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Chapter 8

Chapter 8.docx

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University of Toronto Scarborough
Mahinda Samarakoon

Stats: Data and Models – Canadian Edition Chapter 8 – Linear Regression - A relationship can be modelled with a line and the equation of that line, which will allow us to predict the value of a variable, given a value of the related variable - Linear model – an equation of a straight line through the data - A straight line can summarize the general pattern with only a couple of parameters Residuals - Predicted value/fitted value/fit (y-hat) – the estimate made from a model; distinguished from the true value of y (y) - Residual – the difference between the observed value and its associated predicted value; tells us how far off the model’s prediction is at that point (y – y-hat) o A negative residual means that the predicted value is an overestimate o A positive residual means that the predicted value is an underestimate “Best Fit” Means Least Squares - To assess a line of best fit, square all of the residuals so that the values are all positive - Squaring emphasizes large residuals (as we are more concerned with points far from the line, than those close to the line) - The smaller the sum of the residuals, the better the fit - The line of best fit is the line for which the sum of the squared residuals is smallest; the least- squares line The Linear Model - y-hat = b + b x 0 1 o the predictions from our model follow a straight line o if the model is good, the values will scatter closely around it o b1= slope, b =0y-intercept - Slope is always expressed in y-units per x-unit - The y-intercept is the value the line takes when x = 0 (but sometimes 0 is not a plausible value for x, in which case we use the y-intercept as a starting value for our predictions) The Least-Squares Line - b = r(s /s ), where r is the correlation of the association, and s is the standard deviation 1 y x - If correlation is positive, the scatterplot runs from lower left to upper right, and the slope of the line is positive - Slope uses units – changing the units of the variables affects their standard deviation directly o Units of slop are always the units of y per unit of x - b = ȳ - b x-bar, knowing the slope and the fact that the line goes through the point (x-bar, ȳ), tells 0 1 us how to find the intercept - Least-square lines are commonly called regression lines Correlation and the Line - For standardized values: z-hat = rz y x  Moving one standard deviation from the mean in x, we can expect to move r standard deviations from the mean in y - If r = 0, there is no linear relationship How Big Can Predicted Values Get? - Regression to the mean: Each predicted y tends to be closer to its mean (in standard deviations) than its corresponding x was Residuals Revisited - Residuals are the part of the data that hasn’t been measured, residual = data – model o Or, e = y – y-hat - When a regression model is appropriate, it should model the underlying relationship - A scatterplot of residuals verses x-values should not have shape or direction, it should stretch horizontally and have the same amount of scatter throughout - Often computers plot residuals
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