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© 2003 Pearson Prentice Hall Chapter 11 Simple Linear Regression

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© 2003 Pearson Prentice Hall Learning Objectives 1.Describe the Linear Regression Model 2.State the Regression Modeling Steps 3.Explain Ordinary Least Squares 1. Understand and check model assumptions 4.Compute Regression Coefficients 5.Predict Response Variable 6.Interpret Computer Output

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© 2003 Pearson Prentice Hall Models

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© 2003 Pearson Prentice Hall Models 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 Deterministic Models Probabilistic Models Probabilistic Models

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© 2003 Pearson Prentice Hall Deterministic Models 1.Hypothesize Exact Relationships 2.Suitable When Prediction Error is Negligible 3.Example: Force Is Exactly Mass Times Acceleration F = m·a F = m·a © T/Maker Co.

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© 2003 Pearson Prentice Hall Probabilistic Models 1.Hypothesize 2 Components Deterministic Deterministic Random Error Random Error 2.Example: Sales Volume Is 10 Times Advertising Spending + Random Error Y = 10X + Y = 10X + Random Error May Be Due to Factors Other Than Advertising Random Error May Be Due to Factors Other Than Advertising

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© 2003 Pearson Prentice Hall Types of Probabilistic Models

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© 2003 Pearson Prentice Hall Regression Models

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© 2003 Pearson Prentice Hall Types of Probabilistic Models

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© 2003 Pearson Prentice Hall Regression Models 1.Answer ‘What Is the Relationship Between the Variables?’ 2.Equation Used 1 Numerical Dependent (Response) Variable 1 Numerical Dependent (Response) Variable What Is to Be Predicted What Is to Be Predicted 1 or More Numerical or Categorical Independent (Explanatory) Variables 1 or More Numerical or Categorical Independent (Explanatory) Variables 3.Used Mainly for Prediction & Estimation

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Model Specification

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Specifying the Model 1.Define Variables 2.Hypothesize Nature of Relationship Expected Effects (i.e., Coefficients’ Signs) Expected Effects (i.e., Coefficients’ Signs) Functional Form (Linear or Non-Linear) Functional Form (Linear or Non-Linear) Interactions Interactions

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© 2003 Pearson Prentice Hall Model Specification Is Based on Theory 1.Theory of Field (e.g., Sociology) 2.Mathematical Theory 3.Previous Research 4.‘Common Sense’

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© 2003 Pearson Prentice Hall Thinking Challenge: Which Is More Logical?

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© 2003 Pearson Prentice Hall Types of Regression Models

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models Simple 1 Explanatory Variable

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models 2+ Explanatory Variables Simple Multiple 1 Explanatory Variable

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models Linear 2+ Explanatory Variables Simple Multiple 1 Explanatory Variable

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models Linear Non- Linear 2+ Explanatory Variables Simple Multiple 1 Explanatory Variable

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models Linear Non- Linear 2+ Explanatory Variables Simple Multiple Linear 1 Explanatory Variable

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© 2003 Pearson Prentice Hall Types of Regression Models Regression Models Linear Non- Linear 2+ Explanatory Variables Simple Multiple Linear 1 Explanatory Variable Non- Linear

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© 2003 Pearson Prentice Hall Linear Regression Model

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© 2003 Pearson Prentice Hall Types of Regression Models

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© 2003 Pearson Prentice Hall Linear Equations High School Teacher © T/Maker Co.

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YX iii 01 Linear Regression Model 1.Relationship Between Variables Is a Linear Function Dependent (Response) Variable (e.g., income) Independent (Explanatory) Variable (e.g., education) Population Slope Population Y-Intercept Random Error

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© 2003 Pearson Prentice Hall Population & Sample Regression Models

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© 2003 Pearson Prentice Hall Population & Sample Regression Models Population $ $ $ $ $

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© 2003 Pearson Prentice Hall Population & Sample Regression Models Unknown Relationship Population $ $ $ $ $

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© 2003 Pearson Prentice Hall Population & Sample Regression Models Unknown Relationship Population Random Sample $ $ $ $ $

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© 2003 Pearson Prentice Hall Population & Sample Regression Models Unknown Relationship Population Random Sample $ $ $ $ $

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© 2003 Pearson Prentice Hall Population Linear Regression Model Observed value i = Random error

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© 2003 Pearson Prentice Hall Sample Linear Regression Model Unsampled observation i = Random error Observed value ^

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© 2003 Pearson Prentice Hall Estimating Parameters: Least Squares Method

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall X Y Scattergram 1.Plot of All (X i, Y i ) Pairs 2.Suggests How Well Model Will Fit

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’?

