This chapter discusses regression models, including simple and multiple linear regression. It covers developing regression equations from sample data, measuring the fit of regression models, and assumptions of regression analysis. Key aspects covered include using scatter plots to examine relationships between variables, calculating the slope, intercept, coefficient of determination, and correlation coefficient, and performing hypothesis tests to determine if regression models are statistically significant. The chapter objectives are to help students understand and appropriately apply simple, multiple, and nonlinear regression techniques.
This document discusses various forecasting models and techniques. It begins by describing qualitative models that incorporate subjective factors like the Delphi method, jury of executive opinion, sales force composite, and consumer market surveys. It then covers time-series models like moving averages, exponential smoothing, trend projections, and decomposition that predict the future based on past data. Specific techniques are defined, like simple and weighted moving averages, and exponential smoothing. Examples are provided to illustrate how to apply these techniques to forecast data. Measures of forecast accuracy like mean absolute deviation are also introduced.
The document discusses various forecasting techniques used to predict future values based on historical data patterns. It describes time series models like moving averages, exponential smoothing and trend projections that rely solely on past values to forecast. It also covers decomposition of time series data into trend, seasonality, cycles and random components. The document provides examples of scatter plots to visualize relationships in time series data and defines accuracy measures like MAD, MSE and MAPE to evaluate forecast errors. Overall it provides an overview of quantitative forecasting methods and how to implement them.
This document provides an overview of Chapter 4 from the textbook "Discrete Probability Distributions" by Larson/Farber. The chapter outlines key concepts related to discrete probability distributions including distinguishing between discrete and continuous random variables, constructing probability distributions, and calculating mean, variance, standard deviation, and expected value. It also previews the topics to be covered in Sections 4.1 on probability distributions and 4.2 on binomial distributions.
This chapter discusses simple linear regression analysis. It explains that regression analysis is used to predict the value of a dependent variable based on the value of at least one independent variable. The chapter outlines the simple linear regression model, which involves one independent variable and attempts to describe the relationship between the dependent and independent variables using a linear function. It provides examples to demonstrate how to obtain and interpret the regression equation and coefficients based on sample data. Key outputs from regression analysis like measures of variation, the coefficient of determination, and tests of significance are also introduced.
Discover The Future Of SAP BusinessObjects (BI 4.3 SP02)Wiiisdom
This document discusses the reasons for adopting SAP BusinessObjects BI 4.3. It notes that SAP's BI portfolio includes both on-premise and cloud-based analytics solutions. While cloud adoption is growing, many customers still rely on large on-premise investments, so a hybrid approach is often needed. BI 4.3 aims to provide a modern user experience, tighter integration with SAP Analytics Cloud, and continued enhancements to lower costs and accelerate value. Key focus areas for BI 4.3 include enterprise readiness, the hybrid landscape, and improved usability.
This chapter discusses risk and return, including defining and measuring risk and return through probability distributions and metrics like expected return, standard deviation, and coefficient of variation. It covers investor attitudes toward different levels of risk, how risk and return relate in a portfolio context through diversification, and models like the Capital Asset Pricing Model. The chapter also defines efficient financial markets and the three levels of market efficiency.
The document discusses slope and how to calculate it. It defines slope as the rate of change of a line and provides the formula slope=rise/run. It then explains how to find the slope of a line graph by picking two points and calculating rise over run. Finally, it demonstrates how to find the slope of a line given two points or from a table of x-y values using the same rise over run formula.
Probability Distribution (Discrete Random Variable)Cess011697
Learning Competencies:
- to find the possible values of a random variable.
illustrates a probability distribution for a discrete random variable and its properties.
- to compute probabilities corresponding to a given random variable.
There are some exercises for you to answer.
This document discusses various forecasting models and techniques. It begins by describing qualitative models that incorporate subjective factors like the Delphi method, jury of executive opinion, sales force composite, and consumer market surveys. It then covers time-series models like moving averages, exponential smoothing, trend projections, and decomposition that predict the future based on past data. Specific techniques are defined, like simple and weighted moving averages, and exponential smoothing. Examples are provided to illustrate how to apply these techniques to forecast data. Measures of forecast accuracy like mean absolute deviation are also introduced.
The document discusses various forecasting techniques used to predict future values based on historical data patterns. It describes time series models like moving averages, exponential smoothing and trend projections that rely solely on past values to forecast. It also covers decomposition of time series data into trend, seasonality, cycles and random components. The document provides examples of scatter plots to visualize relationships in time series data and defines accuracy measures like MAD, MSE and MAPE to evaluate forecast errors. Overall it provides an overview of quantitative forecasting methods and how to implement them.
This document provides an overview of Chapter 4 from the textbook "Discrete Probability Distributions" by Larson/Farber. The chapter outlines key concepts related to discrete probability distributions including distinguishing between discrete and continuous random variables, constructing probability distributions, and calculating mean, variance, standard deviation, and expected value. It also previews the topics to be covered in Sections 4.1 on probability distributions and 4.2 on binomial distributions.
This chapter discusses simple linear regression analysis. It explains that regression analysis is used to predict the value of a dependent variable based on the value of at least one independent variable. The chapter outlines the simple linear regression model, which involves one independent variable and attempts to describe the relationship between the dependent and independent variables using a linear function. It provides examples to demonstrate how to obtain and interpret the regression equation and coefficients based on sample data. Key outputs from regression analysis like measures of variation, the coefficient of determination, and tests of significance are also introduced.
Discover The Future Of SAP BusinessObjects (BI 4.3 SP02)Wiiisdom
This document discusses the reasons for adopting SAP BusinessObjects BI 4.3. It notes that SAP's BI portfolio includes both on-premise and cloud-based analytics solutions. While cloud adoption is growing, many customers still rely on large on-premise investments, so a hybrid approach is often needed. BI 4.3 aims to provide a modern user experience, tighter integration with SAP Analytics Cloud, and continued enhancements to lower costs and accelerate value. Key focus areas for BI 4.3 include enterprise readiness, the hybrid landscape, and improved usability.
This chapter discusses risk and return, including defining and measuring risk and return through probability distributions and metrics like expected return, standard deviation, and coefficient of variation. It covers investor attitudes toward different levels of risk, how risk and return relate in a portfolio context through diversification, and models like the Capital Asset Pricing Model. The chapter also defines efficient financial markets and the three levels of market efficiency.
The document discusses slope and how to calculate it. It defines slope as the rate of change of a line and provides the formula slope=rise/run. It then explains how to find the slope of a line graph by picking two points and calculating rise over run. Finally, it demonstrates how to find the slope of a line given two points or from a table of x-y values using the same rise over run formula.
Probability Distribution (Discrete Random Variable)Cess011697
Learning Competencies:
- to find the possible values of a random variable.
illustrates a probability distribution for a discrete random variable and its properties.
- to compute probabilities corresponding to a given random variable.
