Statistics for Business and Economics 6th Edition Chapter 12 Simple Regression Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-1 Chapter Goals After completing this chapter, you should be able to: Explain the correlation coefficient and perform a hypothesis test for zero population correlation Explain the simple linear regression model Obtain and interpret the simple linear regression equation for a set of data Describe R2 as a measure of explanatory power of the regression model Understand the assumptions behind regression analysis

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-2 Chapter Goals (continued) After completing this chapter, you should be able to: Explain measures of variation and determine whether the independent variable is significant Calculate and interpret confidence intervals for the regression coefficients Use a regression equation for prediction Form forecast intervals around an estimated Y value for a given X Use graphical analysis to recognize potential problems in regression analysis Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-3

Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Correlation was first presented in Chapter 3 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-4 Correlation Analysis The population correlation coefficient is denoted (the Greek letter rho) The sample correlation coefficient is r s xy

sxsy where s xy (x x)(y y) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. i i n 1 Chap 12-5 Hypothesis Test for Correlation To test the null hypothesis of no linear association, H0 : 0 the test statistic follows the Students t distribution with (n 2 ) degrees of freedom: t r (n 2) 2 (1 r )

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-6 Decision Rules Hypothesis Test for Correlation Lower-tail test: Upper-tail test: Two-tail test: H0: 0 H1: < 0 H0: 0 H1: > 0 H0: = 0 H1: 0 -t t Reject H0 if t < -tn-2, Where t Reject H0 if t > tn-2, r (n 2) 2

(1 r ) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. has n - 2 d.f. /2 -t/2 /2 t/2 Reject H0 if t < -tn-2, or t > t n-2, Chap 12-7 Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain

(also called the endogenous variable) Independent variable: the variable used to explain the dependent variable (also called the exogenous variable) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-8 Linear Regression Model The relationship between X and Y is described by a linear function Changes in Y are assumed to be caused by changes in X Linear regression population equation model Yi 0 1x i i Where 0 and 1 are the population model coefficients and is a random error term. Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-9 Simple Linear Regression Model

The population regression model: Population Y intercept Dependent Variable Population Slope Coefficient Independent Variable Random Error term Yi 0 1Xi i Linear component Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Random Error component Chap 12-10 Simple Linear Regression Model (continued) Y Yi 0 1Xi i Observed Value of Y for Xi

i Predicted Value of Y for Xi Slope = 1 Random Error for this Xi value Intercept = 0 Xi Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. X Chap 12-11 Simple Linear Regression Equation The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted) y value for observation i Estimate of the regression Estimate of the regression slope intercept y i b0 b1x i

Value of x for observation i The individual random error terms ei have a mean of zero ei ( y i - y i ) y i - (b0 b1x i ) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-12 Least Squares Estimators b0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of the squared differences between y and y : min SSE min ei2 min (y i y i )2 min [y i (b0 b1x i )]2 Differential calculus is used to obtain the coefficient estimators b0 and b1 that minimize SSE Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-13 Least Squares Estimators (continued) The slope coefficient estimator is n (x x)(y y) i b1 i1

n i x 2 (x x ) i rxy sY sX i 1 And the constant or y-intercept is b0 y b1x The regression line always goes through the mean x, y Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-14 Finding the Least Squares Equation

The coefficients b0 and b1 , and other regression results in this chapter, will be found using a computer Hand calculations are tedious Statistical routines are built into Excel Other statistical analysis software can be used Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-15 Linear Regression Model Assumptions The true relationship form is linear (Y is a linear function of X, plus random error) The error terms, i are independent of the x values The error terms are random variables with mean 0 and constant variance, 2 (the constant variance property is called homoscedasticity) 2 E[ i ] 0 and E[ i ] 2

for (i 1, , n) The random error terms, i, are not correlated with one another, so that E[ i j ] 0 for all i j Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-16 Interpretation of the Slope and the Intercept b0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values) b1 is the estimated change in the average value of y as a result of a one-unit change in x Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-17 Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)

A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-18 Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 245 1400 312 1600 279 1700 308 1875 199 1100

219 1550 405 2350 324 2450 319 1425 255 1700 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-19 Graphical Presentation House price model: scatter plot House Price ($1000s) 450 400 350 300 250