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© 2003 Pearson Prentice Hall Least Squares Least Squares 1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative But Positive Differences Off-Set Negative

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© 2003 Pearson Prentice Hall Least Squares Least Squares 1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative But Positive Differences Off-Set Negative

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© 2003 Pearson Prentice Hall Least Squares Least Squares 1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum But Positive Differences Off-Set Negative But Positive Differences Off-Set Negative 2.LS Minimizes the Sum of the Squared Differences (SSE)

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© 2003 Pearson Prentice Hall Least Squares Graphically

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© 2003 Pearson Prentice Hall Coefficient Equations Sample Slope Sample Y-intercept Prediction Equation

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© 2003 Pearson Prentice Hall Computation Table

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© 2003 Pearson Prentice Hall Interpretation of Coefficients

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© 2003 Pearson Prentice Hall Interpretation of Coefficients 1.Slope ( 1 ) Estimated Y Changes by 1 for Each 1 Unit Increase in X 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) If 1 = 2, then Sales (Y) Is Expected to Increase by 2 for Each 1 Unit Increase in Advertising (X) ^ ^ ^

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© 2003 Pearson Prentice Hall Interpretation of Coefficients 1.Slope ( 1 ) Estimated Y Changes by 1 for Each 1 Unit Increase in X 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) 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 Average Value of Y When X = 0 If 0 = 4, then Average Sales (Y) Is Expected to Be 4 When Advertising (X) Is 0 If 0 = 4, then Average Sales (Y) Is Expected to Be 4 When Advertising (X) Is 0 ^ ^ ^ ^ ^

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© 2003 Pearson Prentice Hall Parameter Estimation Example You’re a marketing analyst for Hasbro Toys. You gather the following data: Ad $Sales (Units) What is the relationship between sales & advertising?

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© 2003 Pearson Prentice Hall Scattergram Sales vs. Advertising Sales Advertising

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© 2003 Pearson Prentice Hall Guess The Parameters!

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© 2003 Pearson Prentice Hall Scattergram Sales vs. Advertising Sales Advertising

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© 2003 Pearson Prentice Hall Parameter Estimation Solution Table

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© 2003 Pearson Prentice Hall Parameter Estimation Solution

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution 1.Slope ( 1 ) Sales Volume (Y) Is Expected to Increase by.7 Units for Each $1 Increase in Advertising (X) Sales Volume (Y) Is Expected to Increase by.7 Units for Each $1 Increase in Advertising (X) ^

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution 1.Slope ( 1 ) Sales Volume (Y) Is Expected to Increase by.7 Units for Each $1 Increase in Advertising (X) 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 Average Value of Sales Volume (Y) Is -.10 Units When Advertising (X) Is 0 Difficult to Explain to Marketing Manager Difficult to Explain to Marketing Manager Expect Some Sales Without Advertising Expect Some Sales Without Advertising ^ ^

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© 2003 Pearson Prentice Hall Parameter Estimates Parameter Estimates Parameter Standard T for H0: Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob>|T| INTERCEP ADVERT Parameter Estimation Computer Output 00 ^ 11 ^ kk ^

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© 2003 Pearson Prentice Hall Derivation of Parameter Equations Goal: Minimize squared error

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Derivation of Parameter Equations

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© 2003 Pearson Prentice Hall Parameter Estimation Thinking Challenge You’re an economist for the county cooperative. You gather the following data: Fertilizer (lb.)Yield (lb.) What is the relationship between fertilizer & crop yield? © T/Maker Co.

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© 2003 Pearson Prentice Hall Scattergram Crop Yield vs. Fertilizer* Yield (lb.) Fertilizer (lb.)

