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© 2000 Prentice-Hall, Inc. Statistics Simple Linear Regression Chapter 11 11 - 1 Learning Objectives © 2000 Prentice-Hall, Inc. 1. Describe the Linear Regression Model 2. State the Regression Modeling Steps 3. Explain Ordinary Least Squares 4. Compute Regression Coefficients 5. Predict Response Variable 6. Interpret Computer Output 11 - 2 © 2000 Prentice-Hall, Inc. Models 11 - 3 Models © 2000 Prentice-Hall, Inc. 1. Representation of Some Phenomenon 2. Mathematical Model Is a Mathematical Expression of Some Phenomenon 3. Often Describe Relationships between Variables 4. Types Deterministic Models Probabilistic Models 11 - 4 Deterministic Models © 2000 Prentice-Hall, Inc. 1. Hypothesize Exact Relationships 2. Suitable When Prediction Error is Negligible 3. Example: Force Is Exactly Mass Times Acceleration F = m·a © 1984-1994 T/Maker Co. 11 - 5 Probabilistic Models © 2000 Prentice-Hall, Inc. 1. Hypothesize 2 Components Deterministic Random Error 2. Example: Sales Volume Is 10 Times Advertising Spending + Random Error Y = 10X + Random Error May Be Due to Factors Other Than Advertising 11 - 6 © 2000 Prentice-Hall, Inc. Types of Probabilistic Models Probabilistic Models Regression Models 11 - 7 Correlation Models Other Models © 2000 Prentice-Hall, Inc. Regression Models 11 - 8 © 2000 Prentice-Hall, Inc. Types of Probabilistic Models Probabilistic Models Regression Models 11 - 9 Correlation Models Other Models Regression Models © 2000 Prentice-Hall, Inc. 1. Answer ‘What Is the Relationship Between the Variables?’ 2. Equation Used 1 Numerical Dependent (Response) Variable What Is to Be Predicted 1 or More Numerical or Categorical Independent (Explanatory) Variables 3. Used Mainly for Prediction & Estimation 11 - 10 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 11 © 2000 Prentice-Hall, Inc. Model Specification 11 - 12 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 13 Specifying the Model © 2000 Prentice-Hall, Inc. 1. Define Variables Conceptual (e.g., Advertising, Price) Empirical (e.g., List Price, Regular Price) Measurement (e.g., $, Units) 2. Hypothesize Nature of Relationship Expected Effects (i.e., Coefficients’ Signs) Functional Form (Linear or Non-Linear) Interactions 11 - 14 © 2000 Prentice-Hall, Inc. 1. 2. 3. 4. Model Specification Is Based on Theory Theory of Field (e.g., Sociology) Mathematical Theory Previous Research ‘Common Sense’ 11 - 15 © 2000 Prentice-Hall, Inc. Thinking Challenge: Which Is More Logical? Sales Sales Advertising Sales Advertising Sales Advertising 11 - 16 Advertising © 2000 Prentice-Hall, Inc. 11 - 17 Types of Regression Models © 2000 Prentice-Hall, Inc. Types of Regression Models Regression Models 11 - 18 Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Simple 11 - 19 Regression Models Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Simple 11 - 20 Regression Models 2+ Explanatory Variables Multiple Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Simple Linear 11 - 21 Regression Models 2+ Explanatory Variables Multiple Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Regression Models Multiple Simple Linear 11 - 22 2+ Explanatory Variables NonLinear Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Regression Models 2+ Explanatory Variables Multiple Simple Linear 11 - 23 NonLinear Linear Types of Regression Models © 2000 