200 150 100 50 0 0 500 1000 1500 2000 2500 3000 Square Feet Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-20 Regression Using Excel Tools / Data Analysis / Regression Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-21 Excel Output Regression Statistics Multiple R

0.76211 R Square 0.58082 Adjusted R Square 0.52842 Standard Error The regression equation is: house price 98.24833 0.10977 (square feet) 41.33032 Observations 10 ANOVA df SS MS F 11.0848 Regression 1 18934.9348

18934.9348 Residual 8 13665.5652 1708.1957 Total 9 32600.5000 Coefficients Intercept Square Feet Standard Error t Stat P-value Significance F 0.01039 Lower 95% Upper 95% 98.24833 58.03348

1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-22 Graphical Presentation House price model: scatter plot and regression line 450 House Price ($1000s) Intercept = 98.248

400 350 300 250 200 150 100 50 0 Slope = 0.10977 0 500 1000 1500 2000 2500 3000 Square Feet house price 98.24833 0.10977 (square feet) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-23 Interpretation of the Intercept, b0 house price 98.24833 0.10977 (square feet)

b0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Here, no houses had 0 square feet, so b0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-24 Interpretation of the Slope Coefficient, b1 house price 98.24833 0.10977 (square feet) b1 measures the estimated change in the average value of Y as a result of a oneunit change in X Here, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-25 Measures of Variation Total variation is made up of two parts:

SST SSR SSE Total Sum of Squares Regression Sum of Squares Error Sum of Squares SST (y i y)2 SSR (y i y)2 SSE (y i y i )2 where: y = Average value of the dependent variable yi = Observed values of the dependent variable y = Predicted value of y for the given x value i i Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-26 Measures of Variation (continued)

SST = total sum of squares SSR = regression sum of squares Measures the variation of the yi values around their mean, y Explained variation attributable to the linear relationship between x and y SSE = error sum of squares Variation attributable to factors other than the linear relationship between x and y Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-27 Measures of Variation (continued) Y yi 2 SSE = (yi - yi )

y _ SST = (yi - y)2 y _2 SSR = (yi - y) _ y xi Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. _ y X Chap 12-28 Coefficient of Determination, R2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called R-squared and is denoted as R2 SSR regression sum of squares R

SST total sum of squares 2 note: 2 0 R 1 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-29 Examples of Approximate r2 Values Y r2 = 1 r2 = 1 X 100% of the variation in Y is explained by variation in X Y r =1 2 Perfect linear relationship between X and Y: X Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-30 Examples of Approximate r2 Values Y 0 < r2 < 1 X Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X Y X Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-31 Examples of Approximate r2 Values r2 = 0 Y No linear relationship between X and Y: r2 = 0 X The value of Y does not

depend on X. (None of the variation in Y is explained by variation in X) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-32 Excel Output Multiple R 0.76211 R Square 0.58082 Adjusted R Square 0.52842 Standard Error SSR 18934.9348 R 0.58082 SST 32600.5000 2 Regression Statistics 58.08% of the variation in house prices is explained by variation in square feet 41.33032

Observations 10 ANOVA df SS MS F 11.0848 Regression 1 18934.9348 18934.9348 Residual 8 13665.5652 1708.1957 Total 9 32600.5000

Coefficients Intercept Square Feet Standard Error t Stat P-value Significance F 0.01039 Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938 0.01039

0.03374 0.18580 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-33 Correlation and R2 The coefficient of determination, R2, for a simple regression is equal to the simple correlation squared 2 2 xy R r Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-34 Estimation of Model Error Variance An estimator for the variance of the population model error is n 2 e i

SSE s n 2 n 2 2 2 e i1 Division by n 2 instead of n 1 is because the simple regression model uses two estimated parameters, b0 and b1, instead of one se s2e is called the standard error of the estimate Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-35 Excel Output Regression Statistics Multiple R 0.76211 R Square 0.58082 Adjusted R Square 0.52842 Standard Error

s e 41.33032 41.33032 Observations 10 ANOVA df SS MS F 11.0848 Regression 1 18934.9348 18934.9348 Residual 8 13665.5652 1708.1957 Total 9