There are some exercises for you to answer.
This presentation discusses methods, tools, and best practices for integrating SAP with other platforms. It provides an overview of typical integration scenarios like integrating e-commerce platforms, CRMs, supplier portals, and custom applications. It also outlines the main SAP versions and how they can be integrated, such as using APIs, RFC calls, IDOCs, and oData services. Top considerations for integration include prioritizing real-time over batch sync and proper exception handling. The presentation is meant to help solutions owners, architects and consultants with SAP integration.
The document discusses factorial analysis of variance (ANOVA). It explains how total sums of squares can be partitioned into explained and unexplained components. An example shows an F ratio of 5.0 for one data set, indicating variation between groups is rare. This allows rejecting the null hypothesis with a low probability of Type I error. Finally, it describes how factorial ANOVA can analyze the effects of multiple independent variables on a single dependent variable.
This document summarizes the key topics and concepts covered in Chapter 2 of the 9th edition of the business statistics textbook "Presenting Data in Tables and Charts". The chapter discusses guidelines for analyzing data and organizing both numerical and categorical data. It then covers various methods for tabulating and graphing univariate and bivariate data, including tables, histograms, frequency distributions, scatter plots, bar charts, pie charts, and contingency tables.
This document provides an overview of basic statistics concepts. It defines statistics as the science of collecting, presenting, analyzing, and reasonably interpreting data. Descriptive statistics are used to summarize and organize data through methods like tables, graphs, and descriptive values, while inferential statistics allow researchers to make general conclusions about populations based on sample data. Variables can be either categorical or quantitative, and their distributions and presentations are discussed.
This document provides an overview of forecasting techniques. It begins with the objectives of the chapter, which are to understand various forecasting models and compare methods such as moving averages, exponential smoothing, and time-series models. It also covers qualitatively measuring forecast accuracy. The document then describes different forecasting techniques including qualitative models, time-series models, and causal models. It provides examples of moving averages, weighted moving averages, and exponential smoothing techniques. It concludes with examples of how to implement forecasting models in Excel.
This document discusses limits of functions, including infinite limits, vertical and horizontal asymptotes, and the squeeze theorem. It provides definitions and examples of:
- Infinite limits, where the value of a function increases or decreases without bound as the input approaches a number.
- Vertical and horizontal asymptotes, which are lines that a function approaches but does not meet as the input increases or decreases without limit.
- The squeeze theorem, which can be used to evaluate limits where usual algebraic methods are not effective by "squeezing" the function between two other functions with known limits. Examples demonstrate how to apply this theorem.
This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.
Algebra difficulties among second year bachelor of secondaryJunarie Ramirez
This study examined the difficulties that second year Bachelor of Secondary Education students experienced with algebra. Questionnaires were used to understand what causes difficulties and specific algebraic topics that posed problems. The results showed that ineffective teaching strategies were the primary cause of difficulties. Special products/factoring and rationalizing denominators were the most challenging topics. The researchers concluded that teachers must find better ways to explain concepts and cater their lessons to students' needs.
This document discusses heteroskedasticity in econometric models. It defines heteroskedasticity as non-constant variance of the error term, in contrast to the homoskedasticity assumption of constant variance. It explains that while OLS estimates remain unbiased with heteroskedasticity, the standard errors are biased. Robust standard errors can provide consistent standard errors even with heteroskedasticity. The Breusch-Pagan and White tests are presented as methods to test for the presence of heteroskedasticity based on the residuals. Weighted least squares is also introduced as a method to obtain more efficient estimates than OLS when the form of heteroskedasticity is known.
This document provides an overview of structural equation modeling (SEM) using AMOS. It defines key SEM concepts like latent variables, observed variables, path analysis, and model identification. It also explains how to specify and estimate a SEM model in AMOS, including how to draw path diagrams, name variables, set regression weights, and view output. Model fit is discussed along with potential issues like sample size. Confirmatory factor analysis and other SEM models like path analysis and latent growth models are also introduced.
The document provides instructions for graphing two hyperbolas based on their standard equations. It first gives the equations x^2 - y^2/4 = 1 and y^2 - x^2/9 = 1 and explains how to locate and draw the vertices, foci, asymptotes, and hyperbola shape for each one. It also reviews key properties of hyperbolas such as their transverse axis and the relationship between the foci and constant difference of distances along the hyperbola.
The document provides an overview of annuities and time value of money concepts. It discusses ordinary annuities, including how to calculate future value, payment amount, interest rate, and number of periods for an ordinary annuity. It also covers present value of annuities, amortized loans including loan payments and schedules, and annuities due. Examples are provided for each topic and how to solve them using a financial calculator or Excel spreadsheet.
This chapter discusses numerical descriptive measures used to describe the central tendency, variation, and shape of data. It covers calculating the mean, median, mode, variance, standard deviation, and coefficient of variation for data. The geometric mean is introduced as a measure of the average rate of change over time. Outliers are identified using z-scores. Methods for summarizing and comparing data using these descriptive statistics are presented.
The document discusses techniques for building multiple regression models, including:
- Using quadratic and transformed terms to model nonlinear relationships
- Detecting and addressing collinearity among independent variables
- Employing stepwise regression or best-subsets approaches to select significant variables and develop the best-fitting model
This document provides an overview of key concepts in regression analysis, including simple and multiple linear regression models. It outlines 10 learning objectives for the chapter, which cover topics like developing regression equations from sample data, interpreting regression outputs, assessing model fit, and addressing violations of regression assumptions. The document also includes sample regression calculations and residual plots for a case study on predicting home renovation sales from area payroll levels.
This document provides an introduction and overview of linear equations. It defines key terms like equations, variables, and solutions. It explains that the goal in solving equations is to find the value of the unknown that makes the statement true. The document outlines various properties of equality that can be used to solve equations, such as distributing the same operation to both sides. It also distinguishes between linear and nonlinear equations. Several examples are provided to demonstrate how to solve different types of linear equations, including those with fractions and those that simplify to linear form. The document also briefly introduces solving power equations, which involve variables raised to powers, as well as equations with fractional exponents.
ch02 - Conceptual Framework for Financial Reporting.pptNicolasErnesto2
The conceptual framework establishes fundamental concepts that guide standard-setting and financial reporting more broadly. It is being jointly developed by the IASB and FASB and consists of three levels: the objective of financial reporting, qualitative characteristics, and specific concepts. The objective is to provide useful information to capital providers. Key qualitative characteristics include relevance and faithful representation. The framework also outlines basic elements, assumptions, principles, and constraints that guide accounting practices. It aims to create consistency and coherence in financial reporting standards over time.
This document provides an overview of the SAP Fiori architecture. It discusses the prerequisites, components, deployment options, typical landscape, and future plans. SAP Fiori allows users to access business applications from desktops, tablets, and smartphones using HTML5. It consists of two main components: UI addons and integration addons. The integration addons provide OData services via SAP NetWeaver Gateway while the UI addons contain the apps and UI elements.