Prentice-Hall, Inc. 1 Explanatory Variable Regression Models 2+ Explanatory Variables Multiple Simple Linear 11 - 24 NonLinear Linear NonLinear © 2000 Prentice-Hall, Inc. Linear Regression Model 11 - 25 © 2000 Prentice-Hall, Inc. Types of Regression Models 1 Explanatory Variable Regression Models 2+ Explanatory Variables Multiple Simple Linear 11 - 26 NonLinear Linear NonLinear Linear Equations © 2000 Prentice-Hall, Inc. Y Y = mX + b m = Slope Change in Y Change in X b = Y-intercept X High School Teacher © 1984-1994 T/Maker Co. 11 - 27 Linear Regression Model © 2000 Prentice-Hall, Inc. 1. Relationship Between Variables Is a Linear Function Population Y-Intercept Population Slope Random Error Yi 0 1X i i Dependent (Response) Variable 11 - 28 Independent (Explanatory) Variable © 2000 Prentice-Hall, Inc. 11 - 29 Population & Sample Regression Models Population & Sample Regression Models © 2000 Prentice-Hall, Inc. Population $ $ $ $ $ 11 - 30 Population & Sample Regression Models © 2000 Prentice-Hall, Inc. Population Unknown Relationship $ Yi 0 1X i i $ $ $ $ 11 - 31 Population & Sample Regression Models © 2000 Prentice-Hall, Inc. Population Random Sample Unknown Relationship $ Yi 0 1X i i $ $ $ $ 11 - 32 $ $ Population & Sample Regression Models © 2000 Prentice-Hall, Inc. Population Unknown Relationship $ Yi 0 1X i i $ $ $ $ 11 - 33 Random Sample Yi 0 1X i i $ $ © 2000 Prentice-Hall, Inc. Population Linear Regression Model Y Yi 0 1X i i Observed value i = Random error af E Y 0 1X i X Observed value 11 - 34 © 2000 Prentice-Hall, Inc. Sample Linear Regression Model Y Yi 0 1X i i ^i = Random error Yi 0 1X i Unsampled observation X Observed value 11 - 35 © 2000 Prentice-Hall, Inc. Estimating Parameters: Least Squares Method 11 - 36 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 37 Scattergram © 2000 Prentice-Hall, Inc. 1. Plot of All (Xi, Yi) Pairs 2. Suggests How Well Model Will Fit 60 40 20 0 Y 0 11 - 38 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 39 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 40 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 41 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 42 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 43 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 44 20 40 X 60 Thinking Challenge © 2000 Prentice-Hall, Inc. How would you draw a line through the points? How do you determine which line ‘fits best’? 60 40 20 0 Y 0 11 - 45 20 40 X 60 Least Squares © 2000 Prentice-Hall, Inc. 1. ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative 11 - 46 Least Squares © 2000 Prentice-Hall, Inc. 1. ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative n e i 1 11 - 47 Yi Yi 2 j n 2 i i 1 Least Squares © 2000 Prentice-Hall, Inc. 1. ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative n e i 1 Yi Yi 2 j n 2 i i 1 2. LS Minimizes the Sum of the Squared Differences (SSE) 11 - 48 Least Squares Graphically © 2000 Prentice-Hall, Inc. n 2 2 2 2 2 LS minimizes i 1 2 3 4 i 1 Y2 0 1X 2 2 Y ^4 ^2 ^1 ^3 Yi 0 1X i X 11 - 49 Coefficient Equations © 2000 Prentice-Hall, Inc. Prediction Equation Y X i 0 1 n n i Sample Slope F I F I X JG YJ G H K H K XY 1 i 1 i 1 i i n F I XJ G H K 2 n 2 X i i 1 11 - 50 i i 1 n Sample Y-intercept n 0 Y 1X i 1 n i i Computation Table © 2000 Prentice-Hall, Inc. Xi X1 Yi 2 Xi 2 Yi XiYi Y1 X1 2 Y1 2 X1Y1 2 Y2 2 X2Y2 X2 Y2 X2 : : : : : Xn