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1 BA 275 Quantitative Business Methods Quiz #6 Multiple Linear Regression Adjusted R-squared Prediction Agenda

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2 Simple Linear Regression Model population sample True effect of X on Y Estimated effect of X on Y Key questions: 1. Does X have any effect on Y? 2. If yes, how large is the effect? 3. Given X, what is the estimated Y?

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3 Key Q1: Does X have any effect on Y? Key Q2: How large is the effect? Key Q3: Predict Y for a given X. SE b1 SE b0 b1b1 b0b0

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4 Prediction and Confidence Intervals Confidence interval Prediction interval

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5 Model Comparison: A Good Fit? SS = Sum of Squares = ???

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6 Residual Analysis The three conditions required for the validity of the regression analysis are: the error variable is normally distributed. the error variance is constant for all values of x. the errors are independent of each other. How can we diagnose violations of these conditions? Residual:

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7 Residuals, Standardized Residuals, and Studentized Residuals

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8 Multiple Regression Model

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9 Correlations

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10 Fitted Model Multiple Regression Analysis Dependent variable: Price Standard T Parameter Estimate Error Statistic P-Value CONSTANT -1336.41 173.344 -7.70957 0.0000 Age 12.7351 0.902317 14.1138 0.0000 Bidder 85.8023 8.70515 9.8565 0.0000 ? ? ? ? H 0 : AGE = 0 H a : AGE ≠ 0 H 0 : BIDDER = 0 H a : BIDDER ≠ 0 Q: Effect of AGE? Q: Effect of BIDDER? Degrees of freedom = ?

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11 Fitted Model Multiple Regression Analysis Dependent variable: Price Standard T Parameter Estimate Error Statistic P-Value CONSTANT -1336.41 173.344 -7.70957 0.0000 Age 12.7351 0.902317 14.1138 0.0000 Bidder 85.8023 8.70515 9.8565 0.0000 Fitted Model: Estimated price = -1336.41 + 12.7351 AGE + 85.8023 BIDDER

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12 Prediction and Confidence Intervals Statgraphics demo Fitted Model: Estimated price = -1336.41 + 12.7351 AGE + 85.8023 BIDDER

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13 Analysis of Variance ? ?

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14 Model Selection

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15 Using Dummy Variables

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16 Dummy Variable for LOCATION

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17 Fitted Model

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18 Questions Write down the fitted model. Is the assumed model reliable? Why? What is the value of R 2 ? the adjusted R 2 ? To select a model, why do we prefer adj-R 2 to R 2 ? Predict the amount of money withdrawn from a neighborhood in which the median value of homes is $200,000 for an ATM that is located in a shopping center. If the median value of homes increases by $2,000, then the amount of money withdrawn from an ATM located in a shopping center is expected to increase by. If the median value of homes is $200,000, then the amount of money withdrawn from an ATM located in a shopping center is ???; and the amount of money withdrawn from an ATM located outside a shopping center is ???. What is the difference?

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19 Two Lines with the Same Slopes but Different Intercepts

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20 Two Lines with Different Intercepts and Slopes

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