Statistics for Business and Economics 7th Edition Chapter 11 Simple Regression Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-1 Chapter Goals After completing this chapter, you should be able to: Explain the simple linear regression model Obtain and interpret the simple linear regression equation for a set of data Describe R2 as a measure of explanatory power of the regression model Understand the assumptions behind regression analysis Explain measures of variation and determine whether the independent variable is significant Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-2 Chapter Goals (continued) After completing this chapter, you should be able to: Calculate and interpret confidence intervals for the regression coefficients Use a regression equation for prediction Form forecast intervals around an estimated Y value for a given X Use graphical analysis to recognize potential problems in regression analysis Explain the correlation coefficient and perform a hypothesis test for zero population correlation Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-3 11.1 Overview of Linear Models An equation can be fit to show the best linear relationship between two variables: Y = β0 + β1X Where Y is the dependent variable and X is the independent variable β0 is the Y-intercept β1 is the slope Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-4 Least Squares Regression Estimates for coefficients β0 and β1 are found using a Least Squares Regression technique The least-squares regression line, based on sample data, is yˆ b0 b1x Where b1 is the slope of the line and b0 is the yintercept: Cov(x, y) b1 s2x Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall b y b1x Ch 11-5 Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain (also called the endogenous variable) Independent variable: the variable used to explain the dependent variable (also called the exogenous variable) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-6 11.2 Linear Regression Model The relationship between X and Y is described by a linear function Changes in Y are assumed to be caused by changes in X Linear regression population equation model Yi β0 β1x i ε i Where 0 and 1 are the population model coefficients and is a random error term Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-7 Simple Linear Regression Model The population regression model: Population Y intercept Dependent Variable Population Slope Coefficient Independent Variable Random Error term Yi β0 β1Xi ε i Linear component Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Random Error component Ch 11-8 Simple Linear Regression Model (continued) Y Yi β0 β1Xi ε i Observed Value of Y for Xi εi Predicted Value of Y for Xi Slope = β1 Random Error for this Xi value Intercept = β0 Xi Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall X Ch 11-9 Simple Linear Regression Equation The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted) y value for observation i Estimate of the regression Estimate of the regression slope intercept yˆ i b0 b1x i Value of x for observation i The individual random error terms ei have a mean of zero ei ( y i - yˆ i ) y i - (b0 b1x i ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-10 F-Test for Significance (continued) Test Statistic: H0 : β = MSR F 11.08 MSE H1 : β ≠ = 05 df1= df2 = Decision: Reject H0 at = 0.05 Critical Value: F = 5.32 Conclusion: = 05 Do not reject H0 Reject H0 F There is sufficient evidence that house size affects selling price F.05 = 5.32 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-50 11.6 Prediction The regression equation can be used to predict a value for y, given a particular x For a specified value, xn+1 , the predicted value is yˆ n1 b0 b1x n1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-51 Predictions Using Regression Analysis Predict the price for a house with 2000 square feet: house price 98.25 0.1098 (sq.ft.) 98.25 0.1098(200 0) 317.85 The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-52 Relevant Data Range When using a regression model for prediction, only predict within the relevant range of data Relevant data range Risky to try to extrapolate far beyond the range of observed X’s Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-53 Estimating Mean Values and Predicting Individual Values Goal: Form intervals around y to express uncertainty about the value of y for a given xi Confidence Interval for the expected value of y, given xi Y y y = b0+b1xi Prediction Interval for an single observed y, given xi Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall xi X Ch 11-54 Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular xi Confidence interval for E(Yn1 | Xn1 ) : yˆ n1 t n 2,α/2se (x n1 x)2 2 n (x i x) Notice that the formula involves the term (x n1 x) so the size of interval varies according to the distance xn+1 is from the mean, x Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-55 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular xi Confidence interval for yˆ n1 : yˆ n1 t n 2,α/2 se (x n1 x)2 1 2 n (x i x) This extra term adds to the interval width to reflect the added uncertainty for an individual case Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-56 Estimation of Mean Values: Example Confidence Interval Estimate for E(Yn+1|Xn+1) Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price yi = 317.85 ($1,000s) yˆ n1 t n-2,α/2 se (x i x)2 317.85 37.12 n (xi x) The confidence interval endpoints are 280.66 and 354.90, or from $280,660 to $354,900 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-57 Estimation of Individual Values: Example Confidence Interval Estimate for yn+1 Find the 95% confidence interval for an individual house with 2,000 square feet Predicted Price yi = 317.85 ($1,000s) yˆ n1 t n-1,α/2se (Xi X)2 1 317.85 102.28 n (Xi X) The confidence interval endpoints are 215.50 and 420.07, or from $215,500 to $420,070 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-58 11.7 Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Correlation was first presented in Chapter Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-59 Correlation Analysis The population correlation coefficient is denoted ρ (the Greek letter rho) The sample correlation coefficient is r s xy sxsy where s xy (x x)(y y) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall i i n Ch 11-60 Hypothesis Test for Correlation To test the null hypothesis of no linear association, H0 : ρ 0 the test statistic follows the Student’s t distribution with (n – ) degrees of freedom: t r (n 2) (1 r ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-61 Decision Rules Hypothesis Test for Correlation Lower-tail test: Upper-tail test: Two-tail test: H0: ρ H1: ρ < H0: ρ ≤ H1: ρ > H0: ρ = H1: ρ ≠ -t t Reject H0 if t < -tn-2, Where t Reject H0 if t > tn-2, r (n 2) (1 r ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall /2 -t/2 /2 t/2 Reject H0 if t < -tn-2, or t > tn-2, has n - d.f Ch 11-62 11.9 Graphical Analysis The linear regression model is based on minimizing the sum of squared errors If outliers exist, their potentially large squared errors may have a strong influence on the fitted regression line Be sure to examine your data graphically for outliers and extreme points Decide, based on your model and logic, whether the extreme points should remain or be removed Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-63 Chapter Summary Introduced the linear regression model Reviewed correlation and the assumptions of linear regression Discussed estimating the simple linear regression coefficients Described measures of variation Described inference about the slope Addressed estimation of mean values and prediction of individual values Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11-64 ... Publishing as Prentice Hall Ch 11- 2 Chapter Goals (continued) After completing this chapter, you should be able to: Calculate and interpret confidence intervals for the regression coefficients... intervals for the regression coefficients Use a regression equation for prediction Form forecast intervals around an estimated Y value for a given X Use graphical analysis to recognize potential... correlation coefficient and perform a hypothesis test for zero population correlation Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 11- 3 11. 1 Overview of Linear Models