1 1 Slide The Simple Linear Regression Model n Simple Linear Regression Model y =  0 +  1 x +  n Simple Linear Regression Equation E( y ) =  0 +  1 x n Estimated Simple Linear Regression Equation y = b 0 + b 1 x ^

2 2 Slide Least Squares Method n Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation for the i th observation y i = estimated value of the dependent variable for the i th observation for the i th observation ^

3 3 Slide The Coefficient of Determination n Relationship Among SST, SSR, SSE SST = SSR + SSE n Coefficient of Determination r 2 = SSR/SST where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error ^^

4 4 Slide The Correlation Coefficient n Sample Correlation Coefficient where: b 1 = the slope of the estimated regression b 1 = the slope of the estimated regressionequation

5 5 Slide Model Assumptions Assumptions About the Error Term  Assumptions About the Error Term  The error  is a random variable with mean of zero. The error  is a random variable with mean of zero. The variance of , denoted by  2, is the same for all values of the independent variable. The variance of , denoted by  2, is the same for all values of the independent variable. The values of  are independent. The values of  are independent. The error  is a normally distributed random variable. The error  is a normally distributed random variable.

6 6 Slide Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of  1 is zero. To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of  1 is zero. n Two tests are commonly used t Test t Test F Test F Test Both tests require an estimate of  2, the variance of  in the regression model. Both tests require an estimate of  2, the variance of  in the regression model.

7 7 Slide Testing for Significance An Estimate of  2 An Estimate of  2 The mean square error (MSE) provides the estimate of  2, and the notation s 2 is also used. s 2 = MSE = SSE/(n-2) s 2 = MSE = SSE/(n-2)where:

8 8 Slide Testing for Significance An Estimate of  An Estimate of  To estimate  we take the square root of  2. To estimate  we take the square root of  2. The resulting s is called the standard error of the estimate. The resulting s is called the standard error of the estimate.

9 9 Slide n Hypotheses H 0 :  1 = 0 H 0 :  1 = 0 H a :  1 = 0 H a :  1 = 0 n Test Statistic n Rejection Rule Reject H 0 if t t  where t  is based on a t distribution with where t  is based on a t distribution with n - 2 degrees of freedom. n - 2 degrees of freedom. Testing for Significance: t Test

10 Slide Testing for Significance: F Test n Hypotheses H 0 :  1 = 0 H 0 :  1 = 0 H a :  1 = 0 H a :  1 = 0 n Test Statistic F = MSR/MSE n Rejection Rule Reject H 0 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.

11 Slide n Confidence Interval Estimate of E ( y p ) n Prediction Interval Estimate of y p y p + t  /2 s ind y p + t  /2 s ind where the confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 d.f. Using the Estimated Regression Equation for Estimation and Prediction

12 Slide n Residual for Observation i y i – y i y i – y i n Standardized Residual for Observation i where: Residual Analysis ^^^ ^