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The Simple Linear Regression Model

The Simple Linear Regression Model. Simple Linear Regression Model y =  0 +  1 x +  Simple Linear Regression Equation E( y ) =  0 +  1 x Estimated Simple Linear Regression Equation y = b 0 + b 1 x. ^. Least Squares Method. Least Squares Criterion where:

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The Simple Linear Regression Model

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  1. The Simple Linear Regression Model • Simple Linear Regression Model y = 0 + 1x+  • Simple Linear Regression Equation E(y) = 0 + 1x • Estimated Simple Linear Regression Equation y = b0 + b1x ^

  2. Least Squares Method • Least Squares Criterion where: yi = observed value of the dependent variable for the ith observation yi = estimated value of the dependent variable for the ith observation ^

  3. ^ ^ The Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE • 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

  4. The Correlation Coefficient • Sample Correlation Coefficient where: b1 = the slope of the estimated regression equation

  5. Model Assumptions • Assumptions About the Error Term  • 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 values of  are independent. • The error  is a normally distributed random variable.

  6. 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 s2, the variance of e in the regression model.

  7. Testing for Significance • An Estimate of s2 The mean square error (MSE) provides the estimate of s2, and the notation s2 is also used. s2 = MSE = SSE/(n-2) where:

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

  9. Testing for Significance: t Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Test Statistic • Rejection Rule Reject H0 if t < -tor t > t where tis based on a t distribution with n - 2 degrees of freedom.

  10. Testing for Significance: F Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Test Statistic F = MSR/MSE • 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.

  11. Using the Estimated Regression Equationfor Estimation and Prediction • Confidence Interval Estimate of E(yp) • Prediction Interval Estimate of yp yp+t/2 sind where the confidence coefficient is 1 -  and t/2 is based on a t distribution with n - 2 d.f.

  12. ^ Residual Analysis • Residual for Observation i yi – yi • Standardized Residual for Observation i where: ^ ^ ^

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