Bayesian Analysis LEARNING OBJECTIVES. Calculating Revised Probabilities. Calculating Revised Probabilities. Calculating Revised Probabilities
|
|
- Clementine Thomas
- 5 years ago
- Views:
Transcription
1 Valua%on and pricing (November 5, 2013) LEARNING OBJECTIVES Lecture 7 Decision making (part 3) Regression theory Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA 1. List the steps of the decision-making process. 2. Describe the types of decision-making environments. 3. Make decisions under uncertainty. 4. Use probability values to make decisions under risk. 5. Develop accurate and useful decision trees. 6. Revise probabilities using Bayesian analysis. 7. Use computers to solve basic decision-making problems. 8. Understand the importance and use of utility theory in decision making. Copyright 2015 Pearson Education, Inc. 3 2 Bayesian Analysis Many ways of getting probability data Management s experience and intuition Historical data Computed from other data using Bayes theorem Bayes theorem incorporates initial estimates and information about the accuracy of the sources Allows the revision of initial estimates based on new information Calculating Revised Probabilities Four conditional probabilities for Thompson Lumber P(favorable market(fm) survey results positive) 0.78 P(unfavorable market(um) survey results positive) 0.22 P(favorable market(fm) survey results negative) 0.27 P(unfavorable market(um) survey results negative) 0.73 Prior probabilities P(FM) 0.50 P(UM) Calculating Revised Probabilities Calculating Revised Probabilities TABLE 3.16 Market Survey Reliability Calculating posterior probabilities RESULT OF SURVEY Positive (predicts favorable market for product) Negative (predicts unfavorable market for product) FAVORABLE MARKET (FM) P (survey positive FM) 0.70 P (survey negative FM) 0.30 STATE OF NATURE UNFAVORABLE MARKET (UM) P (survey positive UM) 0.20 P (survey negative UM) 0.80 P(B A) P(A) P(A B) P(B A) P(A)+ P(B A ") P( A ") where A, B any two events A complement of A A favorable market B positive survey
2 Calculating Revised Probabilities P(FM survey positive) P(survey positive FM)P(FM) P(survey positive FM)P(FM)+ P(survey positive UM)P(UM) (0.70)(0.50) (0.70)(0.50)+(0.20)(0.50) P(UM survey positive) P(survey positive UM)P(UM) P(survey positive UM)P(UM)+ P(survey positive FM)P(FM) (0.20)(0.50) (0.20)(0.50)+(0.70)(0.50) Calculating Revised Probabilities TABLE 3.17 Probability Revisions Given a Positive Survey STATE OF NATURE CONDITIONAL PROBABILITY P(SURVEY POSITIVE STATE OF NATURE) PRIOR PROBABILITY POSTERIOR PROBABILITY JOINT PROBABILITY P(STATE OF NATURE SURVEY POSITIVE) FM / UM / P(survey results positive) Calculating Revised Probabilities P(FM survey negative) P(survey negative FM)P(FM) P(survey negative FM)P(FM)+ P(survey negative UM)P(UM) (0.30)(0.50) (0.30)(0.50)+(0.80)(0.50) P(UM survey negative) P(survey negative UM)P(UM) P(survey negative UM)P(UM)+ P(survey negative FM)P(FM) (0.80)(0.50) (0.80)(0.50)+(0.30)(0.50) Calculating Revised Probabilities TABLE 3.18 Probability Revisions Given a Negative Survey STATE OF NATURE CONDITIONAL PROBABILITY P(SURVEY NEGATIVE STATE OF NATURE) PRIOR PROBABILITY POSTERIOR PROBABILITY JOINT PROBABILITY P(STATE OF NATURE SURVEY NEGATIVE) FM / UM / P(survey results positive) Potential Problems Using Survey Results We can not always get the necessary data for analysis Survey results may be based on cases where an action was taken Conditional probability information may not be as accurate as we would like Utility Theory Monetary value is not always a true indicator of the overall value of the result of a decision The overall value of a decision is called utility Economists assume that rational people make decisions to maximize their utility 3 11 Copyright 2015 Pearson Education, Inc. 3 12
3 Utility Theory Utility Theory FIGURE 3.6 Decision Tree for the Lottery Ticket Accept Offer Reject Offer $2,000,000 Heads (0.5) $0 Utility assessment assigns the worst outcome a utility of 0 and the best outcome a utility of 1 A standard gamble is used to determine utility values When you are indifferent, your utility values are equal Tails (0.5) EMV $2,500,000 $5,000,000 Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Utility Theory Utility Theory FIGURE 3.7 Standard Gamble for Utility Assessment Alternative 1 Alternative 2 (p) (1 p) Best Outcome Utility 1 Worst Outcome Utility 0 Other Outcome Utility? Expected utility of alternative 2 Expected utility of alternative 1 Utility of other outcome (p)(utility of best outcome, which is 1) + (1 p)(utility of the worst outcome, which is 0) Utility of other outcome (p)(1) + (1 p)(0) p Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Investment Example Construct a utility curve revealing preference for money between $0 and $10,000 A utility curve plots the utility value versus the monetary value An investment in a bank will result in $5,000 An investment in real estate will result in $0 or $10,000 Unless there is an 80% chance of getting $10,000 from the real estate deal, prefer to have her money in the bank If p 0.80, Jane is indifferent between the bank or the real estate investment Copyright 2015 Pearson Education, Inc Investment Example Figure 3.8 Utility of $5,000 Invest in Real Estate Invest in Bank p 0.80 (1 p) 0.20 $10,000 U($10,000) 1.0 $0 U($0.00) 0.0 $5,000 U($5,000) p 0.80 Utility for $5,000 U($5,000) pu($10,000) + (1 p)u($0) (0.8)(1) + (0.2)(0) 0.8 Copyright 2015 Pearson Education, Inc. 3 18
4 Investment Example Assess other utility values Utility for $7, Utility for $3, Use the three different dollar amounts and assess utilities FIGURE 3.9 Utility Curve U ($10,000) Utility U ($7,000) U ($5,000) U ($3,000) Utility Curve U ($0) 0 Copyright 2015 Pearson Education, Inc $0 $1,000 $3,000 $5,000 $7,000 $10,000 Monetary Value 3 20 Utility Curve Preferences for Risk Typical of a risk avoider Less utility from greater risk Avoids situations where high losses might occur As monetary value increases, utility curve increases at a slower rate A risk seeker gets more utility from greater risk As monetary value increases, the utility curve increases at a faster rate Risk indifferent gives a linear utility curve FIGURE 3.10 Utility Risk Avoider Risk Indifference Risk Seeker Monetary Outcome Utility as a Decision-Making Criteria Utility as a Decision-Making Criteria Once a utility curve has been developed it can be used in making decisions Replaces monetary outcomes with utility values Expected utility is computed instead of the EMV Mark Simkin loves to gamble A game tossing thumbtacks in the air If the thumbtack lands point up, Mark wins $10,000 If the thumbtack lands point down, Mark loses $10,000 Mark believes that there is a 45% chance the thumbtack will land point up Should Mark play the game (alternative 1)?
