ch8 - STAT1306 Introductory Statistics Chapter 8...

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Unformatted text preview: STAT1306 Introductory Statistics Chapter 8 Correlation and Regression Department of Statistics and Actuarial Science The University of Hong Kong Prof. W.K. Li (hrntlwk@hku.hk) STAT1306 (Chapter 8) 2008-09 1st Semester 1 / 25 Correlation Recall in Chapter 5 we introduced the concept of covariance and correlation between two random variables X and Y : covariance : Cov ( X, Y ) = E ( XY )- E ( X ) E ( Y ) correlation coefficient: ρ ( X, Y ) = Cov ( X, Y ) σ X σ Y = ρ for simplicity where σ X and σ Y are the standard deviations of X and Y respectively. Note that ρ does not depend on the scales of X and Y and- 1 ≤ ρ ≤ 1. Prof. W.K. Li (hrntlwk@hku.hk) STAT1306 (Chapter 8) 2008-09 1st Semester 2 / 25 The correlation coefficient ρ measures the strength of a linear relationship with ρ = 0 implies no such relationship and ρ = ± 1 a perfect linear relationship. Given observations of ( X, Y ): ( X 1 , Y 1 ) , ··· , ( X n , Y n ) we can estimate ρ by b ρ = S XY √ S XX √ S Y Y where S XY = 1 n n X i =1 ( X i- X )( Y i- Y ) , S XX = 1 n n X i =1 ( X i- X ) 2 with a similar definition for S Y Y . Prof. W.K. Li (hrntlwk@hku.hk) STAT1306 (Chapter 8) 2008-09 1st Semester 3 / 25 Hence, b ρ = n ∑ i =1 ( X i- X )( Y i- Y ) r n ∑ i =1 ( X i- X ) 2 · n ∑ i =1 ( Y i- Y ) 2 (some text books use r to denote b ρ ) Prof. W.K. Li (hrntlwk@hku.hk) STAT1306 (Chapter 8) 2008-09 1st Semester 4 / 25 Prof. W.K. Li (hrntlwk@hku.hk) STAT1306 (Chapter 8) 2008-09 1st Semester 5 / 25 It is often of interest to know if ρ = 0? That is, we want to test H : ρ = 0 vs H 1 : ρ 6 = 0 ....
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This note was uploaded on 09/06/2010 for the course STAT STAT1306 taught by Professor Prof during the Fall '08 term at HKU.

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ch8 - STAT1306 Introductory Statistics Chapter 8...

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