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Lesson 4 - 2

Lesson 4 - 2. Least-Squares Regression. Objectives. Find the least-squares regression line and use the line to make predictions Interpret the slope and the y-intercept of the least squares regression line Compute the sum of squared residuals. Vocabulary.

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Lesson 4 - 2

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  1. Lesson 4 - 2 Least-Squares Regression

  2. Objectives • Find the least-squares regression line and use the line to make predictions • Interpret the slope and the y-intercept of the least squares regression line • Compute the sum of squared residuals

  3. Vocabulary • Residual – aka, the error; difference between observed value of y and the predicted value of y • Method of Least-squares – minimizes the sum of the residuals squared • Least-squares regression line – line that minimizes the sum of the squared errors • Explanatory variable – the independent variable in the model (x) • Response variable – the dependent variable in the model (y)

  4. Terms • Scope of the Model – the area in which the model applies • B0 – the y-intercept (an offset of the model when the independent variable is zero) • B1 – the slope of the line (in calculus – the first derivative, rate of change)

  5. Least-Squares Regression Line The equation of the least-squares regression line is given by y = b0 + b1x where b0 = y – b1x is the y-intercept of the least-squares regression line and sy b1 = r · ------ is the slope of the least-squares sx regression line ^

  6. Residuals • One difference between math and stat is that statistics assumes that the measurements are not exact, that there is an error or residual • The formula for the residual is always Residual = Observed – Predicted • This relationship is not just for this chapter … it is the general way of defining error in statistics • The least squares regression line minimizes the sum of the square of the residuals

  7. The residual The model line The observed value y The predicted value y The x value of interest Residual on the Scatter Diagram

  8. Least-Squares Regression Model Scope Scope of the model Response Areas outside the scope of the model Explanatory Linear relationship outside the scope of the modelis not guaranteed!

  9. Interpretations A population model for bluegill in a lake is y = 200 x + 11,500 • The slope, b1 = 200, means that the model predicts that, on the average, the population increases by 200 per year • b0 = 11,500 • If 0 is a reasonable value for x, then b0 can be interpreted as the value of y when x is 0 (there were 11,500 bluegill in the lake when we started the model) • If 0 is not a reasonable value for x, then b0 does not have an interpretation

  10. TI-83 Instructions for Linear Reg • With diagnostics turned on and explanatory variable in L1 and response variable in L2 • Press STAT, highlight CALC and select 4: LinReg (ax + b) and hit enter twice • Output: LinReg y = ax + ba = xxx (slope or b1 value)b = xxx (y-intercept or b0 value)r² = xxx (coefficient of determination)r = xxx (correlation coefficient)

  11. Example • Find the least-squares regression line • What does the model predict for x = 5? • What is the residual from the above? • Is this model appropriate for x = 15? Why or why not? y-hat = 88.7327 – 2.8273 x y-hat = 74.5962 residual = -0.69623 No. Out of model’s range

  12. Summary and Homework • Summary • We can find the least-squares regression line that is the “best” linear model for a set of data • The slope can be interpreted as the change in y for every change of 1 in x • The intercept can be interpreted as the value of y when x is 0, as long as a value of 0 for x is reasonable • Homework • pg 221 – 225; 2, 3, 6, 9, 19, 21

  13. Homework Answers • 6 -- True

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