use your equation to predict x when y = 150. ________ (𝒚 − 𝒚 ̅ )𝟐 Bio-Tech Notes p. 3 Correlation Coefficient ‘r’ ____________________________________ 𝑟= 𝑧 𝑠𝑐𝑜𝑟𝑒 = ∑(𝑥 − 𝑥̅ )2 √ 𝑠= 𝑛−1 Notes for finding ‘r’: Notes for key-strokes on calculators: ∑ 𝑧𝑥 𝑧𝑦 𝑛−1 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑒𝑎𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