Yn Xn2 Yn2 XnYn Xi Yi Xi2 Yi2 XiYi 11 - 51 © 2000 Prentice-Hall, Inc. 11 - 52 Interpretation of Coefficients Interpretation of Coefficients © 2000 Prentice-Hall, Inc. ^ 1. Slope (1) Estimated Y Changes by ^1 for Each 1 Unit Increase in X 11 - 53 If ^1 = 2, then Sales (Y) Is Expected to Increase by 2 for Each 1 Unit Increase in Advertising (X) Interpretation of Coefficients © 2000 Prentice-Hall, Inc. ^ 1. Slope (1) Estimated Y Changes by ^1 for Each 1 Unit Increase in X If ^1 = 2, then Sales (Y) Is Expected to Increase by 2 for Each 1 Unit Increase in Advertising (X) ^ 2. Y-Intercept (0) Average Value of Y When X = 0 11 - 54 ^ If 0 = 4, then Average Sales (Y) Is Expected to Be 4 When Advertising (X) Is 0 © 2000 Prentice-Hall, Inc. Parameter Estimation Example You’re a marketing analyst for Hasbro Toys. You gather the following data: Ad $ Sales (Units) 1 1 2 1 3 2 4 2 5 4 What is the relationship between sales & advertising? 11 - 55 Scattergram Sales vs. Advertising © 2000 Prentice-Hall, Inc. Sales 4 3 2 1 0 0 1 2 3 Advertising 11 - 56 4 5 Parameter Estimation Solution Table © 2000 Prentice-Hall, Inc. Xi Yi Xi2 Yi2 XiYi 1 1 1 1 1 2 1 4 1 2 3 2 9 4 6 4 2 16 4 8 5 4 25 16 20 15 10 55 26 37 11 - 57 Parameter Estimation Solution © 2000 Prentice-Hall, Inc. F I F I X JG YJ G H K H K 15 fa 10f a X Y 37 n n 1 i 1 n i i 1 i 1 i i n F I XJ G H K 2 n n i 1 X i2 i i 1 i 5 15 55 5 n a faf 0 Y 1X 2 0.70 3 0.10 11 - 58 af 2 0.70 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution 11 - 59 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution ^ 1. Slope (1) Sales Volume (Y) Is Expected to Increase by .7 Units for Each $1 Increase in Advertising (X) 11 - 60 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution ^ 1. Slope (1) Sales Volume (Y) Is Expected to Increase by .7 Units for Each $1 Increase in Advertising (X) ^ 2. Y-Intercept (0) Average Value of Sales Volume (Y) Is -.10 Units When Advertising (X) Is 0 Difficult to Explain to Marketing Manager Expect Some Sales Without Advertising 11 - 61 Parameter Estimation Computer Output © 2000 Prentice-Hall, Inc. ^ Parameter Estimates k Parameter Standard T for H0: Variable DF Estimate Error Param=0 INTERCEP 1 -0.1000 0.6350 -0.157 ADVERT 1 0.7000 0.1914 3.656 ^0 11 - 62 ^1 Prob>|T| 0.8849 0.0354 © 2000 Prentice-Hall, Inc. Parameter Estimation Thinking Challenge You’re an economist for the county cooperative. You gather the following data: Fertilizer (lb.) Yield (lb.) 4 3.0 6 5.5 10 6.5 12 9.0 What is the relationship between fertilizer & crop yield? © 1984-1994 T/Maker Co. 11 - 63 Scattergram Crop Yield vs. Fertilizer* © 2000 Prentice-Hall, Inc. Yield (lb.) 10 8 6 4 2 0 0 5 10 Fertilizer (lb.) 11 - 64 15 Parameter Estimation Solution Table* © 2000 Prentice-Hall, Inc. Xi Yi 2 Xi 4 3.0 16 9.00 12 6 5.5 36 30.25 33 10 6.5 100 42.25 65 12 9.0 144 81.00 108 32 24.0 296 162.50 218 11 - 65 2 Yi XiYi Parameter Estimation Solution* © 2000 Prentice-Hall, Inc. F I F I X JG YJ G H K H K 32fa 24 f a X Y 218 n n 1 i 1 n i i 1 i 1 i i n F I XJ G H K 2 n n i 1 X i2 i i 1 i 4 32 296 4 n a faf 0 Y 1X 6 0.65 8 0.80 11 - 66 af 2 0.65 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution* 11 - 67 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution* ^ 1. Slope (1) Crop Yield (Y) Is Expected to Increase by .65 lb. for Each 1 lb. Increase in Fertilizer (X) 11 - 68 © 2000 Prentice-Hall, Inc. Coefficient