Statistics for Business and Economics 8 th Edition
Statistics for Business and Economics 8 th Edition Chapter 11 Simple Regression Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -1
Chapter Goals After completing this chapter, you should be able to: n n n Explain the simple linear regression model Obtain and interpret the simple linear regression equation for a set of data Describe R 2 as a measure of explanatory power of the regression model Understand the assumptions behind regression analysis Explain measures of variation and determine whether the independent variable is significant Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -2
Chapter Goals (continued) After completing this chapter, you should be able to: n n n 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 Explain the correlation coefficient and perform a hypothesis test for zero population correlation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -3
11. 1 n Overview of Linear Models An equation can be fit to show the best linear relationship between two variables: Y = β 0 + β 1 X Where Y is the dependent variable and X is the independent variable β 0 is the Y-intercept β 1 is the slope Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -4
Least Squares Regression n Estimates for coefficients β 0 and β 1 are found using a Least Squares Regression technique The least-squares regression line, based on sample data, is Where b 1 is the slope of the line and b 0 is the yintercept: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -5
Introduction to Regression Analysis n Regression analysis is used to: n n 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) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -6
11. 2 Linear Regression Model n n The relationship between X and Y is described by a linear function Changes in Y are assumed to be influenced by changes in X Linear regression population equation model Where 0 and 1 are the population model coefficients and is a random error term. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -7
Simple Linear Regression Model The population regression model: Population Y intercept Dependent Variable Population Slope Coefficient Linear component Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Independent Variable Random Error term Random Error component Ch. 11 -8
Linear Regression Assumptions n n n The true relationship form is linear (Y is a linear function of X, plus random error) The error terms, εi are independent of the x values The error terms are random variables with mean 0 and constant variance, σ2 (the uniform variance property is called homoscedasticity) n The random error terms, εi, are not correlated with one another, so that Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -9
Simple Linear Regression Model (continued) Y Observed Value of Y for xi εi Predicted Value of Y for xi Slope = β 1 Random Error for this Xi value Intercept = β 0 xi Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall X Ch. 11 -10
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 intercept Estimate of the regression slope Value of x for observation i The individual random error terms ei have a mean of zero Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -11
Least Squares Coefficient Estimators 11. 3 n b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared residuals (errors), SSE: Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -12
Least Squares Coefficient Estimators (continued) n The slope coefficient estimator is n And the constant or y-intercept is n The regression line always goes through the mean x, y Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -13
Computer Computation of Regression Coefficients n The coefficients b 0 and b 1 , and other regression results in this chapter, will be found using a computer n Hand calculations are tedious n Statistical routines are built into Excel n Other statistical analysis software can be used Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -14
Interpretation of the Slope and the Intercept n n b 0 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) b 1 is the estimated change in the average value of y as a result of a oneunit change in x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -15
Simple Linear Regression Example n n 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 n Dependent variable (Y) = house price in $1000 s n Independent variable (X) = square feet Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -16
Sample Data for House Price Model House Price in $1000 s (Y) Square Feet (X) 245 1400 312 1600 279 1700 308 1875 199 1100 219 1550 405 2350 324 2450 319 1425 255 1700 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -17
Sample Data for House Price Model House Price in $1000 s (Y) 1 245 2 312 3 279 4 308 5 199 6 219 7 405 8 324 9 319 10 255 Mean Y 286, 5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -18
Scatter Plot with no independent Variable Housing price with no independent variable 450 Housing Price ($1000) 400 350 300 250 200 150 1 2 3 4 5 6 7 8 9 10 House Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -19
Total Sum of Squares House Price in Mean of House Residuals (y$1000 s (y) Price (ybar) Sum of Squares (y-ybar)^2 1 245 286, 5 -41, 5 1722, 3 2 312 286, 5 25, 5 650, 3 3 279 286, 5 -7, 5 56, 3 4 308 286, 5 21, 5 462, 3 5 199 286, 5 -87, 5 7656, 3 6 219 286, 5 -67, 5 4556, 3 7 405 286, 5 118, 5 14042, 3 8 324 286, 5 37, 5 1406, 3 9 319 286, 5 32, 5 1056, 3 10 255 286, 5 -31, 5 992, 3 0 32600, 5 Sum Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -20