5 Utility as a Decision-Making Criteria FIGURE 3.11 Decision Facing Mark Simkin Alternative 1 Mark Plays the Game Alternative 2 Tack Lands Point Up (0.45) Tack Lands Point Down (0.55) $10,000 $10,000 Utility as a Decision-Making Criteria Step 1 Define Mark s utilities U( $10,000) 0.05 U($0) 0.15 U($10,000) 0.30 Utility Mark Does Not Play the Game $0 FIGURE $20,000 $10,000 $0 $10,000 $20,000 Monetary Outcome Utility as a Decision-Making Criteria Utility as a Decision-Making Criteria Step 2 Replace monetary values with utility values E(alternative 1: play the game) (0.45)(0.30) + (0.55)(0.05) E(alternative 2: don t play the game) 0.15 FIGURE 3.13 Using Expected Utilities E Alternative 1 Mark Plays the Game Tack Lands Point Up (0.45) Tack Lands Point Down (0.55) Utility Alternative 2 Don t Play Introduction Regression Models Regression analysis very valuable tool for a manager Understand the relationship between variables Predict the value of one variable based on another variable Simple linear regression models have only two variables Multiple regression models have more than one independent variable Copyright 2015 Pearson Education, Inc. Copyright 2015 Pearson Education, Inc. 4 30
6 Introduction Variable to be predicted is called the dependent variable or response variable Value depends on the value of the independent variable(s) Explanatory or predictor variable Scatter Diagram Scatter diagram or scatter plot often used to investigate the relationship between variables Independent variable normally plotted on axis Dependent variable normally plotted on Y axis Dependent variable Independent + variable Independent variable Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc renovates old homes The dollar volume of renovation work is dependent on the area payroll TABLE 4.1 TRIPLE A S SALES LOCAL PAYROLL ($100,000s) ($100,000,000s) Copyright 2015 Pearson Education, Inc Figure 4.1 Scatter Diagram Sales ($100,000) Payroll ($100 million) Copyright 2015 Pearson Education, Inc Simple Linear Regression Regression models used to test relationships between variables Random error Y β 0 + β 1 +ε where Y dependent variable (response) independent variable (predictor or explanatory) β 0 intercept (value of Y when 0) β 1 slope of the regression line ε random error Simple Linear Regression True values for the slope and intercept are not known Estimated using sample data ˆ Y b 0 + b 1 where ^ Y predicted value of Y b 0 estimate of β 0, based on sample results b 1 estimate of β 1, based on sample results Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc. 4 36
7 Predict sales based on area payroll Y Sales Area payroll The line Figure 4.1 minimizes the errors Error (Actual value) (Predicted value) e Y ˆ Y Regression analysis minimizes the sum of squared errors Least-squares regression Copyright 2015 Pearson Education, Inc Formulas for simple linear regression, intercept and slope ˆ Y b 0 + b 1 average (mean) of values n Y Y average (mean) of Y values n ( )(Y Y ) b 1 ( ) 2 b 0 Y b 1 Copyright 2015 Pearson Education, Inc ΣY 42 TABLE 4.2 Regression calculations Y ( ) 2 ( )(Y Y) 6 3 (3 4) 2 1 (3 4)(6 7) (4 4) 2 0 (4 4)(8 7) (6 4) 2 4 (6 4)(9 7) (4 4) 2 0 (4 4)(5 7) (2 4) 2 4 (2 4)(4.5 7) (5 4) 2 1 (5 4)(9.5 7) 2.5 Y 42/6 7 Σ 24 24/6 4 Σ( ) 2 10 Σ( )(Y Y) 12.5 Regression calculations Y Y ( )(Y Y ) b ( ) b 0 Y b 1 7 (1.25)(4) 2 Therefore ˆ Y Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Regression calculations sales 4 2 Y (payroll) Y If the payroll next ( )(Y year Y ) is $600 b ( ) million Y ˆ (6) 9.5 or $ 950,000 b 0 Y b 1 7 (1.25)(4) 2 Therefore ˆ Y Measuring the Fit of the Regression Model Regression models can be developed for any variables and Y How helpful is the model in predicting Y? With average error positive and negative errors cancel each other out Three measures of variability SST Total variability about the mean SSE Variability about the regression line SSR Total variability that is explained by the model Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc. 4 42
8 Measuring the Fit of the Regression Model Sum of squares total SST (Y Y ) 2 Sum of squares error SSE e 2 (Y Y ˆ ) 2 Sum of squares regression SSR ( Y ˆ Y ) 2 An important relationship SST SSR + SSE Measuring the Fit of the Regression Model TABLE 4.3 Sum of Squares for Y (Y Y) 2 ^ ^ Y (Y Y) 2 ^ (Y Y) (6 7) (3) (8 7) (4) (9 7) (6) (5 7) (4) (4.5 7) (2) (9.5 7) (5) (Y Y) ^ (Y Y) ^ (Y Y) Y 7 SST 22.5 SSE SSR Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Measuring the Fit of the Regression Model Sum of squares total SST For (YTriple Y ) 2 A Construction Sum of squares error SST 22.5 SSE SSE e 2 (Y ˆ Y ) SSR Sum of squares regression SSR ( Y ˆ Y ) 2 An important relationship SST SSR + SSE Measuring the Fit of the Regression Model FIGURE 4.2 Deviations from the Regression Line and from the Mean Sales ($100,000) Y ˆ Y ˆ Y Y Y Y ˆ Y Payroll ($100 million) Y Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Coefficient of Determination The proportion of the variability in Y explained by the regression equation The coefficient of determination is r 2. r 2 SSR SSE 1 SST SST For r Coefficient of Determination The proportion of the variability in Y explained by the regression equation The coefficient of determination is r 2. r 2 SSR About SSE69% of the 1 SST variability SST in Y is explained by the equation based on For payroll () r Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc. 4 48
