 # 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.

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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:

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.

1-3 Equation of the Simple Regression Line

1-4 Ordinary Least Squares (OLS) Analysis

1-5

1-6 Least Squares Analysis

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

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

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

1-10 Analysis of Variance (ANOVA)

1-11 Figure: Measures of variation in regression

1-12 Expectation of b 1

1-13 Variance of b 1

1-14 Expectation of b 0

1-15 Variance of b 0

1-16 Covariance of b 0 and b 1

1-17

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

1-19 Prediction of Y 0

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

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

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.

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

1-24 Covariance p. 493

1-25 Coefficient of regression and correlation

1-26 F and t statistics

1-27 The Simple Regression Model-Matrix Denote

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