Null hypothesis for single linear regressionKen Plummer
The document discusses the null hypothesis for a single linear regression analysis. It explains that the null hypothesis states that there is no effect or relationship between the independent and dependent variables. As an example, if investigating the relationship between hours of sleep and ACT scores, the null hypothesis would be: "There will be no significant prediction of ACT scores by hours of sleep." The document provides a template for writing the null hypothesis in terms of the specific independent and dependent variables being analyzed.
This chapter discusses basic probability concepts, including defining probability, sample spaces, simple and joint events, and assessing probability through classical and subjective approaches. It also covers key probability rules like the general addition rule, computing conditional probabilities, statistical independence, and Bayes' theorem. The goals are to explain these fundamental probability topics, show how to apply common probability rules, and determine if events are statistically independent or dependent.
Bba 3274 qm week 6 part 1 regression modelsStephen Ong
This document provides an overview and outline of regression models and forecasting techniques. It discusses simple and multiple linear regression analysis, how to measure the fit of regression models, assumptions of regression models, and testing models for significance. The goals are to help students understand relationships between variables, predict variable values, develop regression equations from sample data, and properly apply and interpret regression analysis.
This document discusses regression analysis and linear regression models. It provides examples of how regression can be used to understand relationships between variables and predict values. Specifically, it examines a case study of how sales from a home renovation company (Triple A Construction) can be predicted based on area payroll. The regression line that best fits the data is calculated and metrics like the coefficient of determination are used to evaluate how well the regression model fits the data. Assumptions of the linear regression model like independent and normally distributed errors are also covered.
This presentation discusses methods, tools, and best practices for integrating SAP with other platforms. It provides an overview of typical integration scenarios like integrating e-commerce platforms, CRMs, supplier portals, and custom applications. It also outlines the main SAP versions and how they can be integrated, such as using APIs, RFC calls, IDOCs, and oData services. Top considerations for integration include prioritizing real-time over batch sync and proper exception handling. The presentation is meant to help solutions owners, architects and consultants with SAP integration.
The document discusses factorial analysis of variance (ANOVA). It explains how total sums of squares can be partitioned into explained and unexplained components. An example shows an F ratio of 5.0 for one data set, indicating variation between groups is rare. This allows rejecting the null hypothesis with a low probability of Type I error. Finally, it describes how factorial ANOVA can analyze the effects of multiple independent variables on a single dependent variable.
This document summarizes the key topics and concepts covered in Chapter 2 of the 9th edition of the business statistics textbook "Presenting Data in Tables and Charts". The chapter discusses guidelines for analyzing data and organizing both numerical and categorical data. It then covers various methods for tabulating and graphing univariate and bivariate data, including tables, histograms, frequency distributions, scatter plots, bar charts, pie charts, and contingency tables.
This document provides an overview of basic statistics concepts. It defines statistics as the science of collecting, presenting, analyzing, and reasonably interpreting data. Descriptive statistics are used to summarize and organize data through methods like tables, graphs, and descriptive values, while inferential statistics allow researchers to make general conclusions about populations based on sample data. Variables can be either categorical or quantitative, and their distributions and presentations are discussed.
This document provides an overview of forecasting techniques. It begins with the objectives of the chapter, which are to understand various forecasting models and compare methods such as moving averages, exponential smoothing, and time-series models. It also covers qualitatively measuring forecast accuracy. The document then describes different forecasting techniques including qualitative models, time-series models, and causal models. It provides examples of moving averages, weighted moving averages, and exponential smoothing techniques. It concludes with examples of how to implement forecasting models in Excel.
This document discusses limits of functions, including infinite limits, vertical and horizontal asymptotes, and the squeeze theorem. It provides definitions and examples of:
- Infinite limits, where the value of a function increases or decreases without bound as the input approaches a number.
- Vertical and horizontal asymptotes, which are lines that a function approaches but does not meet as the input increases or decreases without limit.
- The squeeze theorem, which can be used to evaluate limits where usual algebraic methods are not effective by "squeezing" the function between two other functions with known limits. Examples demonstrate how to apply this theorem.
This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.
Algebra difficulties among second year bachelor of secondaryJunarie Ramirez
This study examined the difficulties that second year Bachelor of Secondary Education students experienced with algebra. Questionnaires were used to understand what causes difficulties and specific algebraic topics that posed problems. The results showed that ineffective teaching strategies were the primary cause of difficulties. Special products/factoring and rationalizing denominators were the most challenging topics. The researchers concluded that teachers must find better ways to explain concepts and cater their lessons to students' needs.
This document discusses heteroskedasticity in econometric models. It defines heteroskedasticity as non-constant variance of the error term, in contrast to the homoskedasticity assumption of constant variance. It explains that while OLS estimates remain unbiased with heteroskedasticity, the standard errors are biased. Robust standard errors can provide consistent standard errors even with heteroskedasticity. The Breusch-Pagan and White tests are presented as methods to test for the presence of heteroskedasticity based on the residuals. Weighted least squares is also introduced as a method to obtain more efficient estimates than OLS when the form of heteroskedasticity is known.
This document provides an overview of structural equation modeling (SEM) using AMOS. It defines key SEM concepts like latent variables, observed variables, path analysis, and model identification. It also explains how to specify and estimate a SEM model in AMOS, including how to draw path diagrams, name variables, set regression weights, and view output. Model fit is discussed along with potential issues like sample size. Confirmatory factor analysis and other SEM models like path analysis and latent growth models are also introduced.
The document provides instructions for graphing two hyperbolas based on their standard equations. It first gives the equations x^2 - y^2/4 = 1 and y^2 - x^2/9 = 1 and explains how to locate and draw the vertices, foci, asymptotes, and hyperbola shape for each one. It also reviews key properties of hyperbolas such as their transverse axis and the relationship between the foci and constant difference of distances along the hyperbola.
The document provides an overview of annuities and time value of money concepts. It discusses ordinary annuities, including how to calculate future value, payment amount, interest rate, and number of periods for an ordinary annuity. It also covers present value of annuities, amortized loans including loan payments and schedules, and annuities due. Examples are provided for each topic and how to solve them using a financial calculator or Excel spreadsheet.
This chapter discusses numerical descriptive measures used to describe the central tendency, variation, and shape of data. It covers calculating the mean, median, mode, variance, standard deviation, and coefficient of variation for data. The geometric mean is introduced as a measure of the average rate of change over time. Outliers are identified using z-scores. Methods for summarizing and comparing data using these descriptive statistics are presented.
The document discusses techniques for building multiple regression models, including:
- Using quadratic and transformed terms to model nonlinear relationships
- Detecting and addressing collinearity among independent variables
- Employing stepwise regression or best-subsets approaches to select significant variables and develop the best-fitting model
This document provides an overview of key concepts in regression analysis, including simple and multiple linear regression models. It outlines 10 learning objectives for the chapter, which cover topics like developing regression equations from sample data, interpreting regression outputs, assessing model fit, and addressing violations of regression assumptions. The document also includes sample regression calculations and residual plots for a case study on predicting home renovation sales from area payroll levels.
