UASTAT151Ch12 - 12.1 Simple Linear Regression Returns!...

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12.1 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 = a + bx - a is the intercept: the height of the line at x = 0. - b 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 population regression equation : y i = α + β x i + ε i μ ( y i | x i ) = α + β 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 ). α , β , σ are unknown parameters. o α is the “population” intercept. o β 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; α , β , and x i are all fixed. o The conditional distribution of y i given x i is N ( α + β x i , σ ). Basic Assumptions of the SLR Model o The distribution of ε at any particular x has a mean of zero ( 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 . Least squares estimation of α and β : Def’n: A residual is the difference between an observed value and its estimated value. Since ŷ denotes the estimated value of y , then at some observed value of x , say x i , the residual is defined as y i ŷ i = y i – ( a + bx i ) The residual represents the vertical deviation of the point from the line. We want to choose ( a , b ) to minimize the sum of squared deviations (hence “least squares”): 2 ) (
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UASTAT151Ch12 - 12.1 Simple Linear Regression Returns!...

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