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14-2 Confidence intervals for a mean response are intervals constructed about the predicted value of y, at a given level of x, that are used to measure the accuracy of the mean response of all the individuals in the population. Prediction intervals for an individual response are intervals constructed about the predicted value of y that are used to measure the accuracy of a single individual’s predicted value.

14-3 Confidence Interval for the Mean Response of y A (1-  )  100% confidence interval for, the mean response of y for a specified value of x, is given by Lower bound: Upper bound: where x* is the given value of the explanatory variable, n is the number of observations, and t  /2 is the critical value with n-2 degrees of freedom.

14-4 Parallel Example 1: Constructing a Confidence Interval for a Mean Response Construct a 95% confidence interval about the predicted mean time to drill 5 feet for all drillings started at a depth of 110 feet.

14-5 Solution The least squares regression line is. To find the predicted mean time to drill 5 feet for all drillings started at 110 feet, let x * =110 in the regression equation and obtain. Recall: s e =0.5197 t 0.025 =2.228 for 10 degrees of freedom

14-6 Solution Therefore, Lower bound: Upper bound:

14-7 Solution We are 95% confident that the mean time to drill 5 feet for all drillings started at a depth of 110 feet is between 6.45 and 7.15 minutes.

14-8 Prediction Interval for an Individual Response about. A (1-  )  100% prediction interval for the individual response of y, is given by Lower bound: Upper bound: where x* is the given value of the explanatory variable, n is the number of observations, and t  /2 is the critical value with n-2 degrees of freedom.

14-9 Parallel Example 2: Constructing a Prediction Interval for an Individual Response Construct a 95% prediction interval about the predicted time to drill 5 feet for a single drilling started at a depth of 110 feet.

14-10 Solution The least squares regression line is. To find the predicted mean time to drill 5 feet for all drillings started at 110 feet, let x * =110 in the regression equation and obtain. Recall: s e =0.5197 t 0.025 =2.228 for 10 degrees of freedom

14-11 Solution Therefore, Lower bound: Upper bound:

14-12 Solution We are 95% confident that the time to drill 5 feet for a random drilling started at a depth of 110 feet is between 5.59 and 8.01 minutes.

14-13 Model effectiveness Our regression model can be considered effective for producing confidence interval estimates if: Our regression model can be considered effective for producing prediction interval estimates if: This will ensure that the interval estimates do not become too wide to be meaningful to us.