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Linear Regression - Topics Basics of Linear Regression Variation in Linear Regression Linear Regression Analysis Goodness of Fit Standard Error terms for Linear Regression Hypothesis testing

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Regression - Types

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Linear Regression A statistical technique that uses a single, independent variable (X) to estimate a single dependent variable (Y). Based on the equation for a line: Y = b + mX

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Linear Regression - Model i X Y YX YiYi XiXi ? (the actual value of Y i )

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Linear Regression - Model Regression Coefficients for a... Population Sample ˆ Y = b 0 + b 1 X i + e Y = b 0 + b 1 X i ˆ

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ANOVA CRD - Variation SST SSE SSTR SST = SSTR + SSE SST is a measure of the total variation of observations. A measure of the differences in observations. Due to treatments. Random/unexplained.

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Linear Regression - Variation Ice Cream Example Y Y = 2.53

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Linear Regression - Variation Ice Cream Example Sample Regression Line

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Linear Regression - Variation SST SSE SSR SST = SSR + SSE Due to regression. Random/unexplained.

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Linear Regression - Variation XiXi Y X Y SST = (Y i - Y) 2 SSE = (Y i - Y i ) 2 SSR = (Y i - Y) 2 _ _ _

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Determining the Regression Line/Model Use Excel (or any other popular statistical software) Select Tools, Data Analysis, Regression Provide the X range Provide the Y range Output the analysis to a new sheet Manual Calculations

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Determining the Regression Line/Model using Excel

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Determining the Regression Line/Model Manual Calculations SST = (Y i - Y) 2 SSE = (Y i - Y i ) 2 SSR = (Y i - Y) 2 __ SSx = (X i - X ) 2 _ SSy = (Y i - Y) 2 _ SSxy = (X i - X )(X i - Y ) __ b 1 =SSxy/SSx b 0 = Y – b 1 X __ MSE = SSE / df MSR = SSR / df R 2 = SSR/SST YX SSE S n-2 t-test = b 1 / S b1

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Measures of Model Goodness 1. R 2 – Coefficient of Determination 2. F-test > F-crit or p-value less than alpha 3. Standard Error 4. t-test

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Hypothesis testing for Testing to see if the linear relationship between X and Y is significant at the population level. t-test Follow the 5-step process H 0 : H A : t-crit, alpha or alpha/2, n-2 df

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Standard Error Terms in Linear Regression S e (standard error of the estimate) A measure of variation around the regression line If the S e is small… Standard deviation Of the Errors S b1 (standard error of the the sampling distribution of b 1 ) Standard deviation of the slopes A measure of the variation of the slopes from different samples If the S b1 is small…our b 1 estimate is probably very accurate Estimates of … b1b1 b1b1 b1b1

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Linear Regression Example Petfood, Estimate Sales based on Shelf Space Two sets of samples, 12 observations each Perform a Regression Analysis on both sets of data Sample1Sample2

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