32600.5000 Coefficients Intercept Square Feet Standard Error t Stat P-value Significance F 0.01039 Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938

0.01039 0.03374 0.18580 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-36 Comparing Standard Errors se is a measure of the variation of observed y values from the regression line Y Y small se X large se X The magnitude of se should always be judged relative to the size of the y values in the sample data i.e., se = $41.33K is moderately small relative to house prices in the $200 - $300K range Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-37 Inferences About the Regression Model

The variance of the regression slope coefficient (b1) is estimated by 2 2 s s e e s2b1 2 2 (xi x) (n 1)s x where: sb1 = Estimate of the standard error of the least squares slope SSE se n 2 = Standard error of the estimate Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-38 Excel Output Regression Statistics Multiple R 0.76211

R Square 0.58082 Adjusted R Square 0.52842 Standard Error sb1 0.03297 41.33032 Observations 10 ANOVA df SS MS F 11.0848 Regression 1 18934.9348 18934.9348 Residual

8 13665.5652 1708.1957 Total 9 32600.5000 Coefficients Intercept Square Feet Standard Error t Stat P-value Significance F 0.01039 Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892

-35.57720 232.07386 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-39 Comparing Standard Errors of the Slope Sb1 is a measure of the variation in the slope of regression lines from different possible samples Y Y small Sb1 X Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. large Sb1

X Chap 12-40 Inference about the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H0: 1 = 0 H1: 1 0 (no linear relationship) (linear relationship does exist) Test statistic b1 1 t sb1 d.f. n 2 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. where: b1 = regression slope coefficient 1 = hypothesized slope

sb1 = standard error of the slope Chap 12-41 Inference about the Slope: t Test (continued) House Price in $1000s (y) Square Feet (x) 245 1400 312 1600 279 1700 308 1875 199 1100 219 1550

405 2350 324 2450 319 1425 255 1700 Estimated Regression Equation: house price 98.25 0.1098 (sq.ft.) The slope of this model is 0.1098 Does square footage of the house affect its sales price? Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-42 Inferences about the Slope: t Test Example H0: 1 = 0 From Excel output: H1: 1 0 Coefficients

Intercept Square Feet b1 Standard Error sb1 t Stat P-value 98.24833 58.03348 1.69296 0.12892 0.10977 0.03297 3.32938 0.01039 b1 1 0.10977 0 t 3.32938 t sb1 0.03297 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-43 Inferences about the Slope: t Test Example (continued) Test Statistic: t = 3.329 H0: 1 = 0 From Excel output: H1: 1 0 Coefficients Intercept Square Feet d.f. = 10-2 = 8 t8,.025 = 2.3060 /2=.025 Reject H0 /2=.025 Do not reject H0 -tn-2,/2 -2.3060 0 Reject H0 tn-2,/2 2.3060 3.329

b1 Standard Error sb1 t t Stat P-value 98.24833 58.03348 1.69296 0.12892 0.10977 0.03297 3.32938 0.01039 Decision: Reject H0 Conclusion: There is sufficient evidence that square footage affects house price Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-44 Inferences about the Slope: t Test Example (continued) P-value = 0.01039 H0: 1 = 0 From Excel output: H1: 1 0 Coefficients Intercept Square Feet This is a two-tail test, so the p-value is P(t > 3.329)+P(t < -3.329) = 0.01039 (for 8 d.f.) P-value Standard Error t Stat P-value 98.24833 58.03348 1.69296 0.12892

0.10977 0.03297 3.32938 0.01039 Decision: P-value < so Reject H0 Conclusion: There is sufficient evidence that square footage affects house price Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-45 Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: b1 t n 2,/2sb1 1 b1 t n 2,/2sb1 d.f. = n - 2 Excel Printout for House Prices: Coefficients Intercept Square Feet Standard Error t Stat P-value

Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580 At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-46 Confidence Interval Estimate for the Slope

(continued) Coefficients Intercept Square Feet Standard Error t Stat P-value Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938 0.01039 0.03374

