Lecture 10_MultReg-2012-1

Lecture 10_MultReg-2012-1 - Lecture 10 Multiple...

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1 Lecture 10: Multiple Regressions (Introduction) The Multiple Regression Model. Read Chap. 4.1 - Model description - Estimation of the coefficients - Interpretation of the coefficients in the multiple model vs. the simple regression model Inferences about the multiple regression coefficients - Chap 4.2 - Confidence intervals for the coefficients - Hypothesis tests for the coefficients Example : Gasoline Mileage for New Cars (2004)
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2 General Description In multiple regression analysis, we consider more than one independent variable x 1 ,…, x K . We are interested in the conditional mean of y given x 1 ,…, x k -- μ Y x 1 ,.., x K We assume this mean is a linear function of each variable. The coefficients are fit to the data using least-squares Thus, the coefficient b i for x i measures the (linear) effect of changing x i while holding all other variables constant. This is often described as the effect of variable x i after “controlling” for the other variables in the model. The analysis can also be used to predict the value of a new y at chosen values of x 1 ,…, x K .
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3 Example: Gas Mileage A team charged with designing a new automobile is concerned about the gas mileage that they should aim to achieve. They want to aim for the average value that their competitors would achieve for such a car. We’ll use Y = MPG-City . The new car is planned to have the following characteristics: horsepower – 225; weight – 4000 lbs; seating – 5 adults; length - 180 To estimate what the average competitor would achieve for such a vehicle they gathered data on all car models in 2004. The data includes information for each model about gas mileage and about the 4 variables: horsepower, weight, seating, & length. Here are some lines of the data:
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4 Selected Automobile Data Make/Model MPG City MPG Hwy HP WT (1000) Seats Length Width Displ Cyl Acura_RL 18 24 225 3.898 5 196.6 71.6 3.5 6 Acura_TL 20 28 270 3.575 5 189.3 72.2 3.2 6 Acura_TSX 23 32 200 3.318 5 183.3 69.4 2.4 4 Acura_RSX 25 34 160 2.771 4 172.2 67.9 2 4 Buick_Century 20 30 175 3.342 6 194.6 72.7 3.1 6 Buick_LeSabre 20 29 205 3.567 5 200 73.5 3.8 6 Chevrolet_Blazer 16 21 190 3.591 5 176.8 67.8 4.3 6 Chevrolet_Cavalier 24 34 140 2.676 5 180.9 67.9 2.2 4 Chevrolet_Corvette 18 25 350 3.214 2 179.7 73.6 5.7 8 There are 43 car Makes in the data, with a total of 242 Make/Model entries. We viewed 20 Make/Models as “not ordinary”, and will exclude them from future analyses. Thus our analysis only holds for “ordinary” car types.
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5 Answer s via Simple Regression We could use simple regression to derive 4 separate prediction equations. Each could give a predicted value corresponding to their respective design parameter, x . {See demonstration in Lecture.} The following table shows the coefficients of the 4 resulting linear prediction equations, and the corresponding Prediction of Y at each of the given values of our design parameters.
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This note was uploaded on 04/04/2012 for the course STAT 102 taught by Professor Shaman during the Spring '08 term at UPenn.

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Lecture 10_MultReg-2012-1 - Lecture 10 Multiple...

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