Interpretation Solution* ^ 1. Slope (1) Crop Yield (Y) Is Expected to Increase by .65 lb. for Each 1 lb. Increase in Fertilizer (X) ^ 2. Y-Intercept (0) Average Crop Yield (Y) Is Expected to Be 0.8 lb. When No Fertilizer (X) Is Used 11 - 69 © 2000 Prentice-Hall, Inc. Probability Distribution of Random Error 11 - 70 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 71 © 2000 Prentice-Hall, Inc. Linear Regression Assumptions 1. Mean of Probability Distribution of Error Is 0 2. Probability Distribution of Error Has Constant Variance 3. Probability Distribution of Error is Normal 4. Errors Are Independent 11 - 72 © 2000 Prentice-Hall, Inc. Error Probability Distribution ^ f() Y X2 X1 X 11 - 73 Random Error Variation © 2000 Prentice-Hall, Inc. 11 - 74 Random Error Variation © 2000 Prentice-Hall, Inc. 1. Variation of Actual Y from Predicted Y 11 - 75 Random Error Variation © 2000 Prentice-Hall, Inc. 1. Variation of Actual Y from Predicted Y 2. Measured by Standard Error of Regression Model Sample Standard Deviation of ^, s 11 - 76 Random Error Variation © 2000 Prentice-Hall, Inc. 1. Variation of Actual Y from Predicted Y 2. Measured by Standard Error of Regression Model Sample Standard Deviation of ^, s 3. Affects Several Factors Parameter Significance Prediction Accuracy 11 - 77 Measures of Variation in Regression © 2000 Prentice-Hall, Inc. 1. Total Sum of Squares (SSyy) Measures Variation of Observed Yi Around the MeanY 2. Explained Variation (SSR) Variation Due to Relationship Between X&Y 3. Unexplained Variation (SSE) Variation Due to Other Factors 11 - 78 Variation Measures © 2000 Prentice-Hall, Inc. Y Yi Total sum of squares (Yi -Y)2 Unexplained sum ^ )2 of squares (Yi - Y i Yi 0 1X i Explained sum of ^ squares (Yi -Y)2 Y Xi 11 - 79 X Coefficient of Determination © 2000 Prentice-Hall, Inc. 1. Proportion of Variation ‘Explained’ by Relationship Between X & Y 0 r2 1 Explained Variation r Total Variation 2 n i 1 i 1 n Yi Y cYi Y h i 1 11 - 80 n cYi Y h e 2 2 2 j © 2000 Prentice-Hall, Inc. Y Coefficient of Determination Examples Y r2 = 1 r2 = 1 X Y Y r2 = .8 X 11 - 81 X r2 = 0 X © 2000 Prentice-Hall, Inc. Coefficient of Determination Example You’re a marketing analyst for Hasbro Toys. You find 0^ = -0.1 & ^1 = 0.7. Ad $ Sales (Units) 1 1 2 1 3 2 4 2 5 4 Interpret a coefficient of determination of 0.8167. 11 - 82 r 2 Computer Output © 2000 Prentice-Hall, Inc. r2 Root MSE Dep Mean C.V. S 11 - 83 0.60553 2.00000 30.27650 R-square Adj R-sq 0.8167 0.7556 r2 adjusted for number of explanatory variables & sample size © 2000 Prentice-Hall, Inc. Evaluating the Model Testing for Significance 11 - 84 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 85 Test of Slope Coefficient © 2000 Prentice-Hall, Inc. 1. Shows If There Is a Linear Relationship Between X & Y 2. Involves Population Slope 1 3. Hypotheses H0: 1 = 0 (No Linear Relationship) Ha: 1 0 (Linear Relationship) 4. Theoretical Basis Is Sampling Distribution of Slope 11 - 86 © 2000 Prentice-Hall, Inc. 11 - 87 Sampling Distribution of Sample Slopes Sampling Distribution of Sample Slopes © 2000 Prentice-Hall, Inc. Y Sample 1 Line Sample 2 Line Population Line X 11 - 88 Sampling Distribution of Sample Slopes © 2000 Prentice-Hall, Inc. Y Sample 1 Line Sample 2 Line Population Line X 11 - 89 All Possible Sample Slopes Sample 1: 2.5 Sample 2: 1.6 Sample 3: 1.8 Sample 4: 