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© 2003 Pearson Prentice Hall Parameter Estimation Solution Table*

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© 2003 Pearson Prentice Hall Parameter Estimation Solution*

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution*

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution* 1.Slope ( 1 ) Crop Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Fertilizer (X) Crop Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Fertilizer (X) ^

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© 2003 Pearson Prentice Hall Coefficient Interpretation Solution* 1.Slope ( 1 ) Crop Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Fertilizer (X) 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 Average Crop Yield (Y) Is Expected to Be 0.8 lb. When No Fertilizer (X) Is Used ^ ^

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© 2003 Pearson Prentice Hall Probability Distribution of Random Error

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Linear Regression Assumptions 1.Mean of Probability Distribution of Error Is 0 2.Probability Distribution of Error Has Constant Variance 1. Exercise: Constant across what? 3.Probability Distribution of Error is Normal 4. Errors Are Independent

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© 2003 Pearson Prentice Hall Error Probability Distribution ^

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© 2003 Pearson Prentice Hall Random Error Variation

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© 2003 Pearson Prentice Hall Random Error Variation 1.Variation of Actual Y from Predicted Y

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© 2003 Pearson Prentice Hall Random Error Variation 1.Variation of Actual Y from Predicted Y 2.Measured by Standard Error of Regression Model Sample Standard Deviation of , s Sample Standard Deviation of , s ^

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© 2003 Pearson Prentice Hall Random Error Variation 1.Variation of Actual Y from Predicted Y 2.Measured by Standard Error of Regression Model Sample Standard Deviation of , s Sample Standard Deviation of , s 3. Affects Several Factors Parameter Significance Parameter Significance Prediction Accuracy Prediction Accuracy ^

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© 2003 Pearson Prentice Hall Evaluating the Model Testing for Significance

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Test of Slope Coefficient 1.Shows If There Is a Linear Relationship Between X & Y 2.Involves Population Slope 1 3.Hypotheses H 0 : 1 = 0 (No Linear Relationship) H 0 : 1 = 0 (No Linear Relationship) H a : 1 0 (Linear Relationship) H a : 1 0 (Linear Relationship) 4.Theoretical Basis Is Sampling Distribution of Slope

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© 2003 Pearson Prentice Hall Sampling Distribution of Sample Slopes

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© 2003 Pearson Prentice Hall Sampling Distribution of Sample Slopes

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© 2003 Pearson Prentice Hall Sampling Distribution of Sample Slopes 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

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© 2003 Pearson Prentice Hall Sampling Distribution of Sample Slopes 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 1111 1111 S ^ ^

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© 2003 Pearson Prentice Hall Slope Coefficient Test Statistic

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© 2003 Pearson Prentice Hall Test of Slope Coefficient Example You’re a marketing analyst for Hasbro Toys. You find b 0 = -.1, b 1 =.7 & s = Ad $Sales (Units) Is the relationship significant at the.05 level?

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© 2003 Pearson Prentice Hall Solution Table

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© 2003 Pearson Prentice Hall Test of Slope Parameter Solution H 0 : 1 = 0 H a : 1 0 .05 df = 3 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at =.05 There is evidence of a relationship

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© 2003 Pearson Prentice Hall Test Statistic Solution

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© 2003 Pearson Prentice Hall Test of Slope Parameter Computer Output Parameter Estimates Parameter Estimates Parameter Standard T for H0: Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob>|T| INTERCEP ADVERT t = k / S P-Value SS kk k k ^ ^ ^ ^

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© 2003 Pearson Prentice Hall Measures of Variation in Regression 1.Total Sum of Squares (SS yy ) Measures Variation of Observed Y i Around the Mean Y Measures Variation of Observed Y i Around the Mean Y 2.Explained Variation (SSR) Variation Due to Relationship Between X & Y Variation Due to Relationship Between X & Y 3.Unexplained Variation (SSE) Variation Due to Other Factors Variation Due to Other Factors

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© 2003 Pearson Prentice Hall Variation Measures Total sum of squares (Y i - Y) 2 Unexplained sum of squares (Y i - Y i ) 2 ^ Explained sum of squares (Y i - Y) 2 ^ YiYiYiYi

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© 2003 Pearson Prentice Hall 1.Proportion of Variation ‘Explained’ by Relationship Between X & Y Coefficient of Determination 0 r 2 1

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© 2003 Pearson Prentice Hall Coefficient of Determination Examples r 2 = 1 r 2 =.8r 2 = 0

