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Simple linear regression model, verifying assumptions

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STAT 301
Brett Hunter

14 November The Simple Linear Regression Model When conducting a regression analysis on an explanatory variable and a response variable, the following population model is assumed Y = β 0 β x1+ ε β0is the population y-intercept β1is the population slope 2 ε are random errors, where ε ~ N (0, σ ) Without ε, all observed points would fall on the regression line My = β 0 β x 1 Mean of response variable Assumptions in the model The distribution of ε at any particular X value has mean 0. The distribution of ε at any particular X value is normal. The variance of ε, denoted σ , is the same for any particular X value. ε is homoscedastic The random observations ε , ε ,1… 2 assocnated with different observations are independent of each other. We can sum up these assumptions as iid 2 εi N (0, σ ), i = 1, …., n iid = independently and identically distributed In order to perform a regression analysis, we need to avoid any serious violations of the assumptions. We check these assumptions graphically. y Recall that e = i - i i are residuals/errors for our fitted model. We can standardize our residuals ei di= esimateds.d.of e i
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