This document provides an introduction and overview of linear equations. It defines key terms like equations, variables, and solutions. It explains that the goal in solving equations is to find the value of the unknown that makes the statement true. The document outlines various properties of equality that can be used to solve equations, such as distributing the same operation to both sides. It also distinguishes between linear and nonlinear equations. Several examples are provided to demonstrate how to solve different types of linear equations, including those with fractions and those that simplify to linear form. The document also briefly introduces solving power equations, which involve variables raised to powers, as well as equations with fractional exponents.
ch02 - Conceptual Framework for Financial Reporting.pptNicolasErnesto2
The conceptual framework establishes fundamental concepts that guide standard-setting and financial reporting more broadly. It is being jointly developed by the IASB and FASB and consists of three levels: the objective of financial reporting, qualitative characteristics, and specific concepts. The objective is to provide useful information to capital providers. Key qualitative characteristics include relevance and faithful representation. The framework also outlines basic elements, assumptions, principles, and constraints that guide accounting practices. It aims to create consistency and coherence in financial reporting standards over time.
This document provides an overview of the SAP Fiori architecture. It discusses the prerequisites, components, deployment options, typical landscape, and future plans. SAP Fiori allows users to access business applications from desktops, tablets, and smartphones using HTML5. It consists of two main components: UI addons and integration addons. The integration addons provide OData services via SAP NetWeaver Gateway while the UI addons contain the apps and UI elements.
Null hypothesis for single linear regressionKen Plummer
The document discusses the null hypothesis for a single linear regression analysis. It explains that the null hypothesis states that there is no effect or relationship between the independent and dependent variables. As an example, if investigating the relationship between hours of sleep and ACT scores, the null hypothesis would be: "There will be no significant prediction of ACT scores by hours of sleep." The document provides a template for writing the null hypothesis in terms of the specific independent and dependent variables being analyzed.
This chapter discusses basic probability concepts, including defining probability, sample spaces, simple and joint events, and assessing probability through classical and subjective approaches. It also covers key probability rules like the general addition rule, computing conditional probabilities, statistical independence, and Bayes' theorem. The goals are to explain these fundamental probability topics, show how to apply common probability rules, and determine if events are statistically independent or dependent.
Bba 3274 qm week 6 part 1 regression modelsStephen Ong
This document provides an overview and outline of regression models and forecasting techniques. It discusses simple and multiple linear regression analysis, how to measure the fit of regression models, assumptions of regression models, and testing models for significance. The goals are to help students understand relationships between variables, predict variable values, develop regression equations from sample data, and properly apply and interpret regression analysis.
This document discusses regression analysis and linear regression models. It provides examples of how regression can be used to understand relationships between variables and predict values. Specifically, it examines a case study of how sales from a home renovation company (Triple A Construction) can be predicted based on area payroll. The regression line that best fits the data is calculated and metrics like the coefficient of determination are used to evaluate how well the regression model fits the data. Assumptions of the linear regression model like independent and normally distributed errors are also covered.
This document provides an overview of regression models and their use in business analytics. It discusses simple and multiple linear regression models, how to develop regression equations from sample data, and how to interpret key outputs like the slope, intercept, coefficient of determination, and correlation coefficient. Regression analysis is presented as a valuable tool for managers to understand relationships between variables and predict outcomes. The document outlines the key steps in regression including developing scatter plots, calculating regression equations, and measuring the fit of regression models.
This document provides teaching suggestions for regression models:
1) It suggests emphasizing the difference between independent and dependent variables in a regression model using examples.
2) It notes that correlation does not necessarily imply causation and gives an example of variables that are correlated but changing one does not affect the other.
3) It recommends having students manually draw regression lines through data points to appreciate the least squares criterion.
4) It advises selecting random data values to generate a regression line in Excel to demonstrate determining the coefficient of determination and F-test.
5) It suggests discussing the full and shortcut regression formulas to provide a better understanding of the concepts.
This document provides teaching suggestions for regression models:
1) It suggests emphasizing the difference between independent and dependent variables in a regression model using examples.
2) It notes that correlation does not necessarily imply causation and gives an example of variables that are correlated but changing one does not affect the other.
3) It recommends having students manually draw regression lines through data points to appreciate the least squares criterion.
4) It advises selecting random data values to generate a regression line in Excel to demonstrate determining the coefficient of determination and F-test.
5) It suggests discussing the full and shortcut regression formulas to provide a better understanding of the concepts.
The document provides an overview of regression analysis including:
- Regression analysis is a statistical process used to estimate relationships between variables and predict unknown values.
- The document outlines different types of regression like simple, multiple, linear, and nonlinear regression.
- Key aspects of regression like scatter diagrams, regression lines, and the method of least squares are explained.
- An example problem is worked through demonstrating how to calculate the slope and y-intercept of a regression line using the least squares method.
This document discusses multiple regression analysis. It begins by introducing multiple regression as an extension of simple linear regression that allows for modeling relationships between a response variable and multiple explanatory variables. It then covers topics such as examining variable distributions, building regression models, estimating model parameters, and assessing overall model fit and significance of individual predictors. An example demonstrates using multiple regression to build a model for predicting cable television subscribers based on advertising rates, station power, number of local families, and number of competing stations.
Chapter 4 power point presentation Regression modelsJustinXerri
This document outlines key concepts from a chapter on regression models, including:
1. The chapter covers simple and multiple linear regression models, including how to calculate slopes, intercepts, coefficients of determination, and correlation coefficients.
2. Regression can be used to understand relationships between variables and predict the value of dependent variables based on independent variables.
3. The document provides an example of using simple linear regression to predict sales based on payroll for a construction company.
Multiple linear regression allows modeling of relationships between a dependent variable and multiple independent variables. It estimates the coefficients (betas) that best fit the data to a linear equation. The ordinary least squares method is commonly used to estimate the betas by minimizing the sum of squared residuals. Diagnostics include checking overall model significance with F-tests, individual variable significance with t-tests, and detecting multicollinearity. Qualitative variables require preprocessing with dummy variables before inclusion in a regression model.
An econometric model for Linear Regression using StatisticsIRJET Journal
This document discusses linear regression modeling using statistics. It begins by introducing linear regression and its assumptions. Both univariate and multivariate linear regression are covered. The coefficients are derived using statistics in matrix form. Properties of ordinary least squares estimators like their expected values and variances are proven. Hypothesis testing for multiple linear regression is presented in matrix form. The document emphasizes the importance of understanding linear regression for prediction and its application in fields like economics and social sciences. Rigorous statistical analysis is needed to ensure the validity of regression models.
