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Chapter 9: Simple Regression Continued Hypothesis Testing and Confidence Intervals.

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Presentation on theme: "Chapter 9: Simple Regression Continued Hypothesis Testing and Confidence Intervals."— Presentation transcript:

1 Chapter 9: Simple Regression Continued Hypothesis Testing and Confidence Intervals

2 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:

3 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:

4 Hypothesis Testing: Test on  Ho:  =  o Ha:  ≠  o Test Statistic Reject Ho if

5 Hypothesis testing: Test on  Ho:  =  o Ha:  ≠  o Test Statistic Reject Ho if

6 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.

7 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:

8 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:

9 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 ).

10 Confidence intervals on an individual predicted value of Y Can be calculated by the previous confidence interval equations where would be substituted for

11 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


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