A Better Model? n n If there is no independent variable, our best prediction is the mean of the dependent variable, that is the average housing price. We can do a lot better!! I mean our prediction can be much better. We can reduce the sum of squared errors. Let us introduce the independent variable now. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -21
Scatter Plot 450 Housing Price and Square Footage Housing Price ($100) 400 350 300 250 200 150 500 1000 1500 2000 2500 3000 Square Footage Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -22
Calculating b 1 (y -ybar)(xxbar) (Y) (X) 245 1400 -41, 5 -315 13072, 5 99225 312 1600 25, 5 -115 -2932, 5 13225 279 1700 -7, 5 -15 112, 5 225 308 1875 21, 5 160 3440 25600 199 1100 -87, 5 -615 53812, 5 378225 219 1550 -67, 5 -165 11137, 5 27225 405 2350 118, 5 635 75247, 5 403225 324 2450 37, 5 735 27562, 5 540225 319 1425 255 1700 32, 5 -31, 5 -290 -15 -9425 472, 5 84100 225 172500 1571500 y -ybar x-xbar Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall (x-xbar)^2 Ch. 11 -23
Calculating Least Squares Coefficient Estimators n n n The slope coefficient estimator is b 1=172500 /1571500 =0, 109767738 And the constant or y-intercept is b 0=286, 5 - 0, 109767738*1715= 98, 2483296 Our linear regression equation is ycap=98, 2483296+ 0, 109767738 x n Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -24
Graphical Presentation n House price model: scatter plot Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -25
Regression Squared Error Observed Predicted Housing Price (y) (ycap) 245 251, 92633 House Square Footage(X) 1 1400 2 1600 312 273, 88033 3 1700 279 284, 85733 4 1875 308 304, 06708 5 1100 199 218, 99533 6 1550 219 268, 39183 7 2350 405 356, 20783 8 2450 324 367, 18483 9 1425 319 254, 67058 10 1700 255 284, 85733 Error (y-ycap) SSE Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Squared Error (y-ycap)^2 -6, 9263 47, 974047 38, 1197 1453, 1092 -5, 8573 34, 308315 3, 93292 15, 46786 -19, 995 399, 81322 -49, 392 2439, 5529 48, 7922 2380, 6759 -43, 185 1864, 9295 64, 3294 4138, 2743 -29, 857 891, 46015 13665, 565 Ch. 11 -26
Coefficient of Determination n n n n SST=32600, 5 SSE=13665, 565 SSR=SST-SSE= 18934, 9346 R^2= SSR/SST= 0, 58081731 NOT BAD How well does the estimated regression fit the data? Idea: Splitting SST between SSE and SSR 58 % of total sum of squares can be explained by using the regression equation to predict the housing price. The remainder is the error. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -27
Regression Using Excel n Excel will be used to generate the coefficients and measures of goodness of fit for regression n Data / Data Analysis / Regression Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -28
Regression Using Excel n Data / Data Analysis / Regression (continued) Provide desired input: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -29
Excel Output Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -30
Excel Output (continued) Regression Statistics Multiple R 0. 76211 R Square 0. 58082 Adjusted R Square 0. 52842 Standard Error The regression equation is: 41. 33032 Observations 10 ANOVA df SS MS Regression 1 18934. 9348 Residual 8 13665. 5652 1708. 1957 Total 9 32600. 5000 Intercept Square Feet Coefficients Standard Error F 11. 0848 t Stat Significance F 0. 01039 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -31
Graphical Presentation n House price model: scatter plot and regression line Slope = 0. 10977 Intercept = 98. 248 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -32
Interpretation of the Intercept, b 0 n b 0 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) n Here, no houses had 0 square feet, so b 0 = 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -33
Interpretation of the Slope Coefficient, b 1 n b 1 measures the estimated change in the average value of Y as a result of a oneunit change in X n Here, b 1 =. 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -34
11. 4 n Explanatory Power of a Linear Regression Equation Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error (residual) Sum of Squares where: = Average value of the dependent variable yi = Observed values of the dependent variable i = Predicted value of y for the given xi value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -35
Analysis of Variance n SST = total sum of squares n n SSR = regression sum of squares n n 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 n Variation attributable to factors other than the linear relationship between x and y Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -36
Analysis of Variance (continued) Y yi Unexplained variation 2 SSE = (yi - yi ) y _ y SST = (yi - y)2 Explained variation _2 SSR = (yi - y) _ y xi Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall _ y X Ch. 11 -37
Coefficient of Determination, R 2 n n 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 R 2 note: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -38
Examples of Approximate r 2 Values Y r 2 = 1 X 100% of the variation in Y is explained by variation in X Y r 2 = 1 Perfect linear relationship between X and Y: X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -39
Examples of Approximate r 2 Values Y 0 < r 2 < 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -40
Examples of Approximate r 2 Values r 2 = 0 Y No linear relationship between X and Y: r 2 = 0 X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) Ch. 11 -41
