Review of Chapters 9-11

Review of Chapters 9-11 - Review for Final Exam Some...

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Review for Final Exam Some important themes from Chapters 9-11 Final exam covers these chapters, but implicitly tests the entire course, because we use sampling distributions, confidence intervals, significance tests, etc. As usual, exam is mixture of true/false to test concepts and problems (like examples in class and homework), with emphasis on interpreting software output.

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Chap. 9: Linear Regression and Correlation Data: y – a quantitative response variable x – a quantitative explanatory variable We consider: Is there an association? (test of independence using slope) How strong is the association? (uses correlation r and r 2 ) How can we predict y using x ? (estimate a regression equation) Linear regression equation E( y ) = α + β x describes how mean of conditional distribution of y changes as x changes Least squares estimates this and provides a sample prediction equation ˆ y a bx = +
The linear regression equation E(y) = + x is part of a model. The model has another parameter σ that describes the variability of the conditional distributions; that is, the variability of y values for all subjects having the same x- value. For an observation, difference between observed value of y and predicted value of y, is a residual (vertical distance on scatterplot) Least squares method minimizes the sum of squared residuals (errors), which is SSE used also in r 2 and the estimate s of conditional standard deviation of y ˆ y ˆ y y -

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Measuring association: The correlation and its square The correlation is a standardized slope that does not depend on units Correlation r relates to slope b of prediction equation by r = b(s x /s y ) -1 ≤ r ≤ +1, with r having same sign as b and r = 1 or -1 when all sample points fall exactly on prediction line, so r describes strength of linear association The larger the absolute value, the stronger the association Correlation implies that predictions regress toward the mean
The proportional reduction in error in using x to predict y (via the prediction equation) instead of using sample mean of y to predict y is Since -1 ≤ r ≤ +1, 0 ≤ r 2 ≤ 1, and r 2 = 1 when all sample points fall exactly on prediction line r and r 2 do not depend on units, or distinction between x, y The r and r 2 values tend to weaken when we observe x only over a restricted range, and they

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Review of Chapters 9-11 - Review for Final Exam Some...

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