Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Chapter 12 Simple Linear Regression Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation for Estimation and Prediction Computer Solution Residual Analysis: Validating Model Assumptions Simple Linear Regression Model The equation that describes how y is related to x and an error term is called the regression model. The simple linear regression model is: y = b0 + b1x +e • b0 and b1 are called parameters of the model. • e is a random variable called the error term. Simple Linear Regression Equation The simple linear regression equation is: E(y) = b0 + b1x • Graph of the regression equation is a straight line. • b0 is the y intercept of the regression line. • b1 is the slope of the regression line. • E(y) is the expected value of y for a given x value. Simple Linear Regression Equation Positive Linear Relationship E(y) Regression line Intercept b0 Slope b1 is positive x Simple Linear Regression Equation Negative Linear Relationship E(y) Intercept b0 Regression line Slope b1 is negative x Simple Linear Regression Equation No Relationship E(y) Regression line Intercept b0 Slope b1 is 0 x Estimated Simple Linear Regression Equation The estimated simple linear regression equation is: ŷ b0 b1 x • The graph is called the estimated regression line. • b0 is the y intercept of the line. • b1 is the slope of the line. • ŷ is the estimated value of y for a given x value. Estimation Process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1 Sample Data: x y x1 y1 . . . . xn yn Estimated Regression Equation ŷ b0 b1 x Sample Statistics b0, b1 Least Squares Method Least Squares Criterion min (y i y i ) 2 where: yi = observed value of the dependent variable for the ith observation y^i = estimated value of the dependent variable for the ith observation The Least Squares Method Slope for the Estimated Regression Equation xi y i ( xi y i ) / n b1 2 2 xi ( xi ) / n where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation n = total number of observations The Least Squares Method y-Intercept for the Estimated Regression Equation b0 y b1 x where: _ x = mean value for independent variable _ y = mean value for dependent variable n = total number of observations Example: Reed Auto Sales Simple Linear Regression Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide. Example: Reed Auto Sales Simple Linear Regression Number of TV Ads 1 3 2 1 3 Number of Cars Sold 14 24 18 17 27 Example: Reed Auto Sales Slope for the Estimated Regression Equation b1 = 220 - (10)(100)/5 = 5 24 - (10)2/5 y-Intercept for the Estimated Regression Equation b0 = 20 - 5(2) = 10 Estimated Regression Equation y^ = 10 + 5x Example: Reed Auto Sales Scatter Diagram 30 25 Cars Sold 20 ^ = 10 + 5x y 15 10 5 0 0 1 2 TV Ads 3 4 The Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE 2 2 ^ )2 ( y i y ) ( y^i y ) ( y i y i where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error The Coefficient of Determination The coefficient of determination is: r2 = SSR/SST where: SST = total sum of squares SSR = sum of squares due to regression Example: Reed Auto Sales Coefficient of Determination r2 = SSR/SST = 100/114 = .8772 The regression relationship is very strong because 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold. The Correlation Coefficient Sample Correlation Coefficient rxy (sign of b1 ) Coefficien t of Determinat ion rxy (sign of b1 ) r 2 where: b1 = the slope of the estimated regression equation yˆ b0 b1 x Example: Reed Auto Sales Sample Correlation Coefficient rxy (sign of b1 ) r 2 The sign of b1 in the equation yˆ 10 5 x is “+”. rxy = + .8772 rxy = +.9366 Model Assumptions Assumptions About the Error Term e 1. The error e is a random variable with mean of zero. 2. The variance of e , denoted by 2, is the same for all values of the independent variable. 3. The values of e are independent. 4. The error e is a normally distributed random variable. Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero. Two tests are commonly used • t Test • F Test Both tests require an estimate of 2, the variance of e in the regression model. Testing for Significance An Estimate of 2 The mean square error (MSE) provides the estimate of 2, and the notation s2 is also used. s2 = MSE = SSE/(n-2) where: SSE (yi yˆi ) 2 ( yi b0 b1 xi ) 2 Testing for Significance An Estimate of • To estimate we take the square root of 2. • The resulting s is called the standard error of the estimate. SSE s MSE n2 Testing for Significance: t Test Hypotheses H 0 : b1 = 0 H a : b1 = 0 Test Statistic b1 t sb1 Testing for Significance: t Test Rejection Rule Reject H0 if t < -t or t > t where: t is based on a t distribution with n - 2 degrees of freedom Example: Reed Auto Sales t Test • Hypotheses H 0 : b1 = 0 H a : b1 = 0 • Rejection Rule For = .05 and d.f. = 3, t.025 = 3.182 Reject H0 if t > 3.182 Example: Reed Auto Sales t Test • Test Statistics t = 5/1.08 = 4.63 • Conclusions t = 4.63 > 3.182, so reject H0 Confidence Interval for b1 We can use a 95% confidence interval for b1 to test the hypotheses just used in the t test. H0 is rejected if the hypothesized value of b1 is not included in the confidence interval for b1. Confidence Interval for b1 The form of a confidence interval for b1 is: b1 t / 2 sb1 where b1 is the point estimate t / 2 sb1 is the margin of error t / 2 is the t value providing an area of /2 in the upper tail of a t distribution with n - 2 degrees of freedom Example: Reed Auto Sales Rejection Rule Reject H0 if 0 is not included in the confidence interval for b1. 95% Confidence Interval for b1 b1 t / 2 sb1 = 5 +/- 3.182(1.08) = 5 +/- 3.44 or 1.56 to 8.44 Conclusion 0 is not included in the confidence interval. Reject H0 Testing for Significance: F Test Hypotheses H 0 : b1 = 0 H a : b1 = 0 Test Statistic F = MSR/MSE Testing for Significance: F Test Rejection Rule Reject H0 if F > F where: F is based on an F distribution with 1 d.f. in the numerator and n - 2 d.f. in the denominator Example: Reed Auto Sales F Test • Hypotheses • Rejection Rule H 0 : b1 = 0 H a : b1 = 0 For = .05 and d.f. = 1, 3: F.05 = 10.13 Reject H0 if F > 10.13. Example: Reed Auto Sales F Test • Test Statistic F = MSR/MSE = 100/4.667 = 21.43 • Conclusion F = 21.43 > 10.13, so we reject H0. Some Cautions about the Interpretation of Significance Tests Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y. Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. Example: Reed Auto Sales Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: y^ = 10 + 5(3) = 25 cars