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tutorial08 - 1 ISMT 111 B USINESS S TATISTICS – T UTORIAL...

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Unformatted text preview: 1 ISMT 111 B USINESS S TATISTICS – T UTORIAL 8 created by Andrew Yam Simple Linear Regression Model § Aim: Use X (Independent Variable, or Predictor Variable) to predict Y (Dependent Variable, or Response Variable) after drawing n pairs of observations, ( 29 ( 29 n n y x y x , ,..., , 1 1 . § Actual value i i i x Y e b b + + = 1 , n i ,..., 2 , 1 = (also shown as i i i x Y e b a + + = ) where i e is the random error § Predicted value i i x Y 1 ˆ ˆ ˆ b b + = , n i ,..., 2 , 1 = § Assumptions of Regression Model - Linearity ( 29 i i i x x X Y E 1 | b b + = =- Constant Variance; Homoscedasticity 2 2 | s s = = i i x X Y i 2200- Normality ( 29 2 , ~ s e N i i 2200- Independence i e and j e are independent j i ≠ 2200 Estimating β and β 1 We use the least square (LS) method to find b and 1 b by minimizing the total sum of square errors ( SSE ), i.e. minimize ( 29 ( 29 ( 29 ( 29 ∑ ∑ ∑ = = = = +- =- = n i i n i i i n i i i e x y y y SSE 1 2 1 2 1 1 2 ˆ ˆ ˆ b b xy yy S S 1 ˆ b- We get 2 ( 29 ( 29 ( 29 ( 29 n x x n y x y x x x y y x x S S i i i i i i i i i xx xy 2 2 2 1 ˆ ∑ ∑ ∑ ∑ ∑ ∑ ∑-- =--- = = b x y 1 ˆ ˆ b b- = Estimating σ MSE n SSE f d SSE s =- = = = 2 . . ˆ 2 2 s , where 2 ˆ s is an unbiased estimator of 2 s . Testing the Model § Hypothesis Testing for the Slope, 1 b Null Hypothesis 10 1 : b b = H Test Statistics: xx S s t / ˆ 10 1 b b- = ~ t- distribution with 2 d.f.- = n i) when 10 1 : b b < a H , H is rejected when a t t- < , ii) when 10 1 : b b a H , H is rejected when a t t , iii) when 10 1 : b b ≠ a H , H is rejected when 2 / a t t- < or 2 / a t t . Note: If : 1 = b H , rejecting H means X is useful for the prediction of Y . A ( 29 % 100 1 a- C.I. for 1 b is xx n S s t 2 ; 2 1 ˆ a b- ± § Hypothesis Testing for the Population Correlation Coefficient, r Sample Correlation Coefficient, r 3 ( 29 ( 29 ( 29 ( 29 yy xx xy i i i i S S S y y x x y y x x r =---- = ∑ ∑ ∑ 2 2 which is used to measure the linear relationship between X and Y . Note: - r lies between – 1 and 1, i.e. 1 1 ≤ ≤- r . - | | r closes to 1 implies a strong linear relationship between X and Y . - | | r closes to 0 implies a weak linear relationship between X and Y . - r is positive. ⇔ X and Y are positively linearly related. - r is negative. ⇔ X and Y are negatively linearly related. If we want to test the relation between X and Y , we can test about the population correlation coefficient, r : Null Hypothesis : = r H Test Statistics: 2 1 2 r n r t-- = ~ t- distribution with 2 d.f.- = n i) when : < r a H , H is rejected when a t t- < , ii) when : r a H , H is rejected when a t t , iii) when : ≠ r a H , H is rejected when 2 / a t t- < or 2 / a t t ....
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This note was uploaded on 09/15/2008 for the course ISMT 111 taught by Professor Wanxuhu during the Fall '08 term at HKUST.

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tutorial08 - 1 ISMT 111 B USINESS S TATISTICS – T UTORIAL...

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