Excel Output Regression Statistics Multiple R 0. 76211 R Square 0. 58082 Adjusted R Square 0. 52842 Standard Error 58. 08% of the variation in house prices is explained by variation in square feet 41. 33032 Observations 10 ANOVA df SS MS Regression 1 18934. 9348 Residual 8 13665. 5652 1708. 1957 Total 9 32600. 5000 Intercept Square Feet Coefficients Standard Error F 11. 0848 t Stat Significance F 0. 01039 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -42
Correlation and R 2 n The coefficient of determination, R 2, for a simple regression is equal to the simple correlation squared Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -43
Estimation of Model Error Variance n n An estimator for the variance of the population model error is Division by n – 2 instead of n – 1 is because the simple regression model uses two estimated parameters, b 0 and b 1, instead of one is called the standard error of the estimate Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -44
Excel Output Regression Statistics Multiple R 0. 76211 R Square 0. 58082 Adjusted R Square 0. 52842 Standard Error 41. 33032 Observations 10 ANOVA df SS MS Regression 1 18934. 9348 Residual 8 13665. 5652 1708. 1957 Total 9 32600. 5000 Intercept Square Feet Coefficients Standard Error F 11. 0848 t Stat Significance F 0. 01039 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -45
Comparing Standard Errors se is a measure of the variation of observed y values from the regression line Y Y X 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. 33 K is moderately small relative to house prices in the $200 - $300 K range Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -46
11. 5 n Statistical Inference: Hypothesis Tests and Confidence Intervals The variance of the regression slope coefficient (b 1) is estimated by where: = Estimate of the standard error of the least squares slope = Standard error of the estimate Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -47
11. 5 n 1. 2. Statistical Inference: Hypothesis Tests and Confidence Intervals The variance of the regression slope coefficient (b 1) depends on two things: The distance of the points from the regression line measured by s_e. Higher values imply greater variance The total deviation of X values from the mean. The greater the spread in X values, the smaller the variance for the slope coefficient. It follows that smaller variance slope coefficient estimators imply a better regression model. Change in Y resulting from change in X is estimated by b_1. So, Its variance should be as small as possible. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -48
Excel Output Regression Statistics Multiple R 0. 76211 R Square 0. 58082 Adjusted R Square 0. 52842 Standard Error 41. 33032 Observations 10 ANOVA df SS MS Regression 1 18934. 9348 Residual 8 13665. 5652 1708. 1957 Total 9 32600. 5000 Intercept Square Feet Coefficients Standard Error F 11. 0848 t Stat Significance F 0. 01039 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -49
Comparing Standard Errors of the Slope is a measure of the variation in the slope of regression lines from different possible samples Y Y X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall X Ch. 11 -50
Inference about the Slope: t Test n t test for a population slope n n Is there a linear relationship between X and Y? Null and alternative hypotheses H 0: β 1 = 0 H 1: β 1 0 n (no linear relationship) (linear relationship does exist) Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope sb 1 = standard error of the slope Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -51
Inference about the Slope: t Test (continued) House Price in $1000 s (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: The slope of this model is 0. 1098 Does square footage of the house significantly affect its sales price? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -52
Inferences about the Slope: t Test Example H 0: β 1 = 0 H 1: β 1 0 From Excel output: Intercept Square Feet Coefficients b 1 Standard Error t Stat P-value 98. 24833 58. 03348 1. 69296 0. 12892 0. 10977 0. 03297 3. 32938 0. 01039 t Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -53
Inferences about the Slope: t Test Example (continued) Test Statistic: t = 3. 329 H 0: β 1 = 0 H 1: β 1 0 From Excel output: Intercept Square Feet d. f. = 10 -2 = 8 t 8, . 025 = 2. 3060 /2=. 025 Reject H 0 /2=. 025 Do not reject H 0 -tn-2, α/2 -2. 3060 0 Reject H 0 tn-2, α/2 2. 3060 3. 329 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Coefficients b 1 Standard Error t t Stat P-value 98. 24833 58. 03348 1. 69296 0. 12892 0. 10977 0. 03297 3. 32938 0. 01039 Decision: Reject H 0 Conclusion: There is sufficient evidence that square footage affects house price Ch. 11 -54
Inferences about the Slope: t Test Example (continued) P-value = 0. 01039 H 0: β 1 = 0 H 1: β 1 0 P-value From Excel output: 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. ) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Coefficients 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 < α so Reject H 0 Conclusion: There is sufficient evidence that square footage affects house price Ch. 11 -55
Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: d. f. = n - 2 Excel Printout for House Prices: Coefficients Standard Error Intercept 98. 24833 0. 10977 Square Feet t Stat P-value Lower 95% Upper 95% 58. 03348 1. 69296 0. 12892 -35. 57720 232. 07386 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) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -56
