Presentation on theme: "Chapter 9: Simple Regression Continued Hypothesis Testing and Confidence Intervals."— Presentation transcript:
Chapter 9: Simple Regression Continued Hypothesis Testing and Confidence Intervals
Inferences on regression coefficients To place confidence intervals or test hypotheses on and we need to know 2, and 2 which will be estimated by s a 2 and s b 2. Where 2 is estimated by:
Inferences on regression coefficients. If the model is correct, then b/s b and a/s a are distributed as a t distribution with n-2 degrees of freedom. Confidence limits on are: Confidence limits on are:
Hypothesis Testing: Test on Ho: = o Ha: ≠ o Test Statistic Reject Ho if
Hypothesis testing: Test on Ho: = o Ha: ≠ o Test Statistic Reject Ho if
Hypothesis Testing: Significance of regression equation Ho: = 0 (equivalent to Ho: r = 0) Ha: ≠ 0 Test statistic and rejection region same as previous test on . If this hypothesis is not rejected then may be estimated by If r = 0 then s 2 ≈ s y 2 or the regression line does not explain a significant amount of the variation in Y.
Confidence Intervals on the Regression Line Determined by first calculating the variance of,the predicted mean of for a given X k. The standard error of can be estimated by calculated as:
Confidence Intervals on the regression line The variance of depends on the value of X at which the variance is being determined. Var ( ) is a minimum where X k = and increases as X k deviates from. Confidence limits on the regression line are:
Confidence Intervals on the regression line Since increases as X k - increases, the confidence intervals on X k = are at their narrowest and widen as X k deviates from. Confidence limits on an individual predicted value of Y would be wider than the confidence interval on the regression line, since for an individual Y, the Var ( ) or 2 would have to be added to the Var( ). Thus the variance of an individual predicted value of Y would be Var( + 2 ).
Confidence intervals on an individual predicted value of Y Can be calculated by the previous confidence interval equations where would be substituted for
Confidence Intervals on the standard error Can be made by noting that (n-2) 2 / 2 is distributed as a chi-squared distribution with n-2 degrees of freedom. Limits are given by: where