9 Correlation Coefficient An expression of the strength of the linear relationship Always between +1 and 1 The correlation coefficient is r FIGURE 4.3 Four Values of the Correlation Coefficient Y Y r ± r 2 For Y (a) Perfect Positive Correlation: r +1 Y (b) Positive Correlation: 0 < r < 1 r Copyright 2015 Pearson Education, Inc (c) No Correlation: r 0 (d) Perfect Negative Correlation: r 1 Copyright 2015 Pearson Education, Inc Assumptions of the Regression Model With certain assumptions about the errors, statistical tests can be performed to determine the model s usefulness 1. Errors are independent 2. Errors are normally distributed 3. Errors have a mean of zero 4. Errors have a constant variance A plot of the residuals (errors) often highlights glaring violations of assumptions Residual Plots FIGURE 4.4A Pattern of Errors Indicating Randomness Error Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Residual Plots Residual Plots FIGURE 4.4B Nonconstant error variance FIGURE 4.4C Errors Indicate Relationship is not Linear Error Error Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc. 4 54
10 Estimating the Variance Errors are assumed to have a constant variance (σ 2 ), usually unknown Estimated using the mean squared error (MSE), s 2 s 2 MSE SSE n k 1 where n number of observations in the sample k number of independent variables Estimating the Variance For s 2 MSE SSE n k Estimate the standard deviation, s The standard error of the estimate or the standard deviation of the regression s MSE Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Testing the Model for Significance When the sample size is too small, you can get good values for MSE and r 2 even if there is no relationship between the variables Testing the model for significance helps determine if the values are meaningful Performing a statistical hypothesis test Testing the Model for Significance We start with the general linear model Y β 0 + β 1 +ε If β 1 0, the null hypothesis is that there is no relationship between and Y The alternate hypothesis is that there is a linear relationship (β 1 0) If the null hypothesis can be rejected, we have proven there is a relationship We use the F statistic Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Testing the Model for Significance The F statistic is based on the MSE and MSR MSR SSR k where k number of independent variables in the model The F statistic is F MSR MSE Describes an F distribution with: degrees of freedom for the numerator df 1 k degrees of freedom for the denominator df 2 n k 1 Copyright 2015 Pearson Education, Inc Testing the Model for Significance If there is very little error, MSE would be small and the F statistic would be large model is useful If the F statistic is large, the significance level (p-value) will be low, unlikely would have occurred by chance When the F value is large, we can reject the null hypothesis and accept that there is a linear relationship between and Y and the values of the MSE and r 2 are meaningful Copyright 2015 Pearson Education, Inc. 4 60
11 Steps in a Hypothesis Test 1. Specify null and alternative hypotheses H 0 0 H Select the level of significance (α) Common values are 0.01 and Calculate the value of the test statistic F MSR MSE Steps in a Hypothesis Test 4. Make a decision using one of the following methods a) Reject the null hypothesis if the test statistic is greater than the F value from the table in Appendix D. Otherwise, do not reject the null hypothesis: Reject if F calculated > F α,df1,df 2 df 1 k df 2 n k 1 b) Reject the null hypothesis if the observed significance level, or p-value, is less than the level of significance (α). Otherwise, do not reject the null hypothesis: p-value P(F > calculated test statistic) Reject if p-value < α Copyright 2015 Pearson Education, Inc Copyright 2015 Pearson Education, Inc Step 1 H 0 0 (no linear relationship between and Y) H 1 0 (linear relationship exists between and Y) Step 2 MSR SSR k Select α 0.05 Step 3 Calculate the value of the test statistic F MSR MSE Copyright 2015 Pearson Education, Inc Step 4 Reject the null hypothesis if the test statistic is greater than the F value in Appendix D df 1 k 1 df 2 n k The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 is found in Appendix D. F 0.05,1, F calculated 9.09 Reject H 0 because 9.09 > 7.71 Copyright 2015 Pearson Education, Inc FIGURE 4.5 We can conclude there is a statistically significant relationship between and Y The r 2 value of 0.69 means about 69% of the variability in sales (Y) is explained by local payroll () Analysis of Variance (ANOVA) Table With software models, an ANOVA table is typically created that shows the observed significance level (p-value) for the calculated F value This can be compared to the level of significance (α) to make a decision TABLE 4.4 DF SS MS F SIGNIFICANCE Regression k SSR MSR SSR/k MSR/MSE P(F > MSR/MSE) 0.05 F Residual n - k - 1 SSE MSE SSE/(n - k - 1) Total n - 1 SST Copyright 2015 Pearson Education, Inc
