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# chapter12 - Introduction to Probability Introduction to...

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Unformatted text preview: Introduction to Probability Introduction to Probability and Statistics and Statistics Thirteenth Edition Thirteenth Edition Chapter 12 Linear Regression and Correlation Introduction Introduction • In Chapter 11, we used ANOVA to investigate the effect of various factor-level combinations (treatments) on a response x . • Our objective was to see whether the treatment means were different. • In Chapters 12 and 13, we investigate a response y which is affected by various independent variables, x i . • Our objective is to use the information provided by the x i to predict the value of y. Example Example • Let y be a student’s college achievement, measured by his/her GPA. This might be a function of several variables: – x 1 = rank in high school class – x 2 = high school’s overall rating – x 3 = high school GPA 4 Example Example • Let y be the monthly sales revenue for a company. This might be a function of several variables: – x 1 = advertising expenditure – x 2 = time of year – x 3 = state of economy 4 Some Questions Some Questions • Which of the independent variables are useful and which are not? • How could we create a prediction equation to allow us to predict y using knowledge of x 1 , x 2 , x 3 etc? • How good is this prediction? We start with the simplest case, in which the response y is a function of a single independent variable, x . A Simple Linear Model A Simple Linear Model • In Chapter 3, we used the equation of a line to describe the relationship between y and x for a sample of n pairs, (x, y ) . • If we want to describe the relationship between y and x for the whole population , there are two models we can choose • Deterministic Model: y = α + β x • Probabilistic Model: – y = deterministic model + random error – y = α + β x + ε A Simple Linear Model A Simple Linear Model • Since the bivariate measurements that we observe do not generally fall exactly exactly on a straight line, we choose to use: • Probabilistic Model: Probabilistic Model: – y y = = α + β α + β x x + ε + ε – E(y) = E(y) = α + β α + β x x Points deviate from the line of means line of means by an amount ε where ε has a normal distribution with mean 0 and The Random Error The Random Error • The line of means, E(y) = E(y) = α + β α + β x x , , describes average value of y for any fixed value of x. • The population of measurements is generated as y deviates from the population line by ε ε . We estimate α α and β β using sample information. The Method of The Method of Least Squares Least Squares • The equation of the best-fitting line is calculated using a set of n pairs ( x i , y i ). • We choose our estimates a and b to estimate α and β so that the vertical distances of the points from the line, are minimized....
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chapter12 - Introduction to Probability Introduction to...

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