Confidence intervals for the The variance about the line is regression slope residua The slope of the true regression line is usually the most important parameter in a 382(-120+313++5132 regression problem. The slope is the rate of change of the mean response as the explanatory variable increases. We often =-(110233)=30620 want to estimate B. The slope b of the least-squares line is an unbiased estimator of B. a confidence interval is more useful estimate b is likely to be Finally, the standard error about the line The confidence interval for B has the familiar form s=√30620=17.50 estimate± t sEet The standard error s about the line is the Because b is our estimate the confidence key measure of the variability of the interval becomes responses in regression. It is part of the andard error of all the statistics we will b±xSE use for inference21 41 • The variance about the line is: 2 2 22 2 1 residual 2 1 ( 19.20) ( 31.13) ... 51.32 38-2 1 (11023.3) 306.20 36 s n = − = − +− + + ⎡ ⎤ ⎣ ⎦ = = ∑ 42 • Finally, the standard error about the line is: s = 306.20=17.50 The standard error s about the line is the key measure of the variability of the responses in regression. It is part of the standard error of all the statistics we will use for inference. 22 43 Confidence intervals for the regression slope • The slope of the true regression line is usually the most important parameter in a regression problem. The slope is the rate of change of the mean response as the explanatory variable increases. We often want to estimate . The slope b of the least-squares line is an unbiased estimator of . A confidence interval is more useful because it shows how accurate the estimate b is likely to be. β β β 44 • The confidence interval for has the familiar form Because b is our estimate, the confidence interval becomes Here are the details: β estimate SE ± × estimate t SEb b t ± ×