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Chapter 10: Correlation and Regression
10.2: Regression
This document presents a nonparametric approach to multiple regression that uses ranks instead of raw values for both the dependent and independent variables. The key points are:
1. It develops a nonparametric multiple regression model using the ranks of observations on the dependent variable and ranks of observations on the independent variables.
2. The method of least squares is applied to the rank-based model to obtain estimates of the regression coefficients.
3. Prediction equations are presented that allow predicting dependent variable ranks based on independent variable ranks.
This document summarizes an analysis of using Support Vector Regression (SVR) to predict bike rental data from a bike sharing program in Washington D.C. It begins with an introduction to SVR and the bike rental prediction competition. It then shows that linear regression performs poorly on this non-linear problem. The document explains how SVR maps data into higher dimensions using kernel functions to allow for non-linear fits. It concludes by outlining the derivation of the SVR method using kernel functions to simplify calculations for the regression.
Exploring Support Vector Regression - Signals and Systems ProjectSurya Chandra
Our team competed in a Kaggle competition to predict the bike share use as a part of their capital bike share program in Washington DC using a powerful function approximation technique called support vector regression.
Page 1 of 18Part A Multiple Choice (1–11)______1. Using.docxalfred4lewis58146
Page 1 of 18
Part A: Multiple Choice (1–11)
______1. Using the “eyeball” method, the regression line = 2+2x has been fitted to the data points (x = 2, y = 1), (x = 3, y = 8), and (x = 4, y = 7). The sum of the squared residuals will be
a. 7 b. 19 c. 34 d. 8
______2. A computer statistical package has included the following quantities in its output: SST = 50, SSR = 35, and SSE = 15. How much of the variation in y is explained by the regression equation?
a. 49% b. 70% c. 35% d. 15%
______3. In testing the significance of b, the null hypothesis is generally that
a. β = b b. β 0 c. β = 0 d. β = r
______4. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the population _____________ could be zero.
a. standard error of estimate c. y-intercept
b. prediction interval d. coefficient of correlation
______5. A multiple regression equation includes 4 independent variables, and the coefficient of multiple determination is 0.64. How much of the variation in y is explained by the regression equation?
a. 80% b. 16% c. 32% d. 64%
______6. A multiple regression analysis results in the following values for the sum-of-squares terms: SST = 50.0, SSR = 35.0, and SSE = 15.0. The coefficient of multiple determination will be
a. = 0.35 b. = 0.30 c. = 0.70 d. = 0.50
______7. In testing the overall significance of a multiple regression equation in which there are three independent variables, the null hypothesis is
a. :
b. :
c. :
d. :
______8. In a multiple regression analysis involving 25 data points and 4 independent variables, the sum-of-squares terms are calculated as SSR = 120, SSE = 80, and SST = 200. In testing the overall significance of the regression equation, the calculated value of the test statistic will be
a. F = 1.5 c. F = 5.5
b. F = 2.5 d. F = 7.5
______9. For a set of 15 data points, a computer statistical package has found the multiple regression equation to be = -23 + 20+ 5 + 25 and has listed the t-ratio for testing the significance of each partial regression coefficient. Using the 0.05 level in testing whether = 20 differs significantly from zero, the critical t values will be
a. t = -1.960 and t= +1.960
b. t = -2.132 and t = +2.132
c. t = -2.201 and t = +2.201
d. t = -1.796 and t = +1.796
______10. Computer analyses typically provide a p-Value for each partial regression coefficient. In the case of , this is the probability that
a. = 0
b. =
c. the absolute value of could be this large if = 0
d. the absolute value of could be this large if 1
______11. In the multiple regression equation, = 20,000 + 0.05+ 4500 , is the estimated household income, is the amount of life insurance held by the head of the household, and is a dummy variable ( = 1 if the family owns mutual funds, 0 if it doesn’t). The interpretation of = 4500 is that
a. owing mutual funds increases the estimated income by $4500
b. the average value of a mut.
1) The document discusses simple linear regression using a scatter diagram and data from a study of employees' years of working experience and income.
2) It presents the scatter diagram and shows how to draw a trend line to roughly estimate dependent variable (income) values from the independent variable (years experience).
3) Equations for the least squares linear regression line are provided, including how to calculate the standard error of estimate, which is interpreted as the standard deviation around the regression line.
Data Approximation in Mathematical Modelling Regression Analysis and Curve Fi...Dr.Summiya Parveen
Outline of the lecture:
Introduction of Regression
Application of Regression
Regression Techniques
Types of Regression
Goodness of fit
MATLAB/MATHEMATICA implementation with some example
Regression analysis is a form of predictive modelling technique which investigates the relationship between a dependent (target) and independent variable (s) (predictor). This technique is used for forecasting, time series modelling and finding the casual effect relationship between the variables. Regression analysis is an important tool for modelling and analysing data. Here, we fit a curve / line to the data points in such a manner that the differences between the distances of data points from the curve or line is minimized.
By DR. SUMMIYA PARVEEN
Data Approximation in Mathematical Modelling Regression Analysis and curve fi...Dr.Summiya Parveen
The document summarizes a lecture on regression analysis and curve fitting in mathematical modelling. It introduces regression analysis and its applications. It describes different regression techniques like linear, nonlinear, polynomial and multiple regression. It provides examples of fitting linear and polynomial curves to data using MATLAB. It discusses assessing the goodness of fit using metrics like residual norm and coefficient of determination.
Quantitative Analysis for Management new pptkrrish242
This document discusses regression analysis and linear regression models. It provides learning objectives for a chapter on regression models, including developing simple and multiple linear regression equations, interpreting coefficients, and addressing assumptions of regression models. It also includes an example of using regression to model the relationship between a construction company's sales and local payroll. Key terms discussed include dependent and independent variables, the regression line, residuals, and measures of fit such as the coefficient of determination and correlation coefficient.
This document provides an overview of Chapter 10 from the textbook "Quantitative Analysis for Management" which covers transportation and assignment models. The chapter objectives are to teach students how to structure and solve linear programming problems using transportation and assignment models. It introduces the transportation model for distributing goods from suppliers to customers and the assignment model for allocating resources to tasks. The document outlines the chapter and provides examples of setting up and solving a transportation problem using methods like the northwest corner rule and stepping stone method.
The document describes several examples of using linear programming (LP) to solve optimization problems in marketing, manufacturing, and finance. It provides details of LP models for selecting an optimal media mix in advertising, determining the best product mix for a tie manufacturer, and identifying an ideal investment portfolio to maximize returns while meeting risk constraints. Excel solutions are presented for each case study.
The document discusses linear programming models and solution methods. It begins by defining linear programming and its key properties and requirements. It then presents an example problem involving Flair Furniture Company to maximize profit from table and chair production given resource constraints. The problem is formulated mathematically and solved graphically using the feasible region and isoprofit line methods. The isoprofit line method involves moving the objective function line to find the highest profit point touching the feasible region. The corner point method, another solution approach, examines profits at each corner point of the feasible region, where the optimal solution must lie.
