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1-1 Regression Models Population Deterministic Regression Model Y i = 0 + 1 X i u Y i only depends on the value of X i and no other factor can affect Y i. Population Probabilistic Regression Model Y i = 0 + 1 X i + i i n. E(Y |X i )= 0 + 1 X i, That is, Y ij = E(Y |X i ) + ij 0 + 1 X ij + ij i n; j = 1, 2,..., N. 0 and 1 are population parameters 0 and 1 are estimated by sample statistics b 0 and b 1 u Sample Model:

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1-2 Assumptions Underlying Linear Regression– for Y For each value of X, there is a group of Y values, and these Y values are normally distributed. The means of these normal distributions of Y values all lie on the straight line of regression. The error variances of these normal distributions are equal (Homoscedasticity). If the error variances are not constant ( called heteroscedasticity). The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.

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1-3 Equation of the Simple Regression Line

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1-4 Ordinary Least Squares (OLS) Analysis

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1-6 Least Squares Analysis

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1-7 Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate

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1-8 Proof: Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate

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Coefficient of Determination The Coefficient of Determination, r 2 - the proportion of the total variation in the dependent variable Y that is explained or accounted for by the variation in the independent variable X. –The coefficient of determination is the square of the coefficient of correlation, and ranges from 0 to 1. 12-10

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1-10 Analysis of Variance (ANOVA)

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1-11 Figure: Measures of variation in regression

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1-12 Expectation of b 1

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1-13 Variance of b 1

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1-14 Expectation of b 0

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1-15 Variance of b 0

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1-16 Covariance of b 0 and b 1

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Confidence Interval—predict The confidence interval for the mean value of Y for a given value of X is given by: 12-20 p.483

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1-19 Prediction of Y 0

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Prediction Interval of an individual value of Y 0 The prediction interval for an individual value of Y for a given value of X is given by: 12-21 p.484

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1-21 Figure: Confidence Intervals for Estimation Y X=6.5 Confidence Intervals for Y X Confidence Intervals for E(Y X )

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1-22 The Coefficient of Correlation, r The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. –It requires interval or ratio-scaled data (variables). –It can range from -1.00 to 1.00. –Values of -1.00 or 1.00 indicate perfect and strong correlation. –Values close to 0.0 indicate weak correlation. –Negative values indicate an inverse relationship and positive values indicate a direct relationship.

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1-23 (Pearson Product-Moment ) Correlation Coefficient For sampleFor population p.489

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1-24 Covariance p. 493

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1-25 Coefficient of regression and correlation

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1-26 F and t statistics

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1-27 The Simple Regression Model-Matrix Denote

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