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## Session 4

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**Outline for Session 4**• Summary Measures for the Full Model • Top Section of the Output • Interval Estimation • More Multiple Regression • Movers • Nonlinear Regression • Insurance Applied Regression -- Prof. Juran**Top Section: Summary Statistics**Applied Regression -- Prof. Juran**Top Section: Summary Statistics**Applied Regression -- Prof. Juran**As stated earlier R2 is closely related to the correlation**between X and Y, indeed Furthermore, R2 , andthus rX,Y , is closely related to the slope of the regression line via Thus, testing the significance of the slope, testing the significance of R2 and testing the significance of rX,Y are essentially equivalent. Applied Regression -- Prof. Juran**Another Measure of Variability Reduction**Statisticians think in terms of variances, but in business applications the standard deviation is often more meaningful. First recall that Thus, approximately, Applied Regression -- Prof. Juran**Interval Estimation**Applied Regression -- Prof. Juran**An Image of the Residuals**Y yi (xi , yi) X xi (xi , yi) The observed values: The fitted values: The residuals: Recall: The regression line passes through the data so that the sum of squared residuals is as small as possible. Applied Regression -- Prof. Juran**Regression and Prediction**Regression lines are frequently used for predicting future values of Y given future, conjectural or speculative values of X. Suppose we posit a future value of X, say x0. The predicted value, , is Applied Regression -- Prof. Juran**Under our assumptions this is an unbiased estimate of Y**given that x=x0 ,regardless of the value of x0. Let 0 = E(Y(x0)) and thus, since the estimate is unbiased, 0 = b0 + b1x0. However, be alert to the fact that this estimate (prediction) of a future value has a standard error of Furthermore, the standard error of the prediction of the expected (mean) value of Y given x = x0 is Applied Regression -- Prof. Juran**From these facts it follows that a 2-sided “confidence”**interval on the expected value of Y given x= x0, 0, is given by Applied Regression -- Prof. Juran**A 2-sided “prediction”interval on future individual**values of Y given x = x0, y0, is given by Applied Regression -- Prof. Juran**Confidence Interval on E(Y(x0))**Prediction Interval on Y(x0) Applied Regression -- Prof. Juran**Note that both of these intervals are parabolic functions in**x0, have their minimum interval width at x0 = , and their widths depend on and on Sxx The sum of squared x term appears so often in regression equations that it is useful to use the abbreviation Sxx. Note that Sxx can easily be obtained from the variance as computed in most spreadsheets or statistics packages. Applied Regression -- Prof. Juran**An Image of the Prediction and Confidence Intervals**Applied Regression -- Prof. Juran**All-Around Movers**The management question here is whether historical data can be used to create a cost estimation model for intra-Manhattan apartment moves. The dependent variable is the number of labor hours used, which is a proxy for total cost in the moving business. There are two potential independent variables: volume (in cubic feet) and the number of rooms in the apartment being vacated. Applied Regression -- Prof. Juran**Summary Statistics**Applied Regression -- Prof. Juran**The Most Obvious Simple Regression**Applied Regression -- Prof. Juran**An Alternative Simple Regression Model**Applied Regression -- Prof. Juran**A Multiple Regression Model**Applied Regression -- Prof. Juran**Preliminary Observations**• Volume is the best single predictor, but perhaps not useful if customers are to be expected to collect these data and enter them on a web site. • Rooms is a pretty good predictor (not as good as Volume), and may be more useful on a practical basis. Applied Regression -- Prof. Juran**Preliminary Observations**• The multiple regression model makes better predictions, but not much better than either of the simple regression models. • The multiple regression model has problems with multicollinearity. Notice the lack of significance for the Rooms variable (and the strange coefficient). Applied Regression -- Prof. Juran**, corresponding to the estimated number of hours for one**Prediction intervals specific move, given one specific value for the number of rooms. , corresponding to the estimated population average Confidence intervals number of hours over a large number of moves, all with the same number of rooms. Applied Regression -- Prof. Juran**Validity of the Rooms Model**Applied Regression -- Prof. Juran**Analysis of the Residuals**Applied Regression -- Prof. Juran**Comments on the Rooms Model**• Good explanatory power • Statistically Significant • Points fit the line well But… • Small apartments tend to be over-estimated • Large apartments tend to be badly estimated, especially on the high side • Maybe could use more data • Maybe nonlinear Applied Regression -- Prof. Juran**=**B Note: If , then (A) = B. ln e A A Non-linear Model? Applied Regression -- Prof. Juran