12 Multiple Regression Analysis Extensions to the simple linear model Models with more than one independent variable Y β 0 + β β β k k + ε where Y dependent variable (response variable) i i th independent variable (predictor or explanatory variable) β 0 intercept (value of Y when all i 0) β i coefficient of the i th independent variable k number of independent variables ε random error Multiple Regression Analysis To estimate these values, a sample is taken the following equation developed ˆ Y b 0 + b b b k k Where Yˆ predicted value of Y b 0 sample intercept (an estimate of β 0 ) b i sample coefficient of the ith variable (an estimate of β i ) Jenny Wilson Realty Develop a model to determine the suggested listing price for houses based on the size and age of the house Y ˆ b 0 + b b 2 2 where Yˆ predicted value of dependent variable (selling price) b 0 Y intercept 1 and 2 value of the two independent variables (square footage and age) respectively b 1 and b 2 slopes for 1 and 2 respectively Selects a sample of houses that have sold recently and records the data Jenny Wilson Real Estate Data TABLE 4.5 SELLING PRICE ($) SQUARE FOOTAGE AGE 95,000 1, Good CONDITION 119,000 2, Excellent 124,800 1, Excellent 135,000 1, Good 142,000 1, Mint 145,000 1, Mint 159,000 1, Mint 165,000 2, Excellent 182,000 2, Mint 183,000 3, Good 200,000 2, Good 211,000 2, Good 215,000 3, Excellent 219,000 1, Mint Evaluating Multiple Regression Models Evaluating Multiple Regression Models Similar to simple linear regression models The p-value for the F test and r 2 interpreted the same The hypothesis is different because there is more than one independent variable The F test is investigating whether all the coefficients are equal to 0 at the same time To determine which independent variables are significant, tests are performed for each variable H 0 0 H 1 0 The test statistic is calculated and if the p- value is lower than the level of significance (α), the null hypothesis is rejected
13 Jenny Wilson Realty Full model is statistically significant Useful in predicting selling price p-value for F test r Jenny Wilson Realty Both square footage and age are helpful in predicting the Full model is statistically significant selling price Useful in predicting selling price p-value for F test r Are both variables significant? For 1 (square footage) H 0 0 H 1 0 Are both variables significant? For 1 square footage H 0 0 H 1 0 For α 0.05, p-value null hypothesis is rejected For α 0.05, p-value null hypothesis is rejected For 1 (age) For 1 age For α 0.05, p-value null hypothesis is rejected For α 0.05, p-value null hypothesis is rejected Binary or Dummy Variables Binary (or dummy or indicator) variables are special variables created for qualitative data A dummy variable is assigned a value of 1 if a particular condition is met and a value of 0 otherwise The number of dummy variables must equal one less than the number of categories of the qualitative variable Jenny Wilson Realty A better model can be developed if information about the condition of the property is included 3 1 if house is in excellent condition 0 otherwise 4 1 if house is in mint condition 0 otherwise Two dummy variables are used to describe the three categories of condition No variable is needed for good condition since if both 3 and 4 0, the house must be in good condition Model Building The best model is a statistically significant model with a high r 2 and few variables As more variables are added to the model, the r 2 value increases For this reason, the adjusted r 2 value is often used to determine the usefulness of an additional variable The adjusted r 2 takes into account the number of independent variables in the model The formula for r 2 Model Building r 2 SSR SSE 1 SST SST The formula for adjusted r 2 Adjusted r 2 SSE / (n k 1) 1 SST / (n 1) As the number of variables increases, the adjusted r 2 gets smaller unless the increase due to the new variable is large enough to offset the change in k
14 Model Building The formula for r 2 In general, if a new variable r 2 SSR increases the adjusted r SSE 2, it should probably 1 be included in the model SST SST The formula for adjusted r 2 Adjusted r 2 SSE / (n k 1) 1 SST / (n 1) As the number of variables increases, the adjusted r 2 gets smaller unless the increase due to the new variable is large enough to offset the change in k Model Building Stepwise regression systematically adds or deletes independent variables A forward stepwise procedure puts the most significant variable in first, adds the next variable that will improve the model the most Backward stepwise regression begins with all the independent variables and deletes the least helpful Model Building Nonlinear Regression In some cases variables contain duplicate information When two independent variables are correlated, they are said to be collinear When more than two independent variables are correlated, multicollinearity exists When multicollinearity is present, hypothesis tests for the individual coefficients are not valid but the model may still be useful In some situations, variables are not linear Transformations may be used to turn a nonlinear model into a linear model * * * * * * * * * * * * * * * * * * * Linear relationship Nonlinear relationship Colonel Motors Use regression analysis to improve fuel efficiency Study the impact of weight on miles per gallon (MPG) TABLE 4.6 MPG WEIGHT (1,000 LBS.) MPG WEIGHT (1,000 LBS.) Colonel Motors FIGURE 4.6A Linear Model for MPG Data MPG Weight (1,000 lb.)
