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**Unformatted text preview: **October 20, 2010 BUAD 310: Applied Business Statistics 1 Outline for Today Simple linear regression Regression parameters Least-squares regression Correlation coefficient r, interpreting r 2 Residuals and residual plots Significance tests Confidence intervals, prediction intervals Lurking variables 2 Inference where we are now 3 So far we have discussed inference for 1 and 2- sample quantitative variables (e.g. t-tests) and categorical variables (e.g. tests) Now consider inference where we have a single quantitative response variable (Y) and a single quantitative explanatory (predictor) variable (X) We covered descriptive tools (scatterplots and correlation) Now consider least squares regression 2 Least-squares regression 4 Situation: 2 quantitative variables A regression line is a straight line that describes how a response variable (Y) changes as an explanatory variable (X) changes Unlike correlation, regression requires that we have a response variable (Y) and an explanatory variable (also called predictor variable) (X) Fitting a Line to Data Motivating Example: What is the relationship between the price and weight of diamonds? Use regression analysis to find an equation that summarizes the linear association between price and weight The intercept and slope of the line estimate the fixed and variable costs in pricing diamonds 5 Fitting a Line to Data Consider Two Questions about Diamonds: Whats the average price of diamonds that weigh 0.4 carat? How much more do diamonds that weigh 0.5 carat cost? 6 Fitting a Line to Data Equation of a Line Using a sample of diamonds of various weights, regression analysis produces an equation that relates weight to price. Let y denote the response variable (price) and let x denote the explanatory variable (weight). Linear association is evident ( r = 0.66). 7 Fitting a Line to Data Equation of a Line Identify the line fit to the data by an intercept and a slope . The equation of the line is Estimated Price = How to calculate and ? x b b y 1 + = (Weight) 1 b b + b 1 b b 1 b 8 Fitting a Line to Data Least Squares The least-squares regression line is the line that makes the sum of the squares of the vertical distances (Y) of the data points from the line as small as possible The above vertical distance (Y) of the data points from the line is called residual . We denote it as i i i y y e - = 9 The Least Squares Criterion 10 bad fit better fit Want residuals to be small: Minimize the sum of squared residuals = = = +- =- = n i i i n i i i n i i x b b y y y e 1 2 1 1 2 1 2 )) ( ( ) ( 11 To summarize, the best-fitting line is the least squares line, which makes the sum of the squared residuals as small as possible Minimizes the quantity: We want to solve for b and b 1 to make the above...

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