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© 2003 Pearson Prentice Hall Coefficient of Determination Example You’re a marketing analyst for Hasbro Toys. You find 0 = -0.1 & 1 = 0.7. Ad $Sales (Units) Interpret a coefficient of determination of ^ ^

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© 2003 Pearson Prentice Hall r 2 Computer Output Root MSE R-square Root MSE R-square Dep Mean Adj R-sq Dep Mean Adj R-sq C.V C.V r 2 adjusted for number of explanatory variables & sample size S r2r2

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© 2003 Pearson Prentice Hall Using the Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Regression Modeling Steps 1.Hypothesize Deterministic Component 2.Estimate Unknown Model Parameters 3.Specify Probability Distribution of Random Error Term Estimate Standard Deviation of Error Estimate Standard Deviation of Error 4.Evaluate Model 5.Use Model for Prediction & Estimation

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© 2003 Pearson Prentice Hall Prediction With Regression Models 1.Types of Predictions Point Estimates Point Estimates Interval Estimates Interval Estimates 2.What Is Predicted Population Mean Response E(Y) for Given X Population Mean Response E(Y) for Given X Point on Population Regression Line Point on Population Regression Line Individual Response (Y i ) for Given X Individual Response (Y i ) for Given X

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© 2003 Pearson Prentice Hall What Is Predicted

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© 2003 Pearson Prentice Hall Confidence Interval Estimate of Mean Y

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© 2003 Pearson Prentice Hall Factors Affecting Interval Width 1.Level of Confidence (1 - ) Width Increases as Confidence Increases Width Increases as Confidence Increases 2.Data Dispersion (s) Width Increases as Variation Increases Width Increases as Variation Increases 3.Sample Size Width Decreases as Sample Size Increases Width Decreases as Sample Size Increases 4.Distance of X p from Mean X Width Increases as Distance Increases Width Increases as Distance Increases

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© 2003 Pearson Prentice Hall Why Distance from Mean? Greater dispersion than X 1 XXXX

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© 2003 Pearson Prentice Hall Confidence Interval Estimate Example You’re a marketing analyst for Hasbro Toys. You find b 0 = -.1, b 1 =.7 & s = Ad $Sales (Units) Estimate the mean sales when advertising is $4 at the.05 level.

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© 2003 Pearson Prentice Hall Solution Table

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© 2003 Pearson Prentice Hall Confidence Interval Estimate Solution X to be predicted

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© 2003 Pearson Prentice Hall Prediction Interval of Individual Response Note!

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© 2003 Pearson Prentice Hall Why the Extra ‘S ’ ?

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© 2003 Pearson Prentice Hall Interval Estimate Computer Output Dep Var Pred Std Err Low95% Upp95% Low95% Upp95% Dep Var Pred Std Err Low95% Upp95% Low95% Upp95% Obs SALES Value Predict Mean Mean Predict Predict Predicted Y when X = 4 Confidence Interval SYSYSYSY^ Prediction Interval

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© 2003 Pearson Prentice Hall Hyperbolic Interval Bands

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© 2003 Pearson Prentice Hall Correlation Models

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© 2003 Pearson Prentice Hall Types of Probabilistic Models

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© 2003 Pearson Prentice Hall Correlation Models 1.Answer ‘How Strong Is the Linear Relationship Between 2 Variables?’ 2.Coefficient of Correlation Used Population Correlation Coefficient Denoted (Rho) Population Correlation Coefficient Denoted (Rho) Values Range from -1 to +1 Values Range from -1 to +1 Measures Degree of Association Measures Degree of Association 3.Used Mainly for Understanding

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© 2003 Pearson Prentice Hall 1.Pearson Product Moment Coefficient of Correlation, r: Sample Coefficient of Correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values No Correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values Increasing degree of negative correlation No Correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values Perfect Negative Correlation No Correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values Perfect Negative Correlation No Correlation Increasing degree of positive correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Values Perfect Positive Correlation Perfect Negative Correlation No Correlation

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© 2003 Pearson Prentice Hall Coefficient of Correlation Examples r = 1r = -1 r =.89r = 0

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© 2003 Pearson Prentice Hall 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 H 0 : = 0 (No Correlation) H 0 : = 0 (No Correlation) H a : 0 (Correlation) H a : 0 (Correlation)

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© 2003 Pearson Prentice Hall Conclusion 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

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