The document discusses inventory control models and the economic order quantity (EOQ) model. It provides an overview of inventory planning and control processes. The EOQ model aims to minimize total inventory costs by determining the optimal order quantity and reorder point. The model balances ordering, holding, and stockout costs. It provides formulas to calculate the EOQ based on annual demand, ordering costs, and carrying costs. Quantity discount models are also discussed, which incorporate variable unit costs into the total cost calculation.
This document provides an overview of integer programming, goal programming, and nonlinear programming. It begins with learning objectives and an outline of topics to be covered, which include integer programming, modeling with binary variables, goal programming, and nonlinear programming. Several examples are provided to illustrate integer programming problems and how they can be formulated and solved. Mixed-integer and binary variable modeling are explained. Goal programming and how it differs from linear programming in addressing multiple objectives is introduced.
This chapter discusses transportation, assignment, and transshipment models as special cases of linear programming network flow problems. It provides learning objectives and an outline of topics to be covered, which include introducing the transportation problem using an example of distributing office desks from factories to warehouses, formulating it as a linear program, and solving it using the transportation algorithm. The chapter also discusses the assignment problem using an example of assigning workers to repair jobs and the transshipment problem using an example of shipping snow blowers through distribution centers. It describes developing initial feasible solutions using the northwest corner rule and improving solutions using the stepping stone method.
This document discusses linear programming applications in marketing, manufacturing, and other areas. It provides examples to demonstrate how to model and solve linear programming problems involving media mix optimization, production scheduling, inventory management, and other scenarios. Specifically, it presents sample problems and solutions involving marketing mix optimization for a gambling club, sampling costs for a market research firm, production planning for a tie manufacturer, and multi-period production scheduling for an electric motor company. The chapter aims to illustrate how to apply linear programming to optimize objectives subject to constraints across various business applications.
This chapter discusses linear programming models and their graphical and computer-based solutions. It begins by outlining the learning objectives and chapter contents. Key points covered include:
- The basic assumptions and requirements of linear programming problems
- How to formulate an LP problem by defining variables, objectives and constraints
- Graphically representing constraints and determining the feasible region
- Using isoprofit lines and the corner point method to solve LP problems graphically
- An example problem involving determining optimal product mix for Flair Furniture is presented and solved graphically.
Here are the steps to solve this example:
1) Set up the production run model equations:
- Optimal production quantity equation:
(p - d)C/DC√(h/s) = Q*
- Annual setup cost equation:
sC/Q = D
2) Plug in the values:
- p = 80 units/day
- d = 60 units/day
- C = $0.50/unit/year
- D = 10,000 units/year
- Cs = $100
- Ch = $0.50/unit/year
3) Solve the optimal production quantity equation for Q*:
Q* = √(80-60
This document outlines the key steps and concepts in decision making. It discusses the six steps in decision making: 1) define the problem, 2) list alternatives, 3) identify outcomes, 4) list payoffs, 5) select a decision model, and 6) make a decision. It also describes three types of decision environments: decision making under certainty, uncertainty, and risk. Finally, it discusses approaches for decision making under uncertainty like maximax and maximin, and the use of expected monetary value for decision making under risk.
This document outlines the key concepts and steps involved in decision analysis and decision making under uncertainty. It discusses the six steps in decision making, types of decision making environments, and methods for making decisions under uncertainty, risk, and with imperfect information. These methods include maximax, maximin, Hurwicz criterion, equally likely, minimax regret, expected monetary value, expected value of perfect information, and expected opportunity loss. An example involving a company called Thompson Lumber is used to illustrate applying these decision making techniques. Sensitivity analysis is also discussed as a way to examine how the optimal decision may change with different input data.
This document provides an overview of key concepts in probability analysis that will be covered in a quantitative analysis chapter. It includes learning objectives, a chapter outline, and explanations of fundamental probability concepts such as mutually exclusive and collectively exhaustive events, independent and dependent events, and how to calculate probabilities using formulas like Bayes' theorem. Examples are provided to illustrate concepts like determining probabilities from data, adding probabilities of events, and revising probabilities with new information.
This chapter introduces quantitative analysis and its applications. Quantitative analysis is a scientific approach to managerial decision making that processes raw data into meaningful information. The chapter outlines the quantitative analysis approach, which includes defining the problem, developing a model, acquiring data, developing a solution, testing the solution, analyzing results, and implementing. It also discusses using computers and spreadsheets to develop models and potential problems with the quantitative analysis approach.
This document outlines concepts related to decision making under uncertainty and risk. It discusses six steps for decision making, including defining the problem, listing alternatives and outcomes, and identifying payoffs. It then covers various models for decision making under uncertainty, like maximax, maximin, and expected monetary value. Sensitivity analysis is presented as a way to examine how decisions may change with different input data or probabilities. The document uses examples like Thompson Lumber Company to illustrate concepts step-by-step.
This document outlines concepts related to decision making under uncertainty and risk. It discusses six steps to decision making, including defining the problem, listing alternatives and outcomes, and identifying payoffs. It then covers various decision making models for uncertainty, like maximax, maximin, and expected monetary value. Sensitivity analysis is introduced as a way to examine how decisions may change with different input data. The document uses examples and tables to illustrate key concepts in decision analysis.
Quantitative analysis is a scientific approach to managerial decision making that involves defining problems, acquiring data, developing models, testing solutions, analyzing results, and implementing solutions. It uses mathematical tools to process raw data into meaningful information. Potential problems include conflicting viewpoints, outdated solutions, difficulties acquiring and understanding models, and resistance to change. Implementation is critical to success but challenging to achieve.
This document provides an overview of chapter 9 from a quantitative analysis textbook. It discusses the simplex method for solving linear programming problems. The chapter objectives are to convert linear programming constraints to equalities using slack and surplus variables, set up and solve problems using simplex tableaus, interpret simplex tableau numbers, handle special cases, and conduct sensitivity analysis and construct the dual problem. The document then outlines the chapter and provides examples to introduce the initial simplex tableau, identify the pivot row and column, and describe the five steps to compute the next simplex tableau iteration.
This document discusses the Hungarian method for solving assignment problems. It begins by explaining the assignment problem and providing an example with 2 machines and 2 operators. It then describes the Hungarian method, which finds the optimal assignment in polynomial time. The method is explained through a 4x4 example, with the key steps being: 1) row and column reductions, 2) finding a complete assignment or using adjustments to enable it. The document also notes the method works for balanced or minimization problems, and how maximization problems can be converted. It concludes by providing a professor-subject example and walking through applying the Hungarian method to find the optimal assignment.