Confidence Interval Estimate for the Slope (continued) Coefficients Standard Error Intercept 98. 24833 0. 10977 Square Feet t Stat P-value Lower 95% Upper 95% 58. 03348 1. 69296 0. 12892 -35. 57720 232. 07386 0. 03297 3. 32938 0. 01039 0. 03374 0. 18580 Since the units of the house price variable is $1000 s, 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -57
Hypothesis Test for Population Slope Using the F Distribution n F Test statistic: 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) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -58
Hypothesis Test for Population Slope Using the F Distribution (continued) n An alternate test for the hypothesis that the slope is zero: H 0: β 1 = 0 H 1: β 1 0 n n Use the F statistic The decision rule is reject H 0 if F ≥ F 1, n-2, α Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -59
Excel Output Regression Statistics Multiple R 0. 76211 R Square 0. 58082 Adjusted R Square 0. 52842 Standard Error 41. 33032 Observations 10 With 1 and 8 degrees of freedom P-value for the F-Test ANOVA df SS MS Regression 1 18934. 9348 Residual 8 13665. 5652 1708. 1957 Total 9 32600. 5000 Intercept Square Feet Coefficients Standard Error F 11. 0848 t Stat Significance F 0. 01039 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -60
F-Test for Significance (continued) Test Statistic: H 0: β 1 = 0 H 1: β 1 ≠ 0 =. 05 df 1= 1 df 2 = 8 Decision: Reject H 0 at = 0. 05 Critical Value: F 1, 8, 0. 05 = 5. 32 Conclusion: =. 05 0 Do not reject H 0 Reject H 0 F There is sufficient evidence that house size affects selling price F. 05 = 5. 32 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -61
11. 6 n n 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -62
Predictions Using Regression Analysis Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317. 85($1, 000 s) = $317, 850 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -63
Relevant Data Range n When using a regression model for prediction, only predict within the relevant range of data Relevant data range Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Risky to try to extrapolate far beyond the range of observed x values Ch. 11 -64
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 = b 0+b 1 xi Prediction Interval for an single observed y, given xi Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall xi X Ch. 11 -65
Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular xi Notice that the formula involves the term so the size of interval varies according to the distance xn+1 is from the mean, x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -66
Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular xi This extra term adds to the interval width to reflect the added uncertainty for an individual case Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -67
Example: Confidence Interval for the Average Y, Given X 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, 000 s) The confidence interval endpoints are 280. 73 and 354. 97, or from $280, 730 to $354, 970 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -68
Example: Prediction Interval for an Individual Y, Given X 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, 000 s) The confidence interval endpoints are 215. 57 and 420. 13, or from $215, 570 to $420, 130 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -69
Correlation Analysis 11. 7 n Correlation analysis is used to measure strength of the association (linear relationship) between two variables n Correlation is only concerned with strength of the relationship n No causal effect is implied with correlation n Correlation was first presented in Chapter 4 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -70
Correlation Analysis n n The population correlation coefficient is denoted ρ (the Greek letter rho) The sample correlation coefficient is where Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -71
Test for Zero Population Correlation n To test the null hypothesis of no linear association, the test statistic follows the Student’s t distribution with (n – 2 ) degrees of freedom: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -72
Decision Rules Hypothesis Test for Correlation Lower-tail test: Upper-tail test: Two-tail test: H 0: ρ 0 H 1: ρ < 0 H 0: ρ ≤ 0 H 1: ρ > 0 H 0: ρ = 0 H 1: ρ ≠ 0 -t t Reject H 0 if t < -tn-2, Reject H 0 if t > tn-2, Where Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall /2 -t /2 Reject H 0 if t < -tn-2, /2 or t > tn-2, /2 has n - 2 d. f. Ch. 11 -73
11. 8 n n Beta Measure of Financial Risk A Beta Coefficient is a measure of how the returns of a particular firm respond to the returns of a broad stock index (such as the S&P 500) For a specific firm, the Beta Coefficient is the slope coefficient from a regression of the firm’s returns compared to the overall market returns over some specified time period Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -74
Beta Coefficient Example n Slope coefficient is the Beta Coefficient Information about the quality of the regression model that provides the estimate of beta Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -75
11. 9 n n 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 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -76
Chapter Summary n n n Introduced the linear regression model Reviewed correlation and the assumptions of linear regression Discussed estimating the simple linear regression coefficients n Described measures of variation n Described inference about the slope n Addressed estimation of mean values and prediction of individual values Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -77
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11 -78
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