STP 226 ELEMENTARY STATISTICS

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1 TP 6 TP 6 ELEMENTARY TATITIC CHAPTER 4 DECRIPTIVE MEAURE IN REGREION AND CORRELATION Liear Regressio ad correlatio allows us to examie the relatioship betwee two or more quatitative variables. 4.1 Liear Equatios with oe Idepedet Variable y = b 0 + b 1 x is a straight lie where b 0 ad b 1 are costats, b 0 is the y-itercept ad b 1 is the slope of the lie. lope (b 1 ) = for every 1 uit horizotal icrease there is a b 1 uit vertical icrease/decrease depedig o the lie. The straight-lie graph of the liear equatio y = b 0 + b 1 x slopes upward if b 1 > 0, slopes dowward if b 1 < 0, ad is horizotal if b 1 = The Regressio Equatio Ofte, i real life situatios, it is ot likely to have data that follow some straight lie perfectly. A scatterplot (scatter diagram) is useful i visualizig apparet relatioships betwee two variables. 1

2 TP 6 Example(Table 4.) (Age ad price of a Orio) Car (Orio) Age (yr): x Price ($100): y If the poits seem to follow a straight lie, the a straight lie ca be used to approximate the relatioship. May differet lies ca be draw to approximate the relatioship, however, the least squares criterio method gives the best lie to fit the data (relatioship betwee the two variables). Least quares Criterio The straight lie that best fits a set of data poits is the oe havig the smallest possible sum of squared errors. Fid the differece (error) betwee every poit ad its correspodig poit o the best fit lie, square those errors, ad sum them up. We wat this sum to be miimum. mi di = mi( yi ( b0 + b1 xi )) = mi b0, b e 1 Regressio Lie The straight lie that best fits a set of data poits accordig to the lease squares criterio.

3 TP 6 Regressio equatio The equatio of the regressio lie: yˆ = x Notatio used i Regressio ad Correlatio Defiitio: Computatioal xx = ( x x) xx ( x) = x xy yy = ( x x)( y y) = ( y y) yy xy ( x)( y) = xy ( y = y ) Formula 4.1 Regressio Equatio for a set of data poits is yˆ = b0 + b1 x, where xy 1 b1 = ad b0 = ( y b1 x) = y b1 x xx 3

4 TP 6 Predictor Variable ad Respose Variable For the liear regressio equatio, y = b0 + b1 x y respose variable or depedet variable x predictor variable/explaatory variable or idepedet variable Example (Orio Data): respose variable=price, predictor variable=age Extrapolatio - makig predictios for values of the predictor variable outside the rage of the observed values of the predictor variable. Grossly icorrect predictios ca result from extrapolatio. Outlier a data poit that lies far from the regressio lie, relative to other data poits Ifluetial observatio a data poit whose removal causes the regressio to chage cosiderably. It is usually separated i the x-directio from the other data poits. It pulls the regressio lie towards itself. Warigs o the use of Liear Regressio Draw scatter diagram first Predict withi the rage of the data. Watch out for the ifluetial observatio 4.3 The Coefficiet of Determiatio ( r ) Oe way of measurig the utility of regressio equatio determie the percetage of variatio i the observed values of the respose variable explaied by the regressio (or predictor variable). Example (Orio Data) 4

5 TP 6 x y ŷ y y yˆ y y yˆ Coefficiet of determiatio, r : is the proportio of variatio is the observed values of xy the respose variable that is explaied by the regressio. r = Eg. (cotd.) r = 885.0/ = (85.3%) xx yy 4.4 Liear Correlatio Liear Correlatio Coefficiet, r (Pearso product momet correlatio coefficiet): A statistic used to measure the stregth of liear relatioship betwee two variables. DEFINITION 4.6 The liear correlatio coefficiet, r, of data poits is defied by r = 1 1 ( x x)( y y) x y, or r = xx xy yy Eg. (cotd.) x = 58, y = 975, xy = 473, x = 36, y = 9619 xy ( )( ) / = = xy x y r = 0.94 [ x ( x) / ][ y ( y) / ] xx yy strog egative liear correlatio betwee the age ad price of Orios. Uderstadig the Liear Correlatio Coefficiet a. r reflects the slope of the scatter diagram b. the magitude of r idicates the stregth of the liear relatioship c. The sig of r suggests the type of liear relatioship d. r =0 meas o liear relatio, r >0 meas positive relatio, r <0 mea egative relatio betwee the two variables. e. free of data uits.(eg. correlatio betwee height(i) ad weight(lb)) Note: coefficiet of determiatio, r is the square of the liear correlatio coefficiet. Eg. (cotd.) (-0.94) =

6 TP 6 Warigs o the use of liear correlatio coefficiet. a. measures oly liear relatio betwee the variables b. watch out for the spurious correlatio (lurkig variables). c. Affected by the extreme observatios d. Watch out for separate groups 6

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