11-5 CONFIDENCE INTERVALS 421 11-5 CONFIDENCE INTERVALS 11-5.1 Confidence Intervals on the Slope and Intercept In addition to point estimates of the slope and intercept, it is possible to obtain confidence interval estimates of these parameters. The width of these confidence intervals is a measure of the overall quality of the regression line. If the error terms, i , in the regression model are normally and independently distributed, 1 ^ 1 1 2 2 ^ 2 S x x and 1 ^ 0 0 2 B ^ 2 c 1 n x 2 d S x x are both distributed as t random variables with n 2 degrees of freedom. This leads to the following definition of 100(1 )% confidence intervals on the slope and intercept. Confidence Intervals on Parameters Under the assumption that the observations are normally and independently distributed, a 100(1 )% confidence interval on the slope 1 in simple linear regression is ^ 1 t 2, n 2 B S x x ^ 2 1 ^ 1 t 2, n 2 B S x x ^ 2 (11-29) is Similarly, a 100(1 ^ 0 t 2, n 2 B )% confidence interval on the intercept 1 n x 2 d S x x 0 0 ^ 2 c ^ 0 t 2, n 2 B ^ 2 c 1 n x 2 d S x x (11-30) EXAMPLE 11-4 Oxygen Purity Confidence Interval on the Slope This simplifies to 12.181 1 We will find a 95% confidence interval on the slope of the re- gression line using the data in Example 11-1. Recall that ^ 14.947, S xx 0.68088, and ^ 2 1.18 (see Table 11-2). 1 Then, from Equation 11-29 we find ^ 1 t 0.025,18 B S xx ^ 2 1 ^ 1 t 0.025,18 or 14.947 2.101 2.101 1.18 A 0.68088 1.18 A 0.68088 B S xx ^ 2 17.713 Practical Interpretation: This CI does not include zero, so there is strong evidence (at 0.05) that the slope is not zero. The CI is reasonably narrow ( 2.766) because the error vari- ance is fairly small. 1 14.947 11-5.2 Confidence Interval on the Mean Response A confidence interval may be constructed on the mean response at a specified value of x, say, x 0 . This is a confidence interval about E(Y x 0 ) Y x 0 and is often called a confidence interval