Predicted Price Yi= 317.85 ($1,000s) Confidence Interval Estimate for E(y)|xp37.12317.85)x(x)x(xn1styˆ22pεα/2The confidence interval endpoints are 280.66 -- 354.90, or from $280,660 -- $354,900
Chap 14-70 Estimation of Individual Values: Example Find the 95% confidence interval for an individual house with 2,000 square feet Predicted Price Yi= 317.85 ($1,000s) Prediction Interval Estimate for y|xp102.28317.85)x(x)x(xn11styˆ22pεα/2The prediction interval endpoints are 215.50 -- 420.07, or from $215,500 -- $420,070
Chap 14-71 Finding Confidence and Prediction Intervals PHStat In Excel, use PHStat | regression | simple linear regression …Check the “confidence and prediction interval for X=”box and enter the x-value and confidence level desired
Chap 14-72 Input values Finding Confidence and Prediction Intervals PHStat (continued) Confidence Interval Estimate for E(y)|xpPrediction Interval Estimate for y|xp
Chap 14-73 Residual Analysis Purposes Examine for linearity assumption Examine for constant variance for all levels of x Evaluate normal distribution assumption Graphical Analysis of Residuals Can plot residuals vs. x Can create histogram of residuals to check for normality
Chap 14-74 Residual Analysis for Linearity Not Linear Linear x residuals x y x y x residuals
Chap 14-75 Residual Analysis for Constant Variance Non-constant variance Constant variance x x y x x y residuals residuals
Chap 14-76 House Price Model Residual Plot-60-40-200204060800100020003000Square FeetResidualsExcel Output RESIDUAL OUTPUT Predicted House Price Residuals1 251.92316 -6.923162 2 273.87671 38.12329 3 284.85348 -5.853484 4 304.06284 3.937162 5 218.99284 -19.99284 6 268.38832 -49.38832 7 356.20251 48.79749 8 367.17929 -43.17929 9 254.6674 64.33264 10 284.85348 -29.85348
Chap 14-77 Leverage The leverage of a case measures its ability to move the regression model all by itself by just moving in the ydirection. For example, one high leverage case hid a strong relationship between the number of long guns found and the number of times false information was discovered in airport screening.
Chap 14-78 Leverage
Chap 14-79 Leverage
Chap 14-80 Leverage (cont.) Notice how the point marked with the x changes the regression slope quite a bit:
Chap 14-81 Leverage (cont.) The leverage of a particular case can take on any value between 0.0 and 1.0. The closer the leverage of a case is to 1.0, the more impact that case has on the regression line. Note that a point with zero leverage has no effect on the regression model other than being used in calculations for the model.
Chap 14-82 Leverage (cont.) A case can have leverage in two ways: It might be extraordinary in one or more individual variables. It might be extraordinary in a combination of variables. There are no tests to decide if the leverage of a case is too large. You can, however, make a histogram of the leverages and give special attention to those cases with leverage values that stand out.
Chap 14-83 Influential Points
Chap 14-84 Influential Points
Chap 14-85 Influential Cases An influential case is one that has bothhigh leverage
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