This chapter discusses three network models: the minimal spanning tree technique, which finds a path through a network connecting all points while minimizing total distance; the maximal flow technique, which determines the maximum flow of a substance through a network; and the shortest route technique, which finds the shortest path through a network. It provides examples of applying each technique to solve problems faced by construction, transportation, and utility companies.
Progress Report - Qualcomm AI Workshop - AI available - everywhereAI summit 1...Holger Mueller
Qualcomm invited analysts and media for an AI workshop, held at Qualcomm HQ in San Diego, June 26th. My key takeaways across the different offerings is that Qualcomm us using AI across its whole portfolio. Remarkable to other analyst summits was 50% of time being dedicated to demos / hands on exeriences.
L'indice de performance des ports à conteneurs de l'année 2023SPATPortToamasina
Une évaluation comparable de la performance basée sur le temps d'escale des navires
L'objectif de l'ICPP est d'identifier les domaines d'amélioration qui peuvent en fin de compte bénéficier à toutes les parties concernées, des compagnies maritimes aux gouvernements nationaux en passant par les consommateurs. Il est conçu pour servir de point de référence aux principaux acteurs de l'économie mondiale, notamment les autorités et les opérateurs portuaires, les gouvernements nationaux, les organisations supranationales, les agences de développement, les divers intérêts maritimes et d'autres acteurs publics et privés du commerce, de la logistique et des services de la chaîne d'approvisionnement.
Le développement de l'ICPP repose sur le temps total passé par les porte-conteneurs dans les ports, de la manière expliquée dans les sections suivantes du rapport, et comme dans les itérations précédentes de l'ICPP. Cette quatrième itération utilise des données pour l'année civile complète 2023. Elle poursuit le changement introduit l'année dernière en n'incluant que les ports qui ont eu un minimum de 24 escales valides au cours de la période de 12 mois de l'étude. Le nombre de ports inclus dans l'ICPP 2023 est de 405.
Comme dans les éditions précédentes de l'ICPP, la production du classement fait appel à deux approches méthodologiques différentes : une approche administrative, ou technique, une méthodologie pragmatique reflétant les connaissances et le jugement des experts ; et une approche statistique, utilisant l'analyse factorielle (AF), ou plus précisément la factorisation matricielle. L'utilisation de ces deux approches vise à garantir que le classement des performances des ports à conteneurs reflète le plus fidèlement possible les performances réelles des ports, tout en étant statistiquement robuste.
The Key Summaries of Forum Gas 2024.pptxSampe Purba
The Gas Forum 2024 organized by SKKMIGAS, get latest insights From Government, Gas Producers, Infrastructures and Transportation Operator, Buyers, End Users and Gas Analyst
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2. 4-2
Learning Objectives
1. Identify variables and use them in a regression
model.
2. Develop simple linear regression equations.
from sample data and interpret the slope and
intercept.
3. Compute the coefficient of determination and
the coefficient of correlation and interpret their
meanings.
4. Interpret the F-test in a linear regression model.
5. List the assumptions used in regression and
use residual plots to identify problems.
After completing this chapter, students will be able to:After completing this chapter, students will be able to:
3. 4-3
Learning Objectives
6. Develop a multiple regression model and use it
for prediction purposes.
7. Use dummy variables to model categorical
data.
8. Determine which variables should be included
in a multiple regression model.
9. Transform a nonlinear function into a linear
one for use in regression.
10. Understand and avoid common mistakes made
in the use of regression analysis.
After completing this chapter, students will be able to:After completing this chapter, students will be able to:
4. 4-4
Chapter Outline
4.1 Introduction
4.2 Scatter Diagrams
4.3 Simple Linear Regression
4.4 Measuring the Fit of the Regression
Model
4.5 Using Computer Software for Regression
4.6 Assumptions of the Regression Model
5. 4-5
Chapter Outline
4.7 Testing the Model for Significance
4.8 Multiple Regression Analysis
4.9 Binary or Dummy Variables
4.10 Model Building
4.11 Nonlinear Regression
4.12 Cautions and Pitfalls in Regression
Analysis
6. 4-6
Introduction
Regression analysisRegression analysis is a very valuable
tool for a manager.
Regression can be used to:
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
variables.
7. 4-7
Introduction
The variable to be predicted is called
the dependent variabledependent variable.
This is sometimes called the responseresponse
variable.variable.
The value of this variable depends on
the value of the independent variable.independent variable.
This is sometimes called the explanatoryexplanatory
or predictor variable.predictor variable.
Independent
variable
Dependent
variable
Independent
variable
= +
8. 4-8
Scatter Diagram
A scatter diagramscatter diagram or scatter plotscatter plot is
often used to investigate the
relationship between variables.
The independent variable is normally
plotted on the X axis.
The dependent variable is normally
plotted on the Y axis.
9. 4-9
Triple A Construction
Triple A Construction renovates old homes.
Managers have found that the dollar volume of
renovation work is dependent on the area
payroll.
TRIPLE A’S SALES
($100,000s)
LOCAL PAYROLL
($100,000,000s)
6 3
8 4
9 6
5 4
4.5 2
9.5 5
Table 4.1
11. 4-11
Simple Linear Regression
where
Y = dependent variable (response)
X = independent variable (predictor or explanatory)
β0 = intercept (value of Y when X = 0)
β1 = slope of the regression line
ε = random error
Regression models are used to test if there is a
relationship between variables.
There is some random error that cannot be
predicted.
εββ ++= XY 10
12. 4-12
Simple Linear Regression
True values for the slope and intercept are not
known so they are estimated using sample data.
XbbY 10 +=ˆ
where
Y = predicted value of Y
b0 = estimate of β0, based on sample results
b1 = estimate of β1, based on sample results
^
13. 4-13
Triple A Construction
Triple A Construction is trying to predict sales
based on area payroll.
Y = Sales
X = Area payroll
The line chosen in Figure 4.1 is the one that
minimizes the errors.
Error = (Actual value) – (Predicted value)
YYe ˆ−=
14. 4-14
Triple A Construction
For the simple linear regression model, the values
of the intercept and slope can be calculated using
the formulas below.
XbbY 10 +=ˆ
valuesof(mean)average X
n
X
X ==
∑
valuesof(mean)average Y
n
Y
Y ==
∑
∑
∑
−
−−
= 21
)(
))((
XX
YYXX
b
XbYb 10 −=
16. 4-16
Triple A Construction
4
6
24
6
===
∑ X
X
7
6
42
6
===
∑Y
Y
251
10
512
21 .
.
)(
))((
==
−
−−
=
∑
∑
XX
YYXX
b
24251710 =−=−= ))(.(XbYb
Regression calculations
XY 2512 .ˆ +=Therefore
17. 4-17
Triple A Construction
4
6
24
6
===
∑ X
X
7
6
42
6
===
∑Y
Y
251
10
512
21 .
.