15 Colonel Motors FIGURE 4.6B Nonlinear Model for MPG Data MPG Weight (1,000 lb.) Colonel Motors The nonlinear model is a quadratic model The easiest approach develop a new variable New model 2 (weight) 2 ˆ Y b 0 + b b Cautions and Pitfalls Cautions and Pitfalls If the assumptions are not met, the statistical test may not be valid Correlation does not necessarily mean causation Multicollinearity makes interpreting coefficients problematic, but the model may still be good Using a regression model beyond the range of is questionable, as the relationship may not hold outside the sample data A t-test for the intercept (b 0 ) may be ignored as this point is often outside the range of the model A linear relationship may not be the best relationship, even if the F test returns an acceptable value A nonlinear relationship can exist even if a linear relationship does not Even though a relationship is statistically significant it may not have any practical value Remember Chapter 3: Where Prices Come From: The Interaction of Demand and Supply Read Quantitative Methods-module guide. Any questions please olivier.edu@gmail.com and make notes as you do so, in whatever way works best for you in terms of remembering information (your performance on this course is only assessed by exam). Copyright 2010 Pearson Education, Inc. Economics R. Glenn Hubbard, Anthony Patrick O Brien, 3e. 89 of 46
Chapter 4. Regression Models. Learning Objectives
Chapter 4 Regression Models To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna Power Point slides created by Brian Peterson Learning Objectives After completing
More informationChapter 4: Regression Models
Sales volume of company 1 Textbook: pp. 129-164 Chapter 4: Regression Models Money spent on advertising 2 Learning Objectives After completing this chapter, students will be able to: Identify variables,
More informationRegression Models. Chapter 4. Introduction. Introduction. Introduction
Chapter 4 Regression Models Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna 008 Prentice-Hall, Inc. Introduction Regression analysis is a very valuable tool for a manager
More informationRegression Models. Chapter 4
Chapter 4 Regression Models To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna Power Point slides created by Brian Peterson Introduction Regression analysis
More informationThe Multiple Regression Model
Multiple Regression The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & or more independent variables (X i ) Multiple Regression Model with k Independent Variables:
More informationChapter 3 Multiple Regression Complete Example
Department of Quantitative Methods & Information Systems ECON 504 Chapter 3 Multiple Regression Complete Example Spring 2013 Dr. Mohammad Zainal Review Goals After completing this lecture, you should be
More informationChapter 14 Student Lecture Notes Department of Quantitative Methods & Information Systems. Business Statistics. Chapter 14 Multiple Regression
Chapter 14 Student Lecture Notes 14-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 14 Multiple Regression QMIS 0 Dr. Mohammad Zainal Chapter Goals After completing
More informationCorrelation Analysis
Simple Regression 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
More informationChapter 14 Student Lecture Notes 14-1
Chapter 14 Student Lecture Notes 14-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 14 Multiple Regression Analysis and Model Building Chap 14-1 Chapter Goals After completing this
More informationChapter 7 Student Lecture Notes 7-1
Chapter 7 Student Lecture Notes 7- Chapter Goals QM353: Business Statistics Chapter 7 Multiple Regression Analysis and Model Building After completing this chapter, you should be able to: Explain model
More informationChapter 13. Multiple Regression and Model Building
Chapter 13 Multiple Regression and Model Building Multiple Regression Models The General Multiple Regression Model y x x x 0 1 1 2 2... k k y is the dependent variable x, x,..., x 1 2 k the model are the
More informationStatistics for Managers using Microsoft Excel 6 th Edition
Statistics for Managers using Microsoft Excel 6 th Edition Chapter 13 Simple Linear Regression 13-1 Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of
More informationBasic Business Statistics, 10/e
Chapter 4 4- Basic Business Statistics th Edition Chapter 4 Introduction to Multiple Regression Basic Business Statistics, e 9 Prentice-Hall, Inc. Chap 4- Learning Objectives In this chapter, you learn:
More informationRegression Models REVISED TEACHING SUGGESTIONS ALTERNATIVE EXAMPLES
M04_REND6289_10_IM_C04.QXD 5/7/08 2:49 PM Page 46 4 C H A P T E R Regression Models TEACHING SUGGESTIONS Teaching Suggestion 4.1: Which Is the Independent Variable? We find that students are often confused
More informationBasic Business Statistics 6 th Edition
Basic Business Statistics 6 th Edition Chapter 12 Simple Linear Regression Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationBusiness Statistics. Chapter 14 Introduction to Linear Regression and Correlation Analysis QMIS 220. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 14 Introduction to Linear Regression and Correlation Analysis QMIS 220 Dr. Mohammad Zainal Chapter Goals After completing
More informationKeller: Stats for Mgmt & Econ, 7th Ed July 17, 2006
Chapter 17 Simple Linear Regression and Correlation 17.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationEcon 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu Sampling Distribution of b 1 Expected value of b 1 : Variance of b 1 : E(b 1 ) = 1 Var(b 1 ) = σ 2 /SS x Estimate of
More informationChapter 16. Simple Linear Regression and Correlation
Chapter 16 Simple Linear Regression and Correlation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationChapter Learning Objectives. Regression Analysis. Correlation. Simple Linear Regression. Chapter 12. Simple Linear Regression
Chapter 12 12-1 North Seattle Community College BUS21 Business Statistics Chapter 12 Learning Objectives In this chapter, you learn:! How to use regression analysis to predict the value of a dependent
More informationMathematics for Economics MA course
Mathematics for Economics MA course Simple Linear Regression Dr. Seetha Bandara Simple Regression Simple linear regression is a statistical method that allows us to summarize and study relationships between
More informationChapter 16. Simple Linear Regression and dcorrelation
Chapter 16 Simple Linear Regression and dcorrelation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationRegression Analysis II
Regression Analysis II Measures of Goodness of fit Two measures of Goodness of fit Measure of the absolute fit of the sample points to the sample regression line Standard error of the estimate An index
More informationCh 13 & 14 - Regression Analysis
Ch 3 & 4 - Regression Analysis Simple Regression Model I. Multiple Choice:. A simple regression is a regression model that contains a. only one independent variable b. only one dependent variable c. more
More informationInference for Regression
Inference for Regression Section 9.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 13b - 3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationInference for Regression Inference about the Regression Model and Using the Regression Line
Inference for Regression Inference about the Regression Model and Using the Regression Line PBS Chapter 10.1 and 10.2 2009 W.H. Freeman and Company Objectives (PBS Chapter 10.1 and 10.2) Inference about
More informationTrendlines Simple Linear Regression Multiple Linear Regression Systematic Model Building Practical Issues
Trendlines Simple Linear Regression Multiple Linear Regression Systematic Model Building Practical Issues Overfitting Categorical Variables Interaction Terms Non-linear Terms Linear Logarithmic y = a +
More informationRegression Analysis. BUS 735: Business Decision Making and Research. Learn how to detect relationships between ordinal and categorical variables.