0.18580 Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the .05 level of significance Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-47 F-Test for Significance MSR F MSE F Test statistic: where MSR MSE SSR k SSE n k 1 where F follows an F distribution with k numerator and (n k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model) Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-48 Excel Output Regression Statistics Multiple R 0.76211 R Square 0.58082 Adjusted R Square 0.52842 Standard Error MSR 18934.9348 F 11.0848 MSE 1708.1957 41.33032 Observations 10 With 1 and 8 degrees of freedom P-value for the F-Test ANOVA

df SS MS F 11.0848 Regression 1 18934.9348 18934.9348 Residual 8 13665.5652 1708.1957 Total 9 32600.5000 Coefficients Intercept Square Feet Standard Error

t Stat P-value Significance F 0.01039 Lower 95% Upper 95% 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-49 F-Test for Significance (continued) Test Statistic: H0: 1 = 0 MSR F 11.08 MSE H1: 1 0 = .05 df1= 1 df2 = 8 Decision: Reject H0 at = 0.05 Critical Value: F = 5.32 Conclusion: = .05 0 Do not reject H0 Reject H0

F There is sufficient evidence that house size affects selling price F.05 = 5.32 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-50 Prediction The regression equation can be used to predict a value for y, given a particular x For a specified value, xn+1 , the predicted value is y n1 b0 b1x n1 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-51 Predictions Using Regression Analysis Predict the price for a house with 2000 square feet: house price 98.25 0.1098 (sq.ft.) 98.25 0.1098(2000) 317.85

The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-52 Relevant Data Range When using a regression model for prediction, only predict within the relevant range of data House Price ($1000s) Relevant data range 450 400 350 300 250 200 150 100 50 0 0 500 1000 1500 2000 2500

3000 Risky to try to extrapolate far beyond the range of observed Xs Square Feet Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-53 Estimating Mean Values and Predicting Individual Values Goal: Form intervals around y to express uncertainty about the value of y for a given xi Confidence Interval for the expected value of y, given xi Y y y = b0+b1xi Prediction Interval for an single observed y, given xi Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. xi

X Chap 12-54 Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular xi Confidence interval for E(Yn1 | Xn1 ) : y n1 t n 2,/2se 1 (x n1 x)2 2 n (x i x) Notice that the formula involves the term (x n1 x) 2 so the size of interval varies according to the distance xn+1 is from the mean, x Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-55 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular xi Confidence interval for y n1 : y n1 t n 2,/2 se 1 (x n1 x)2 1 2 n (x i x)

This extra term adds to the interval width to reflect the added uncertainty for an individual case Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-56 Estimation of Mean Values: Example Confidence Interval Estimate for E(Yn+1|Xn+1) Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price yi = 317.85 ($1,000s) y n1 t n-2,/2 se 1 (x i x)2 317.85 37.12 2 n (xi x) The confidence interval endpoints are 280.66 and 354.90, or from $280,660 to $354,900 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-57 Estimation of Individual Values: Example Confidence Interval Estimate for yn+1 Find the 95% confidence interval for an individual house with 2,000 square feet

Predicted Price yi = 317.85 ($1,000s) y n1 t n-1,/2se 1 (Xi X)2 1 317.85 102.28 2 n (Xi X) The confidence interval endpoints are 215.50 and 420.07, or from $215,500 to $420,070 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-58 Finding Confidence and Prediction Intervals in Excel In Excel, use PHStat | regression | simple linear regression Check the confidence and prediction interval for x= box and enter the x-value and confidence level desired Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-59 Finding Confidence and Prediction Intervals in Excel (continued)

Input values y Confidence Interval Estimate for E(Yn+1|Xn+1) Confidence Interval Estimate for individual yn+1 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-60 Graphical Analysis The linear regression model is based on minimizing the sum of squared errors If outliers exist, their potentially large squared errors may have a strong influence on the fitted regression line Be sure to examine your data graphically for outliers and extreme points Decide, based on your model and logic, whether the extreme points should remain or be removed Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Chap 12-61 Chapter Summary Introduced the linear regression model Reviewed correlation and the assumptions of linear regression Discussed estimating the simple linear regression coefficients Described measures of variation Described inference about the slope Addressed estimation of mean values and prediction of individual values Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 12-62