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sta108_handout11 - Handout 11 Multiple regression For the...

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Handout 11 Multiple regression For the Electric bill data we have Y = annual electric bill, X 1 = monthly household income, X 2 =number of persons, X 3 =living area. The model is Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ε . The fitted regression is ˆ Y = - 358 . 4 + . 0751 X 1 + 55 . 09 X 2 + . 2811 X 3 , SSE ( X 1 , X 2 , X 3 ) = 550163 , SSTO = 3701668 . The correlation matrix is: Bill Income Persons Area Bill 1 .837 .494 .905 Income 1 .143 .961 Person 1 .366 Area 1 Three testing problems. I. An overall test for the regression: H 0 : β 1 = β 2 = β 3 = 0 against H 1 : at least one of β 1 , β 2 , β 3 is nonzero. Note that ” H 0 is true” is equivalent to ”knowing X 1 , X 2 , X 3 does not help in predicting Y ”. II. Can a predictor variable, say X 1 , can be dropped from the model? This is equivalent to asking ”does adding variable X 1 to the model Y = β 0 + β 2 X 2 + β 3 X 3 + ε improve prediction of Y ”. In order to settle the question ”can we can drop variable X 1 from the full model Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ε ” we need to test H 0 : β 1 = 0 against H 1 : β 1 6 = 0. There are two (equivalent) tests for this: a) t-test, b) (partial) F-test. III. Can we drop more than one variable, say X 1 and X 3 , from the model Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ε ? An equivalent question is: does the addition of variables X 1 and X 3 to the model Y = β 0 + β 2 X 2 + ε significantly improve the prediction of Y ? Statistically this is equivalent to testing H 0 : β 1 = β 3 = 0 against H 1 : not both of β 1 and β 3 are zero.
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