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn how to estimate
More informationChapter 14 Simple Linear Regression (A)
Chapter 14 Simple Linear Regression (A) 1. Characteristics Managerial decisions often are based on the relationship between two or more variables. can be used to develop an equation showing how the variables
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
What is Multiple Linear Regression Several independent variables may influence the change in response variable we are trying to study. When several independent variables are included in the equation, the
More informationEstimating σ 2. We can do simple prediction of Y and estimation of the mean of Y at any value of X.
Estimating σ 2 We can do simple prediction of Y and estimation of the mean of Y at any value of X. To perform inferences about our regression line, we must estimate σ 2, the variance of the error term.
More informationRegression Analysis. BUS 735: Business Decision Making and Research
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals and Agenda Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn
More informationST430 Exam 2 Solutions
ST430 Exam 2 Solutions Date: November 9, 2015 Name: Guideline: You may use one-page (front and back of a standard A4 paper) of notes. No laptop or textbook are permitted but you may use a calculator. Giving
More informationIntroduction. Formulating LP Problems LEARNING OBJECTIVES. Requirements of a Linear Programming Problem. LP Properties and Assumptions
Valua%on and pricing (November 5, 2013) LEARNING OBJETIVES Lecture 10 Linear Programming (part 1) Olivier J. de Jong, LL.M., MM., MBA, FD, FFA, AA www.olivierdejong.com 1. Understand the basic assumptions
More informationChapte The McGraw-Hill Companies, Inc. All rights reserved.
12er12 Chapte Bivariate i Regression (Part 1) Bivariate Regression Visual Displays Begin the analysis of bivariate data (i.e., two variables) with a scatter plot. A scatter plot - displays each observed
More informationSimple Linear Regression
9-1 l Chapter 9 l Simple Linear Regression 9.1 Simple Linear Regression 9.2 Scatter Diagram 9.3 Graphical Method for Determining Regression 9.4 Least Square Method 9.5 Correlation Coefficient and Coefficient
More informationLecture 10 Multiple Linear Regression
Lecture 10 Multiple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: 6.1-6.5 10-1 Topic Overview Multiple Linear Regression Model 10-2 Data for Multiple Regression Y i is the response variable
More informationLI EAR REGRESSIO A D CORRELATIO
CHAPTER 6 LI EAR REGRESSIO A D CORRELATIO Page Contents 6.1 Introduction 10 6. Curve Fitting 10 6.3 Fitting a Simple Linear Regression Line 103 6.4 Linear Correlation Analysis 107 6.5 Spearman s Rank Correlation
More informationLECTURE 6. Introduction to Econometrics. Hypothesis testing & Goodness of fit
LECTURE 6 Introduction to Econometrics Hypothesis testing & Goodness of fit October 25, 2016 1 / 23 ON TODAY S LECTURE We will explain how multiple hypotheses are tested in a regression model We will define
More informationBusiness Statistics. Lecture 10: Correlation and Linear Regression
Business Statistics Lecture 10: Correlation and Linear Regression Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. It displays the Form
More informationTHE ROYAL STATISTICAL SOCIETY 2008 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE (MODULAR FORMAT) MODULE 4 LINEAR MODELS
THE ROYAL STATISTICAL SOCIETY 008 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE (MODULAR FORMAT) MODULE 4 LINEAR MODELS The Society provides these solutions to assist candidates preparing for the examinations
More informationChapter 13 Student Lecture Notes Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 13 Student Lecture Notes 13-1 Department of Quantitative Methods & Information Sstems Business Statistics Chapter 14 Introduction to Linear Regression and Correlation Analsis QMIS 0 Dr. Mohammad
More informationRegression analysis is a tool for building mathematical and statistical models that characterize relationships between variables Finds a linear
Regression analysis is a tool for building mathematical and statistical models that characterize relationships between variables Finds a linear relationship between: - one independent variable X and -
More informationSIMPLE REGRESSION ANALYSIS. Business Statistics
SIMPLE REGRESSION ANALYSIS Business Statistics CONTENTS Ordinary least squares (recap for some) Statistical formulation of the regression model Assessing the regression model Testing the regression coefficients
More informationCh14. Multiple Regression Analysis
Ch14. Multiple Regression Analysis 1 Goals : multiple regression analysis Model Building and Estimating More than 1 independent variables Quantitative( 量 ) independent variables Qualitative( ) independent
More informationReview of Statistics 101
Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods
More informationMultiple Regression Methods
Chapter 1: Multiple Regression Methods Hildebrand, Ott and Gray Basic Statistical Ideas for Managers Second Edition 1 Learning Objectives for Ch. 1 The Multiple Linear Regression Model How to interpret
More informationEcon 3790: Statistics Business and Economics. Instructor: Yogesh Uppal
Econ 3790: Statistics Business and Economics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 14 Covariance and Simple Correlation Coefficient Simple Linear Regression Covariance Covariance between
More informationThe simple linear regression model discussed in Chapter 13 was written as
1519T_c14 03/27/2006 07:28 AM Page 614 Chapter Jose Luis Pelaez Inc/Blend Images/Getty Images, Inc./Getty Images, Inc. 14 Multiple Regression 14.1 Multiple Regression Analysis 14.2 Assumptions of the Multiple
More informationSection 3: Simple Linear Regression
Section 3: Simple Linear Regression Carlos M. Carvalho The University of Texas at Austin McCombs School of Business http://faculty.mccombs.utexas.edu/carlos.carvalho/teaching/ 1 Regression: General Introduction
More informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS Page 1 MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level
More informationFinding Relationships Among Variables
Finding Relationships Among Variables BUS 230: Business and Economic Research and Communication 1 Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions, hypothesis
More informationVariance Decomposition and Goodness of Fit
Variance Decomposition and Goodness of Fit 1. Example: Monthly Earnings and Years of Education In this tutorial, we will focus on an example that explores the relationship between total monthly earnings
More informationSTA121: Applied Regression Analysis
STA121: Applied Regression Analysis Linear Regression Analysis - Chapters 3 and 4 in Dielman Artin Department of Statistical Science September 15, 2009 Outline 1 Simple Linear Regression Analysis 2 Using
More informationInference for Regression Simple Linear Regression
Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression p Statistical model for linear regression p Estimating
More informationWe like to capture and represent the relationship between a set of possible causes and their response, by using a statistical predictive model.
