08C_F10 - The Hong Kong University of Science Technology...

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The Hong Kong University of Science & Technology ISMT 111: Business Statistics Tutorial Set 8: Linear Regression Simple Linear Regression Model Actual Value y i = β o + β 1 x i + ε i i = 1 , 2 , . . . , n Predicted Value b y i = b β o + b β 1 x i i = 1 , 2 , . . . , n 1. Estimating β o and β 1 We use the least square (LS) method to fi nd β o and β 1 by minimizing the total sum of square errors ( SSE ), i.e. minimizes SSE = n X i =1 ( y i b y i ) 2 = n X i =1 ( y i ( b β o + b β 1 x i )) 2 SSE = S yy b β 1 S xy b β 1 = S xy S xx = P n i =1 ( x i x )( y i y ) P n i =1 ( x i x ) 2 = P n i =1 x i y i P n i =1 x i P n i =1 y i n P n i =1 x 2 i P n i =1 ( x i ) 2 n b β o = y b β 1 x 2. Estimating σ 2 b σ 2 = s 2 = SSE d.f. = SSE n 2 = MSE b σ 2 is the unbiased estimator of σ 2 1
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3. Testing the Model (a) Hypothesis Testing for the Slope, β 1 Null Hypothesis, H o : β 1 = β 1 o Test Statistic: t = b β 1 β 1 o s/ S xx t with d.f. = n 2 under H o Alternative Hypothesis Rejection Region H a : β 1 6 = β 1 o t > t α 2 or t < t α 2 H a : β 1 > β 1 o t > t α H a : β 1 < β 1 o t < t α If the Null Hypothesis is H o : β 1 = 0 Rejecting H o means x is useful for the prediction of y . (1 α )100% C.I. for β 1 is b β 1 ± t α 2 s S xx (b) Hypothesis Testing for the Population Correlation Coe cient, ρ Sample Correlation Coe cient, r r = P n i =1 ( x i x )( y i y ) p P n i =1 ( x i x ) 2 P n i =1 ( y i y ) 2 = S xy p S xx S yy which is used to measure the linear relationship between x and y .
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