Lesson 4 2 LeastSquares Regression Objectives Find the

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 • 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)

Terms • Scope of the Model – the area in which the model applies • B 0 – the y-intercept (an offset of the model when the independent variable is zero) • B 1 – the slope of the line (in calculus – the first derivative, rate of change)

Least-Squares Regression Line The equation of the least-squares regression line is given by where y^ = b 0 + b 1 x b 0 = y – b 1 x is the y-intercept of the least-squares regression line and sy b 1 = r · ------ is the slope of the least-squares sx regression line

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

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

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

Interpretations A population model for bluegill in a lake is y = 200 x + 11, 500 ● The slope, b 1 = 200, means that the model predicts that, on the average, the population increases by 200 per year ● b 0 = 11, 500 § If 0 is a reasonable value for x, then b 0 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 b 0 does not have an interpretation

TI-83 Instructions for Linear Reg • With diagnostics turned on and explanatory variable in L 1 and response variable in L 2 • Press STAT, highlight CALC and select 4: Lin. Reg (ax + b) and hit enter twice • Output: Lin. Reg y = ax + b a = xxx (slope or b 1 value) b = xxx (y-intercept or b 0 value) r² = xxx (coefficient of determination) r = xxx (correlation coefficient)

Example x 0 1 2 3 4 5 6 7 8 9 y 89. 2 86. 4 83. 5 81. 1 78. 2 73. 9 64. 3 71. 8 65. 6 66. 2 • Find the least-squares regression line y-hat = 88. 7327 – 2. 8273 x • What does the model predict for x = 5? y-hat = 74. 5962 • What is the residual from the above? residual = -0. 69623 • Is this model appropriate for x = 15? Why or why not? No. Out of model’s range

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

Homework Answers • 6 -- True