14.2 The Regression Equation

Transcription

1 14.2 The Regression Equation Tom Lewis Fall Term 2009 Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Outline 1 Exact and inexact linear relationships 2 Fitting lines to data 3 Formulas 4 Technology Tom Lewis () 14.2 The Regression Equation Fall Term / 12

2 Exact and inexact linear relationships Exact linear relationships A flooring crew charges $50.00 plus $3.50 per square foot of hardwood flooring installed. Let x denote the square feet of floor to be installed. This variable is called the independent or explanatory variable. Let y denote the cost of installing a floor. This variable is called the dependent or response variable. In our example, y = 3.5x + 50 and the response variable is completely predicted from the explanatory variable. In other words, x gives a complete explanation for y. Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Exact and inexact linear relationships Inexact linear relationships The length of the ulna and the height of 17 statistics students were measured in centimeters; see the data set UlnaHeight.txt. Can we explain the height of an individual in response to the length of their ulna? Let x denote the length of the ulna of the subject, the explanatory variable. Let y denote the height of the subject, the response variable. We might expect that y b 1 x + b 0, for some parameters b 1 (slope) and b 0 (intercept), but we would be very surprised if this relationship were exact! In reality, the relationship is certainly more complex, such as y = b 1 x + b 0 }{{} best linear approximation + e }{{} other factors Tom Lewis () 14.2 The Regression Equation Fall Term / 12

3 Fitting lines to data Problem The purpose of this problem is to introduce you to the sum of squares error (abbreviated by SSE). Consider the data set: (1, 2), (3, 5), and (4, 8). Estimate the line of best fit for the data set. Draw a line that does not fit the data very well and compute the SSE. Estimate the line of best fit for the data set. Draw a line that does fit the data set and compute the SSE. Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Fitting lines to data Fitting lines In general, we will not be able to exactly fit a line ŷ = b 1 x + b 0 to a set of data; there will be some error. For a particular value of the explanatory variable x, let y = the true value of the response variable ŷ = the value of y predicted from the linear model, b 1 x + b 0 The difference between these values is called the error in the approximation: e = ŷ y e 2 is the squared error. Tom Lewis () 14.2 The Regression Equation Fall Term / 12

4 Fitting lines to data Least Squares Criterion The straight line that best fits a set of data points is the one having the smallest possible sum of squared errors. Regression Line and Regression Equation Regression Line: The straight line that best fits a set of data points according to the least squares criterion. Regression Equation: The equation of the regression line. Problem Consider the data set: (1, 2), (3, 5), and (4, 8). Find the regression equation. Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Fitting lines to data The regression line Figure: The regression line for the data set Tom Lewis () 14.2 The Regression Equation Fall Term / 12

5 Formulas Regression formulas Given a set of n ordered pairs (x 1, y 1 ),..., (x n, y n ), let S xx = (x i x) 2 = ( xi 2 xi ) 2 n S yy = (y i y) 2 = ( yi 2 yi ) 2 n ( i x i) ( ) y i S xy = (x i x)(y i y) = i x i y i n The regression equation The regression equation for a set of n data points is ŷ = b 1 x + b 0, where b 1 = S xy S xx and b 0 = y b 1 x Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Formulas Problem Use the special formulas to find the regression equation for the data set (1, 2), (3, 5), and (4, 8). Tom Lewis () 14.2 The Regression Equation Fall Term / 12

6 Technology Using R Commander To view your data set and the regression line, use Fill in the dialog box as desired. Graphs Scatterplot... To find the regression equation and more, use Statistics Fit Models Linear regression... Fill in the dialog box as desired. The intercept, b 0, and the slope, b 1, will be found in the column headed by Estimate. Tom Lewis () 14.2 The Regression Equation Fall Term / 12 Technology An application This next story is from Chromatography.html. Results of a study of gas chromatography, a technique which is used to detect very small amounts of a substance. Five measurements were taken for each of four specimens containing different amounts of the substance. The amount of the substance in each specimen was determined before the experiment. The response variable is the output reading from the gas chromatograph. The purpose of the study is to calibrate the chromatograph by relating the actual amount of the substance to the chromatograph reading. Use R Commander to find the regression equation for the data in chromatograph.txt. Tom Lewis () 14.2 The Regression Equation Fall Term / 12