UASTAT141Ch27 - Ch. 27 - Simple Linear Regression Returns!...

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Ch. 27 - Simple Linear Regression Returns! Def’n: The regression line predicts the value for the response variable y as a straight-line function of the value x of the explanatory variable. Equation for the regression line: ŷ = b 0 + b 1 x - b 0 is the intercept: the height of the line at x = 0. - b 1 is the slope: the amount by which y increases when x increases by 1 unit. - ŷ (“y-hat”) denotes the predicted value of y (or mean y for a given value of x ). Simple Linear Regression (SLR) model : Now, we introduce the idealized regression line : y i = β 0 + β 1 x i + ε i μ ( y i | x i ) = β 0 + β 1 x i , i = 1, …, n ( x 1 , y 1 ), ( x 2 , y 2 ), …, ( x n , y n ) are the observed data. ε 1 , …, ε n are unobserved “errors”, assumed to be a random sample from N (0, σ ). β 0 , β 1 , σ are unknown parameters. o β 0 is the “population” intercept. o β 1 is the average change in y associated with a 1-unit increase in x . o σ determines the extent to which points deviate from the line y 1 , …, y n are random variables. The conditional distribution of y i given x i is N ( β 0 + β 1 x i , σ ). Basic Assumptions of the SLR Model o The relationship between x and y is sufficiently linear. ± Presuming linearity, this means, at any x , μ ε = 0. o The std. dev. of ε is the same for any particular x (i.e. it’s constant). o The distribution of ε at any particular x is normal. o The random deviations ε 1 , ε 2 , …, ε n associated with different observations are independent of one another. The equation merely approximates the relation, it is a model .
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This note was uploaded on 07/31/2011 for the course STAT 141 taught by Professor Paulcartledge during the Winter '10 term at University of Alberta.

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UASTAT141Ch27 - Ch. 27 - Simple Linear Regression Returns!...

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