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# chap20_2010 - Chapter 20 Linear and Multiple Regression...

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Chapter 20 Linear and Multiple Regression

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Empirical Models Study of relationship between two or more variables Response variable: (dependent, output) Predictor or explanatory variables: (independent, input) Deterministic relationship: The outcomes can be predicted precisely (physics, chemistry, etc.) Regression Analysis: statistical tools used to model and explore relationships between variables
Regression Analysis Simple regression models: one explanatory variable Linear Non-linear Multiple regression models: two or more explanatory variables Linear Non-linear

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Simple Linear Regression Model Population regression line y = response variable A = y-intercept (population parameter) B = slope (population parameter) x = explanatory variable  = random error Missing or omitted variables Random variation Estimated regression equation ŷ = estimated value of y for a given x ε + + = Bx A y bx a y + = ˆ
Scatterplots and Least Squares Line  (residual): difference between the actual value y and the predicted value of y for population data e: error for the estimated equation Sum of Squared Errors (SSE) y y e ˆ - = 0 ) ˆ ( = - = y y e - = = 2 2 ) ˆ ( y y e SSE

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Scatterplots and Least Squares Line Least squares method finds a and b to minimize SSE a and b are called the least squares estimates of A and B - = - - = n y x xy y y x x SS xy ) )( ( ) ( ) ( - = - = n x x x x SS xx 2 2 2 ) ( ) ( xx xy SS SS b = x b y a - = Excel slope(y, x) intercept(y, x) forecast
Scatterplots and Least Squares Line –Example 20.1 2296 . 71 . 2585 = = = xx SS b 6483 . 3 = - = x b y a x y 63 16 73.4694 0.0204 -1.2245 88 25 1127.0408 78.4490 297.3469 38 13 269.8980 9.8776 51.6327 70 19 242.4694 8.1633 44.4898 27 9 752.3265 51.0204 195.9184 51 15 11.7551 1.3061 3.9184 44 16 108.7551 0.0204 1.4898 Mean 54.4286 16.1429 Std Dev 20.7594 4.9809 Sum 2585.7143 148.8571 593.5714 SSxx SSyy SSxy 2 ) ( x x - 2 ) ( y y - ) )( ( y y x x - -

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Scatterplots and Least Squares Line –Example 20.1 Minitab Graph Scatterplot
Scatterplots and Least Squares Line –Example 20.1 Regression Analysis: y versus x The regression equation is y = 3.65 + 0.230 x S = 1.58733 R-Sq = 91.5% R-Sq(adj) = 89.8% Analysis of Variance Predictor Coef SE Coef T P Constant 3.648 1.802 2.02 0.099 x 0.22956 0.03122 7.35 0.001 Source DF SS MS F P Regression 1 136.26 136.26 54.08 0.001 Residual Error 5 12.60 2.52 Total 6 148.86 Minitab Stat Regression Regression

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Interpretations of a and b Interpretation of a Intercept on y axis at x=0 Caution on extrapolation Interpretation of b Slope Change in y due to an increase of one unit in x Positive relationship when b>0 Negative relationship when b<0
Assumptions of the Regression Model y = A + Bx +  The random error  has a mean equal to zero. y|x = A + Bx The errors associated with different observations are independent For any given x, the distribution of errors is normal The distribution of population errors for each x has the same standard deviation, 

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Standard Deviation of Random Errors For the population, y = A + Bx +   is the std. dev. of all  Since  is unknown, it is estimated by the std. dev.
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## This note was uploaded on 12/03/2011 for the course EIN 5226 taught by Professor Staff during the Fall '11 term at FIU.

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chap20_2010 - Chapter 20 Linear and Multiple Regression...

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