11-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

11-2 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 11 Simple Linear Regression

11-3 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Contents 1.Probabilistic Models 2.Fitting the Model: The Least Squares Approach

11-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Contents 5.The Coefficients of Correlation and Determination

11-5 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives Introduce the straight-line (simple linear regression) model as a means of relating one quantitative variable to another quantitative variable Assess how well the simple linear regression model fits the sample data

11-6 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives Introduce the correlation coefficient as a means of relating one quantitative variable to another quantitative variable Employ the simple linear regression model for predicting the value of one variable from a specified value of another variable

11-7 Copyright © 2014, 2011, and 2008 Pearson Education, Inc Probabilistic Models

11-8 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Models Representation of some phenomenon Mathematical model is a mathematical expression of some phenomenon Often describe relationships between variables Types –Deterministic models –Probabilistic models

11-9 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Deterministic Models Hypothesize exact relationships Suitable when prediction error is negligible Example: force is exactly mass times acceleration –F = m·a

11-10 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Probabilistic Models Hypothesize two components –Deterministic –Random error Example: sales volume (y) is 10 times advertising spending (x) + random error –y = 10x +  –Random error may be due to factors other than advertising

11-11 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. General Form of Probabilistic Models y = Deterministic component + Random error where y is the variable of interest. We always assume that the mean value of the random error equals 0. This is equivalent to assuming that the mean value of y, E(y), equals the deterministic component of the model; that is, E(y) = Deterministic component

11-12 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. A First-Order (Straight Line) Probabilistic Model y =  0 +  1 x +  where y = Dependent or response variable (variable to be modeled) x = Independent or predictor variable (variable used as a predictor of y) E(y) =  0 +  1 x = Deterministic component  (epsilon) = Random error component

11-13 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. A First-Order (Straight Line) Probabilistic Model y =  0 +  1 x +   0 (beta zero) = y-intercept of the line, that is, the point at which the line intercepts or cuts through the y-axis  1 (beta one) = slope of the line, that is, the change (amount of increase or decrease) in the deterministic component of y for every 1-unit increase in x

11-14 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. [Note: A positive slope implies that E(y) increases by the amount  1 for each unit increase in x. A negative slope implies that E(y) decreases by the amount  1.] A First-Order (Straight Line) Probabilistic Model

11-15 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Five-Step Procedure Step 1:Hypothesize the deterministic component of the model that relates the mean, E(y), to the independent variable x. Step 2:Use the sample data to estimate unknown parameters in the model. Step 3:Specify the probability distribution of the random error term and estimate the standard deviation of this distribution. Step 4:Statistically evaluate the usefulness of the model. Step 5:When satisfied that the model is useful, use it for prediction, estimation, and other purposes.

11-16 Copyright © 2014, 2011, and 2008 Pearson Education, Inc Fitting the Model: The Least Squares Approach

11-17 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Scatterplot 1.Plot of all (x i, y i ) pairs 2.Suggests how well model will fit x y

11-18 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge x y How would you draw a line through the points? How do you determine which line ‘fits best’?

11-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. The least squares line is one that has the following two properties: 1.The sum of the errors equals 0, i.e., mean error = 0. 2.The sum of squared errors (SSE) is smaller than for any other straight-line model, i.e., the error variance is minimum. Least Squares Line

11-20 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Formula for the Least Squares Estimates n = Sample size

11-21 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. y-intercept: represents the predicted value of y when x = 0 (Caution: This value will not be meaningful if the value x = 0 is nonsensical or outside the range of the sample data.) slope: represents the increase (or decrease) in y for every 1-unit increase in x (Caution: This interpretation is valid only for x-values within the range of the sample data.) Interpreting the Estimates of  0 and  1 in Simple Liner Regression

11-22 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Least Squares Graphically  2 y x  1  3  4 ^ ^ ^ ^

11-23 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Least Squares Example You’re a marketing analyst for Hasbro Toys. You gather the following data: Ad Expenditure (100$) Sales (Units) Find the least squares line relating sales and advertising.

11-24 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Scatterplot Sales vs. Advertising Sales Advertising

11-25 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Parameter Estimation Solution

11-26 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Parameter Estimation Solution The slope of the least squares line is:

11-27 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Parameter Estimation Computer Output Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob>|T| INTERCEP ADVERT 00 ^ 11 ^

11-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient Interpretation Solution 1.Slope (  1 ) Sales Volume (y) is expected to increase by $700 for each $100 increase in advertising (x), over the sampled range of advertising expenditures from $100 to $500 ^ 2. y-Intercept (  0 ) Since 0 is outside of the range of the sampled values of x, the y-intercept has no meaningful interpretation^

11-29 Copyright © 2014, 2011, and 2008 Pearson Education, Inc The Coefficients of Correlation and Determination

11-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Correlation Models Answers ‘How strong is the linear relationship between two variables?’ Coefficient of correlation –Sample correlation coefficient denoted r –Values range from –1 to +1 –Measures degree of association –Does not indicate cause–effect relationship

11-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation where

11-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation

11-33 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation

11-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation

11-35 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation Example You’re a marketing analyst for Hasbro Toys. Ad Expenditure (100$) Sales (Units) Calculate the coefficient of correlation.

11-36 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation Solution

11-37 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. A Test for Linear Correlation

11-38 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Condition Required for a Valid Test of Correlation The sample of (x, y) values is randomly selected from a normal population.

11-39 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation Thinking Challenge You’re an economist for the county cooperative. You gather the following data: Fertilizer (lb.)Yield (lb.) Find the coefficient of correlation. © T/Maker Co.

11-40 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Correlation Solution

11-41 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Determination It represents the proportion of the total sample variability around y that is explained by the linear relationship between y and x. 0  r 2  1 r 2 = (coefficient of correlation) 2

11-42 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Determination Example You’re a marketing analyst for Hasbro Toys. You know r =.904. Ad Expenditure (100$) Sales (Units) Calculate and interpret the coefficient of determination.

11-43 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Coefficient of Determination Solution r 2 = (coefficient of correlation) 2 r 2 = (.904) 2 r 2 =.817 Interpretation: About 81.7% of the sample variation in Sales (y) can be explained by using Ad $ (x) to predict Sales (y) in the linear model.

11-44 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. r 2 Computer Output Root MSE R-square Dep Mean Adj R-sq C.V r 2 adjusted for number of explanatory variables & sample size r2r2

11-45 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Simple Linear Regression Variables y = Dependent variable (quantitative) x = Independent variable (quantitative) Method of Least Squares Properties 1. average error of prediction = 0 2. sum of squared errors is minimum

11-46 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Practical Interpretation of y-intercept predicted y value when x = 0 (no practical interpretation if x = 0 is either nonsensical or outside range of sample data) Practical Interpretation of Slope Increase or decrease in y for every 1-unit increase in x

11-47 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas First-Order (Straight Line) Model E(y) =  0 +  1 x where E(y) = mean of y  0 = y-intercept of line (point where line intercepts the y-axis)  1 = slope of line (change in y for every 1-unit change in x)

11-48 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Coefficient of Correlation, r 1. Ranges between –1 and 1 2. Measures strength of linear relationship between y and x Coefficient of Determination, r 2 1. Ranges between 0 and 1 2. Measures proportion of sample variation in y explained by the model

11-49 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Practical Interpretation of Model Standard Deviation, s Ninety-five percent of y-values fall within 2s of their respected predicted values Width of confidence interval for E(y) will always be narrower than width of prediction interval for y