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STA291 Statistical Methods Lecture 29 Standard Errors for Mean Values Confidence Interval for the Mean Response Last time, we said we were modeling our line to infer the “line of means”—the expected value of our response variable for each given value of the explanatory variable. The confidence interval for the mean response, v, at a value xv, is: ˆyv tn*2 SE ˆ v where: 2 e s SEˆ v SE b1 xv x n 2 2 SE, and the confidence interval, becomes smaller with increasing n. SE, and the confidence interval, are larger for samples with more spread around the line (when se is larger). Standard Errors for Mean Values Confidence Interval for the Mean Response The confidence interval for the mean response, v, at a value xv, is: y ˆ t * SE ˆ v v n 2 where: SEˆ v 2 e s SE b1 xv x n 2 SE becomes larger the further xν gets from x . That is, the confidence interval broadens as you move away from x . (See figure at right.) 2 Standard Errors for Predicted Values Prediction Interval for an Individual Response Now, we tackle the more difficult (as far as additional variability) of predicting a single value at a value xv. When conditions are met, that interval is: where: * yˆv ± tn-2 ´ SE ( yˆv ) 2 s 2 2 e ˆ SE ( yv ) = SE ( b1 ) ´ ( xv - x ) + + se n 2 2 s Because of the extra term e, the confidence interval for individual values is broader that those for the predicted mean value. Difference Between Confidence and Prediction Intervals Confidence interval for a mean: yˆ v t * n2 2 e s SE b1 xv x n 2 2 The result ˆ 10.1 4.55 0.15 at 95% means: “We are 95% confident that the mean value of y is between 4.40 and 4.70 when x = 10.1.” Difference Between Confidence and Prediction Intervals Prediction interval for an individual value: 2 e s yˆ v t SE b1 xv x se2 n The result yˆ 10.1 4.55 0.60 at 95% means: * n2 2 2 “We are 95% confident that a single measurement of y will be between 3.95 and 5.15 when x = 10.1.” Using Confidence and Prediction Intervals Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity se = 17.95, and SE(b1) = 0.0051 Find the predicted Price of a 1000 GB hard drive. Find the 95% confidence interval for the mean Price of all 1000 GB hard drives. Find the 95% prediction interval for the Price of one 1000 GB hard drive. Using Confidence and Prediction Intervals Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity se = 17.95, and SE(b1) = 0.0051 Find the predicted Price of a 1000 GB hard drive. Price = 18.64 + 0.104(1000) = 122.64 Using Confidence and Prediction Intervals Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity se = 17.95, and SE(b1) = 0.0051 Find the 95% confidence interval for the mean Price of all 1000 GB hard drives. yˆ v t n* 2 2 s SE 2 b1 xv x e n 2 2 17 . 95 2 $122.64 2.571 0.00512 1000 1110 7 $122.64 $17.50 $105.14,$140.14 Using Confidence and Prediction Intervals Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity se = 17.95, and SE(b1) = 0.0051 Find the 95% prediction interval for the price of one 1000 GB hard drive. * yˆv ± tn-2 2 s SE 2 ( b1 ) ´ ( xv - x ) + e + se2 n 2 2 17.95 = $122.64 ± 2.571 0.00512 ´ (1000 -1110 ) + +17.952 7 = $122.64 ± $49.36 = [ 73.28,172.00 ]. CI: [ $105.14, $140.14] 2 Looking back o Construct and interpret a confidence interval for the mean value o Construct and interpret a prediction interval for an individual value