Statistical Methods in Business Lecture 5. Linear Regression We like to capture and represent the relationship between a set of possible causes and their response, by using a statistical predictive model.
More informationMultiple Regression. Inference for Multiple Regression and A Case Study. IPS Chapters 11.1 and W.H. Freeman and Company
Multiple Regression Inference for Multiple Regression and A Case Study IPS Chapters 11.1 and 11.2 2009 W.H. Freeman and Company Objectives (IPS Chapters 11.1 and 11.2) Multiple regression Data for multiple
More informationWhat is a Hypothesis?
What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill in this city is μ = $42 population proportion Example:
More informationCorrelation and regression
1 Correlation and regression Yongjua Laosiritaworn Introductory on Field Epidemiology 6 July 2015, Thailand Data 2 Illustrative data (Doll, 1955) 3 Scatter plot 4 Doll, 1955 5 6 Correlation coefficient,
More information(ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box.
FINAL EXAM ** Two different ways to submit your answer sheet (i) Use MS-Word and place it in a drop-box. (ii) Scan your answer sheets INTO ONE FILE only, and submit it in the drop-box. Deadline: December
More informationLecture 15 Multiple regression I Chapter 6 Set 2 Least Square Estimation The quadratic form to be minimized is
Lecture 15 Multiple regression I Chapter 6 Set 2 Least Square Estimation The quadratic form to be minimized is Q = (Y i β 0 β 1 X i1 β 2 X i2 β p 1 X i.p 1 ) 2, which in matrix notation is Q = (Y Xβ) (Y
More informationCorrelation and Regression
Correlation and Regression Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University 1 Learning Objectives Upon successful completion of this module, the student should
More informationReview of Statistics
Review of Statistics Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and
More informationSTAT 350 Final (new Material) Review Problems Key Spring 2016
1. The editor of a statistics textbook would like to plan for the next edition. A key variable is the number of pages that will be in the final version. Text files are prepared by the authors using LaTeX,
More informationInteractions. Interactions. Lectures 1 & 2. Linear Relationships. y = a + bx. Slope. Intercept
Interactions Lectures 1 & Regression Sometimes two variables appear related: > smoking and lung cancers > height and weight > years of education and income > engine size and gas mileage > GMAT scores and
More informationMidterm 2 - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis February 24, 2010 Instructor: John Parman Midterm 2 - Solutions You have until 10:20am to complete this exam. Please remember to put
More informationStatistics and Quantitative Analysis U4320
Statistics and Quantitative Analysis U3 Lecture 13: Explaining Variation Prof. Sharyn O Halloran Explaining Variation: Adjusted R (cont) Definition of Adjusted R So we'd like a measure like R, but one
More informationCh. 1: Data and Distributions
Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and
More informationLECTURE 10. Introduction to Econometrics. Multicollinearity & Heteroskedasticity
LECTURE 10 Introduction to Econometrics Multicollinearity & Heteroskedasticity November 22, 2016 1 / 23 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists
More informationFormal Statement of Simple Linear Regression Model
Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor
More informationSimple Linear Regression: One Qualitative IV
Simple Linear Regression: One Qualitative IV 1. Purpose As noted before regression is used both to explain and predict variation in DVs, and adding to the equation categorical variables extends regression
More informationApplied Regression Analysis. Section 2: Multiple Linear Regression
Applied Regression Analysis Section 2: Multiple Linear Regression 1 The Multiple Regression Model Many problems involve more than one independent variable or factor which affects the dependent or response
More informationSTA441: Spring Multiple Regression. This slide show is a free open source document. See the last slide for copyright information.