2.1 : : Very large number of sample slopes Sampling Distribution of Sample Slopes © 2000 Prentice-Hall, Inc. Y Sample 1 Line Sample 2 Line Population Line X Sampling Distribution S^1 1 11 - 90 ^ 1 All Possible Sample Slopes Sample 1: 2.5 Sample 2: 1.6 Sample 3: 1.8 Sample 4: 2.1 : : Very large number of sample slopes © 2000 Prentice-Hall, Inc. Slope Coefficient Test Statistic t n2 1 1 S 1 where S S 1 n 2 X i i 1 11 - 91 F I X J G H K 2 n i i 1 n © 2000 Prentice-Hall, Inc. Test of Slope Coefficient Example You’re a marketing analyst for Hasbro Toys. You find b0 = -.1, b1 = .7 & s = .60553. Ad $ Sales (Units) 1 1 2 1 3 2 4 2 5 4 Is the relationship significant at the .05 level? 11 - 92 Solution Table © 2000 Prentice-Hall, Inc. Xi Yi 2 Xi 1 1 1 1 1 2 1 4 1 2 3 2 9 4 6 4 2 16 4 8 5 4 25 16 20 15 10 55 26 37 11 - 93 2 Yi XiYi © 2000 Prentice-Hall, Inc. Test of Slope Parameter Solution H0: 1 = 0 Ha: 1 0 .05 df 5 - 2 = 3 Critical Value(s): Reject .025 t Reject .025 -3.1824 0 3.1824 11 - 94 Test Statistic: t 1 1 S 0.70 0 3.656 0.1915 1 Decision: Reject at = .05 Conclusion: There is evidence of a relationship Test Statistic Solution © 2000 Prentice-Hall, Inc. t n2 1 1 S 0.70 0 3.656 0.1915 1 where S S 1 n i 1 11 - 95 F I X J G H K 2 n 2 Xi i i 1 n 0.60553 15 f a 55 3 5 0.1915 Test of Slope Parameter Computer Output © 2000 Prentice-Hall, Inc. Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob>|T| INTERCEP 1 -0.1000 0.6350 -0.157 0.8849 ADVERT 1 0.7000 0.1914 3.656 0.0354 ^ k S^ k t = ^k / S^ k P-Value 11 - 96 © 2000 Prentice-Hall, Inc. Using the Model for Prediction & Estimation 11 - 97 Regression Modeling Steps © 2000 Prentice-Hall, Inc. 1. Hypothesize Deterministic Component 2. Estimate Unknown Model Parameters 3. Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error 4. Evaluate Model 5. Use Model for Prediction & Estimation 11 - 98 Prediction With Regression Models © 2000 Prentice-Hall, Inc. 1. Types of Predictions Point Estimates Interval Estimates 2. What Is Predicted Population Mean Response E(Y) for Given X Point on Population Regression Line Individual Response (Yi) for Given X 11 - 99 What Is Predicted © 2000 Prentice-Hall, Inc. Y YIndividual Mean Y, E(Y) ^ 0 + ^Y i= ^ 1X E(Y) = 0 + 1X Prediction,^Y XP 11 - 100 X © 2000 Prentice-Hall, Inc. Confidence Interval Estimate of Mean Y Y t n 2, / 2 SY E (Y ) Y t n 2, / 2 SY where cX X h cX X h 2 SY S 1 n p n i 1 11 - 101 2 i Factors Affecting Interval Width © 2000 Prentice-Hall, Inc. 1. Level of Confidence (1 - ) Width Increases as Confidence Increases 2. Data Dispersion (s) Width Increases as Variation Increases 3. Sample Size Width Decreases as Sample Size Increases 4. Distance of Xp from MeanX Width Increases as Distance Increases 11 - 102 Why Distance from Mean? © 2000 Prentice-Hall, Inc. Y m a S _ Y 1 e l p e n i L Sample 2 X1 11 - 103 X Greater dispersion than X1 Line X2 X © 2000 Prentice-Hall, Inc. Confidence Interval Estimate Example You’re a marketing analyst for Hasbro Toys. You find b0 = -.1, b1 = .7 & s = .60553. Ad $ Sales (Units) 1 1 2 1 3 2 4 2 5 4 Estimate the mean sales when advertising is $4 at the .05 level. 