MGMT305-Lec9

MGMT305-Lec9 - Chapter 14 Simple Linear Regression s s s s...

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Unformatted text preview: Chapter 14 Simple Linear Regression s s s s s s s s 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 Slide 1 POLAROID CORPORATION s Polaroid uses Regression Analysis to produce films with the performance levels its customers desire: ˆ y = −19.8 − 7.6 x where ˆ y = change in film speed x = film age in months s Average decrease in film speed = Slide 2 The Simple Linear Regression Model s Simple Linear Regression Model y = β 0 + β 1x + ε Simple Linear Regression Equation E(y) = β 0 + β 1x Estimated Simple Linear Regression Equation ^ y = b0 + b1x s s Slide 3 Least Squares Method s Least Squares Criterion m in ∑ ( y i − y i ) 2 where: yi = observed value of the dependent variable for the ith observation ^ yi = estimated value of the dependent variable for the ith observation Slide 4 The Least Squares Method s Slope for the Estimated Regression Equation b1 ∑x y − (∑x ∑ y ) / n = ∑( x − x )( y − y ) = ∑x − (∑x ) / n ∑( x − x ) i i i i i i 2 i 2 2 i i y­Intercept for the Estimated Regression Equation _ _ b0 = y ­ b1x where: xi = value of independent variable for ith observation y_ = value of dependent variable for ith observation i x = mean value for independent variable _ y = mean value for dependent variable n = total number of observations s Slide 5 Example: Reed Auto Sales s Simple Linear Regression Reed Auto periodically has a special week­long 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 below. Number of TV Ads Number of Cars Sold 1 14 3 24 2 18 1 17 3 27 Slide 6 Example: Reed Auto Sales s Slope for the Estimated Regression Equation s y­Intercept for the Estimated Regression Equation Estimated Regression Equation s Slide 7 Example: Reed Auto Sales s Scatter Diagram 30 25 Cars Sold 20 15 10 5 0 0 1 2 TV Ads 3 4 Slide 8 The Coefficient of Determination s Relationship Among SST, SSR, SSE SST = SSR + SSE 2 2 ^ ^2 ∑(y i − y ) = ∑(y i − y ) + ∑(y i − y i ) s Coefficient of Determination r2 = SSR/SST where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error Slide 9 Example: Reed Auto Sales s Coefficient of Determination The regression relationship is strong/moderate/weak since 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. Slide 10 The Correlation Coefficient s Sample Correlation Coefficient rxy = (sign of b1 ) Coefficient of Determination rxy = (sign of b1 ) r 2 where: b1 = the slope of the estimated regression ˆ equation y = b0 + b1 x Slide 11 Example: Reed Auto Sales s Sample Correlation Coefficient rxy = (sign of b1 ) r 2 Slide 12 Model Assumptions s Assumptions About the Error Term ε • The error ε is a random variable with mean of zero. • The variance of ε , denoted by σ 2, is the same for all values of the independent variable. are independent. • The values of ε are independent. • The error ε is a normally distributed random variable. Slide 13 Testing for Significance s To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of β 1 is zero. Two tests are commonly used • t Test • F Test Both tests require an estimate of σ 2, the variance of ε in the regression model. s s Slide 14 Testing for Significance s An Estimate of σ 2 The mean square error (MSE) provides the estimate of σ 2, and the notation s2 is also used. s2 = MSE = SSE/(n­2) where:SSE = ˆ ( yi − yi ) 2 = ∑ ( yi − b0 − b1 xi ) 2 ∑ Slide 15 Testing for Significance s An Estimate of σ • To estimate σ we take the square root of σ 2. • The resulting s is called the standard error of the estimate. SSE s = MSE = n−2 Q. Find s for Reed Auto Sales example. s Slide 16 Testing for Significance: t Test s Hypotheses H0: β 1 = 0 Ha: β 1 = 0 s s Test Statistic b1 where, t= sb = sb1 1 s Rejection Rule Reject H0 if t < ­tα/2 or t > tα/2 or where tα/2 is based on a t distribution with ∑ (x − x ) i 2 • n ­ 2 degrees of freedom. Or, Reject Ho if p­value < alpha. Slide 17 Example: Reed Auto Sales s t Test • Hypotheses • • • Rejection Rule Test Statistic Conclusion Slide 18 Confidence Interval for β 1 s s We can use a 95% confidence interval for β 1 to test the hypotheses just used in the t test. H0 is rejected if the hypothesized value of β 1 is not included in the confidence interval for β 1. Slide 19 Confidence Interval for β 1 s The form of a confidence interval for β 1 is: b1 ± tα / 2 sb1 where tα / 2 sb1 is the margin of error tα / 2 is the t value providing an area of α /2 in the upper tail of a t distribution with n ­ 2 degrees of freedom b1 is the point estimate Slide 20 Example: Reed Auto Sales s Rejection Rule s 95% Confidence Interval for β 1 s Conclusion Slide 21 Testing for Significance: F Test s Hypotheses Test Statistic Rejection Rule H0: β 1 = 0 Ha: β 1 = 0 F = MSR/MSE Reject H0 if F > Fα s s where Fα is based on an F distribution with 1 d.f. in the numerator and n ­ 2 d.f. in the denominator. Or, Reject Ho if p­value < alpha. Slide 22 Example: Reed Auto Sales s F Test • Hypotheses • • • Rejection Rule Test Statistic Conclusion Slide 23 Using the Estimated Regression Equation for Estimation and Prediction s Confidence Interval Estimate of E(y ): ˆ y p ± tα / 2 s y p ˆ p ˆ y p =b0 +b1 x p , syp =s ˆ ( x p −x )2 1 + , 2 n ∑ xi − x ) ( s = MSE s Prediction Interval Estimate of y : ˆ y p ± tα / 2 sind sind ( x p − x )2 1 = s 1+ + n ∑ ( xi − x ) 2 p where the confidence coefficient is 1 ­ α and Slide 24 Two Important Questions s Q1. Which interval is wider, CI or PI? Why? s xp Q2. At what value of both the intervals do have the smallest width? Slide 25 Example: Reed Auto Sales s Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: Confidence Interval for E(y ) p s 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: Prediction Interval for y p s 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: Slide 26 Residual Analysis s Purpose: Validating model assumptions. Residual for Observation i ^ yi – yi Standardized Residual for Observation i s s ^ yi − yi s yi −^yi where: ^ sy i − y i = s 1 − hi ( xi − x ) 2 1 hi = + , 2 n ∑ ( xi − x ) s = MSE Slide 27 Example: Reed Auto Sales s Residuals Observation 1 2 3 4 5 Predicted Cars Sold 15 25 20 15 25 Residuals -1 -1 -2 2 2 Slide 28 Example: Reed Auto Sales s Residual Plot 3 2 TV Ads Residual Plot Residuals 1 0 -1 -2 -3 0 1 2 3 4 TV Ads Slide 29 ...
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This note was uploaded on 12/04/2009 for the course MGMT 305 taught by Professor Priya during the Summer '08 term at Purdue.

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