STA441: Spring 2018 Multiple Regression This slide show is a free open source document. See the last slide for copyright information. 1 Least Squares Plane 2 Statistical MODEL There are p-1 explanatory
More informationECON 450 Development Economics
ECON 450 Development Economics Statistics Background University of Illinois at Urbana-Champaign Summer 2017 Outline 1 Introduction 2 3 4 5 Introduction Regression analysis is one of the most important
More informationUnit 10: Simple Linear Regression and Correlation
Unit 10: Simple Linear Regression and Correlation Statistics 571: Statistical Methods Ramón V. León 6/28/2004 Unit 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regression analysis is a method for
More informationLectures on Simple Linear Regression Stat 431, Summer 2012
Lectures on Simple Linear Regression Stat 43, Summer 0 Hyunseung Kang July 6-8, 0 Last Updated: July 8, 0 :59PM Introduction Previously, we have been investigating various properties of the population
More information(4) 1. Create dummy variables for Town. Name these dummy variables A and B. These 0,1 variables now indicate the location of the house.
Exam 3 Resource Economics 312 Introductory Econometrics Please complete all questions on this exam. The data in the spreadsheet: Exam 3- Home Prices.xls are to be used for all analyses. These data are
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #6
STA 8 Applied Linear Models: Regression Analysis Spring 011 Solution for Homework #6 6. a) = 11 1 31 41 51 1 3 4 5 11 1 31 41 51 β = β1 β β 3 b) = 1 1 1 1 1 11 1 31 41 51 1 3 4 5 β = β 0 β1 β 6.15 a) Stem-and-leaf
More informationVariance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017
Variance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017 PDF file location: http://www.murraylax.org/rtutorials/regression_anovatable.pdf
More informationRegression Analysis. Regression: Methodology for studying the relationship among two or more variables
Regression Analysis Regression: Methodology for studying the relationship among two or more variables Two major aims: Determine an appropriate model for the relationship between the variables Predict the
More informationDraft Proof - Do not copy, post, or distribute. Chapter Learning Objectives REGRESSION AND CORRELATION THE SCATTER DIAGRAM
1 REGRESSION AND CORRELATION As we learned in Chapter 9 ( Bivariate Tables ), the differential access to the Internet is real and persistent. Celeste Campos-Castillo s (015) research confirmed the impact
More informationF-tests and Nested Models
F-tests and Nested Models Nested Models: A core concept in statistics is comparing nested s. Consider the Y = β 0 + β 1 x 1 + β 2 x 2 + ǫ. (1) The following reduced s are special cases (nested within)
More informationMULTIPLE REGRESSION ANALYSIS AND OTHER ISSUES. Business Statistics
MULTIPLE REGRESSION ANALYSIS AND OTHER ISSUES Business Statistics CONTENTS Multiple regression Dummy regressors Assumptions of regression analysis Predicting with regression analysis Old exam question
More informationIntroduction to Probability
Introduction to Probability Content Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 2
More informationSTATISTICS 110/201 PRACTICE FINAL EXAM
STATISTICS 110/201 PRACTICE FINAL EXAM Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. In other words, the SS is built up as each variable
More informationCS 5014: Research Methods in Computer Science
Computer Science Clifford A. Shaffer Department of Computer Science Virginia Tech Blacksburg, Virginia Fall 2010 Copyright c 2010 by Clifford A. Shaffer Computer Science Fall 2010 1 / 207 Correlation and
More informationInference for the Regression Coefficient
Inference for the Regression Coefficient Recall, b 0 and b 1 are the estimates of the slope β 1 and intercept β 0 of population regression line. We can shows that b 0 and b 1 are the unbiased estimates
More informationSwarthmore Honors Exam 2012: Statistics
Swarthmore Honors Exam 2012: Statistics 1 Swarthmore Honors Exam 2012: Statistics John W. Emerson, Yale University NAME: Instructions: This is a closed-book three-hour exam having six questions. You may
More informationLinear Regression. Simple linear regression model determines the relationship between one dependent variable (y) and one independent variable (x).
Linear Regression Simple linear regression model determines the relationship between one dependent variable (y) and one independent variable (x). A dependent variable is a random variable whose variation
More informationWhat Is ANOVA? Comparing Groups. One-way ANOVA. One way ANOVA (the F ratio test)
What Is ANOVA? One-way ANOVA ANOVA ANalysis Of VAriance ANOVA compares the means of several groups. The groups are sometimes called "treatments" First textbook presentation in 95. Group Group σ µ µ σ µ
More informationApplied Regression Analysis
Applied Regression Analysis Chapter 3 Multiple Linear Regression Hongcheng Li April, 6, 2013 Recall simple linear regression 1 Recall simple linear regression 2 Parameter Estimation 3 Interpretations of
More informationChapter 8. Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis
Chapter 8 Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1 Content 1. Identifying the Target Parameter 2.
More informationLecture 9: Linear Regression
Lecture 9: Linear Regression Goals Develop basic concepts of linear regression from a probabilistic framework Estimating parameters and hypothesis testing with linear models Linear regression in R Regression
More informationREVIEW 8/2/2017 陈芳华东师大英语系
REVIEW Hypothesis testing starts with a null hypothesis and a null distribution. We compare what we have to the null distribution, if the result is too extreme to belong to the null distribution (p
More informationA discussion on multiple regression models
A discussion on multiple regression models In our previous discussion of simple linear regression, we focused on a model in which one independent or explanatory variable X was used to predict the value
More informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,
More informationChapter 15 Multiple Regression
Multiple Regression Learning Objectives 1. Understand how multiple regression analysis can be used to develop relationships involving one dependent variable and several independent variables. 2. Be able
More informationMBA Statistics COURSE #4
MBA Statistics 51-651-00 COURSE #4 Simple and multiple linear regression What should be the sales of ice cream? Example: Before beginning building a movie theater, one must estimate the daily number of
More informationSection 4: Multiple Linear Regression
Section 4: Multiple Linear Regression Carlos M. Carvalho The University of Texas at Austin McCombs School of Business http://faculty.mccombs.utexas.edu/carlos.carvalho/teaching/ 1 The Multiple Regression
More information