11 - 104 Solution Table © 2000 Prentice-Hall, Inc. Xi Yi Xi2 Yi2 XiYi 1 1 1 1 1 2 1 4 1 2 3 2 9 4 6 4 2 16 4 8 5 4 25 16 20 15 10 55 26 37 11 - 105 © 2000 Prentice-Hall, Inc. Confidence Interval Estimate Solution Y t n 2, / 2 SY E (Y ) Y t n 2, / 2 SY a faf 4 3f 1 a .60553 Y 0.1 0.7 4 2.7 X to be predicted 2 SY a 5 fa 10 f 0.3316 a fa 2.7 3.1824 0.3316 E (Y ) 2.7 3.1824 0.3316 1.6445 E (Y ) 3.7553 11 - 106 f © 2000 Prentice-Hall, Inc. Prediction Interval of Individual Response Y t n 2, / 2 S Y Y YP Y t n 2, / 2 S Y Y e j e j where cX X h cX X h 2 1 S Y Y S 1 e j n P n i 1 Note! 11 - 107 2 i Why the Extra ‘S’? © 2000 Prentice-Hall, Inc. Y Y we're trying to predict Expected (Mean) Y + ^ ^= 0 ^ 1X i Yi E(Y) = 0 + 1X Prediction, ^ Y XP 11 - 108 X Interval Estimate Computer Output © 2000 Prentice-Hall, Inc. Dep Var Obs SALES 1 1.000 2 1.000 3 2.000 4 2.000 5 4.000 Pred Std Err Low95% Upp95% Low95% Upp95% Value Predict Mean Mean Predict Predict 0.600 0.469 -0.892 2.092 -1.837 3.037 1.300 0.332 0.244 2.355 -0.897 3.497 2.000 0.271 1.138 2.861 -0.111 4.111 2.700 0.332 1.644 3.755 0.502 4.897 3.400 0.469 1.907 4.892 0.962 5.837 Predicted Y when X = 4 11 - 109 SY^ Confidence Interval Prediction Interval Hyperbolic Interval Bands © 2000 Prentice-Hall, Inc. Y ^ ^= 0 Xi ^ 1 + Yi _ X 11 - 110 X XP © 2000 Prentice-Hall, Inc. Correlation Models 11 - 111 © 2000 Prentice-Hall, Inc. Types of Probabilistic Models Probabilistic Models Regression Models 11 - 112 Correlation Models Other Models Correlation Models © 2000 Prentice-Hall, Inc. 1. Answer ‘How Strong Is the Linear Relationship Between 2 Variables?’ 2. Coefficient of Correlation Used Population Correlation Coefficient Denoted (Rho) Values Range from -1 to +1 Measures Degree of Association 3. Used Mainly for Understanding 11 - 113 Sample Coefficient of Correlation © 2000 Prentice-Hall, Inc. 1. Pearson Product Moment Coefficient of Correlation, r: r Coefficient of Determination n cYi Y h cX i X h i 1 n i 1 11 - 114 n cX i X h cYi Y h 2 i 1 2 © 2000 Prentice-Hall, Inc. 11 - 115 Coefficient of Correlation Values © 2000 Prentice-Hall, Inc. -1.0 11 - 116 Coefficient of Correlation Values -.5 0 +.5 +1.0 © 2000 Prentice-Hall, Inc. Coefficient of Correlation Values No Correlation -1.0 11 - 117 -.5 0 +.5 +1.0 © 2000 Prentice-Hall, Inc. Coefficient of Correlation Values No Correlation -1.0 -.5 Increasing degree of negative correlation 11 - 118 0 +.5 +1.0 © 2000 Prentice-Hall, Inc. Coefficient of Correlation Values Perfect Negative Correlation -1.0 11 - 119 No Correlation -.5 0 +.5 +1.0 © 2000 Prentice-Hall, Inc. Coefficient of Correlation Values Perfect Negative Correlation -1.0 No Correlation -.5 0 +.5 +1.0 Increasing degree of positive correlation 11 - 120 © 2000 Prentice-Hall, Inc. Coefficient of Correlation Values Perfect Negative Correlation -1.0 11 - 121 Perfect Positive Correlation No Correlation -.5 0 +.5 +1.0 © 2000 Prentice-Hall, Inc. Y Coefficient of Correlation Examples r=1 Y r = -1 X Y r = .89 Y X 11 - 122 X r=0 X © 2000 Prentice-Hall, Inc. Test of Coefficient of Correlation 1. Shows If There Is a Linear Relationship Between 2 Numerical Variables 2. Same Conclusion as Testing Population Slope 1 3. Hypotheses H0: = 0 (No Correlation) Ha: 0 (Correlation) 11 - 123 Conclusion © 2000 Prentice-Hall, Inc. 1. Described the Linear Regression Model 2. Stated the Regression Modeling Steps 3. Explained Ordinary Least Squares 4. Computed Regression Coefficients 5. Predicted Response Variable 6. Interpreted Computer Output 11 - 124 End of Chapter Any blank slides that follow are blank intentionally.
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