)(
))((
==
−
−−
=
∑
∑
XX
YYXX
b
24251710 =−=−= ))(.(XbYb
Regression calculations
XY 2512 .ˆ +=Therefore
sales = 2 + 1.25(payroll)
If the payroll next
year is $600 million
000950$or5962512 ,.)(.ˆ =+=Y
18. 4-18
Measuring the Fit
of the Regression Model
Regression models can be developed
for any variables X and Y.
How do we know the model is actually
helpful in predicting Y based on X?
We could just take the average error, but
the positive and negative errors would
cancel each other out.
Three measures of variability are:
SST – Total variability about the mean.
SSE – Variability about the regression line.
SSR – Total variability that is explained by
the model.
19. 4-19
Measuring the Fit
of the Regression Model
Sum of the squares total:
2
)(∑ −= YYSST
Sum of the squared error:
∑ ∑ −== 22
)ˆ( YYeSSE
Sum of squares due to regression:
∑ −= 2
)ˆ( YYSSR
An important relationship:
SSESSRSST +=
20. 4-20
Measuring the Fit
of the Regression Model
Y X (Y – Y)2
Y (Y – Y)2
(Y – Y)2
6 3 (6 – 7)2
= 1 2 + 1.25(3) = 5.75 0.0625 1.563
8 4 (8 – 7)2
= 1 2 + 1.25(4) = 7.00 1 0
9 6 (9 – 7)2
= 4 2 + 1.25(6) = 9.50 0.25 6.25
5 4 (5 – 7)2
= 4 2 + 1.25(4) = 7.00 4 0
4.5 2 (4.5 – 7)2
= 6.25 2 + 1.25(2) = 4.50 0 6.25
9.5 5 (9.5 – 7)2
= 6.25 2 + 1.25(5) = 8.25 1.5625 1.563
∑(Y – Y)2
= 22.5 ∑(Y – Y)2
= 6.875
∑(Y – Y)2
=
15.625
Y = 7 SST = 22.5 SSE = 6.875 SSR = 15.625
^
^^
^^
Table 4.3
Sum of Squares for Triple A Construction
21. 4-21
Sum of the squares total
2
)(∑ −= YYSST
Sum of the squared error
∑ ∑ −== 22
)ˆ( YYeSSE
Sum of squares due to regression
∑ −= 2
)ˆ( YYSSR
An important relationship
SSESSRSST +=
Measuring the Fit
of the Regression Model
For Triple A Construction
SST = 22.5
SSE = 6.875
SSR = 15.625
22. 4-22
Measuring the Fit
of the Regression Model
Figure 4.2
Deviations from the Regression Line and from the Mean
23. 4-23
Coefficient of Determination
The proportion of the variability in Y explained by
the regression equation is called the coefficientcoefficient
of determination.of determination.
The coefficient of determination is r2
.
SST
SSE
SST
SSR
r −== 12
For Triple A Construction:
69440
522
625152
.
.
.
==r
About 69% of the variability in Y is explained by
the equation based on payroll (X).
24. 4-24
Correlation Coefficient
The correlation coefficientcorrelation coefficient is an expression of the
strength of the linear relationship.
It will always be between +1 and –1.
The correlation coefficient is r.
2
rr ±=
For Triple A Construction:
8333069440 .. ==r
25. 4-25
Four Values of the Correlation
Coefficient
*
*
*
*
(a) Perfect Positive
Correlation:
r = +1
X
Y
*
* *
*
(c) No Correlation:
r = 0
X
Y
* *
*
*
* *
* **
*
(d) Perfect Negative
Correlation:
r = –1
X
Y
*
*
*
*
* *
*
*
*
(b) Positive
Correlation:
0 < r < 1
X
Y
*
*
*
*
*
*
*
Figure 4.3
29. 4-29
Assumptions of the Regression Model
1. Errors are independent.
2. Errors are normally distributed.
3. Errors have a mean of zero.
4. Errors have a constant variance.
If we make certain assumptions about the errors
in a regression model, we can perform statistical
tests to determine if the model is useful.
A plot of the residuals (errors) will often highlight
any glaring violations of the assumption.
33. 4-33
Estimating the Variance
Errors are assumed to have a constant
variance (σ 2
), but we usually don’t know
this.
It can be estimated using the meanmean
squared errorsquared error (MSEMSE), s2.
1
2
−−
==
kn
SSE
MSEs
where
n = number of observations in the sample
k = number of independent variables
34. 4-34
Estimating the Variance
For Triple A Construction:
71881
4
87506
116
87506
1
2
.
..
==
−−
=
−−
==
kn
SSE
MSEs
We can estimate the standard deviation, s.
This is also called the standard error of thestandard error of the
estimateestimate or the standard deviation of thestandard deviation of the
regression.regression.
31171881 .. === MSEs
35. 4-35
Testing the Model for Significance
When the sample size is too small, you
can get good values for MSE and r2
even if
there is no relationship between the
variables.
Testing the model for significance helps
determine if the values are meaningful.
We do this by performing a statistical
hypothesis test.
36. 4-36
Testing the Model for Significance
We start with the general linear model
εββ ++= XY 10
If β1 = 0, the null hypothesis is that there is
nono relationship between X and Y.
The alternate hypothesis is that there isis a
linear relationship (β1 ≠ 0).
If the null hypothesis can be rejected, we
have proven there is a relationship.
We use the F statistic for this test.
37. 4-37
Testing the Model for Significance
The F statistic is based on the MSE and MSR:
k
SSR
MSR =
where
k = number of independent variables in the
model
The F statistic is:
MSE
MSR
F =
This describes an F distribution with:
degrees of freedom for the numerator = df1 = k
degrees of freedom for the denominator = df2 = n – k – 1
38. 4-38
Testing the Model for Significance
If there is very little error, the MSE would
be small and the F-statistic would be large
indicating the model is useful.
If the F-statistic is large, the significance
level (p-value) will be low, indicating it is
unlikely this would have occurred by
chance.
So when the F-value is large, we can reject
the null hypothesis and accept that there is
a linear relationship between X and Y and
the values of the MSE and r2
are
meaningful.
39. 4-39
Steps in a Hypothesis Test
1. Specify null and alternative hypotheses:
010 =β:H
011 ≠β:H
2. Select the level of significance (α). Common
values are 0.01 and 0.05.
3. Calculate the value of the test statistic using the
formula:
MSE
MSR
F =
40. 4-40
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:
21
ifReject dfdfcalculated FF ,,α>
kdf =1
12 −−= kndf
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:
)( statistictestcalculatedvalue- >= FPp
α<value-ifReject p
41. 4-41
Triple A Construction
Step 1.Step 1.
H0: β1 = 0 (no linear
relationship between X and Y)
H1: β1 ≠ 0 (linear relationship
exists between X and Y)
Step 2.Step 2.
Select α = 0.05
625015
1
625015
.
.
===
k
SSR
MSR
099
71881
625015
.
.
.
===
MSE
MSR
F
Step 3.Step 3.
Calculate the value of the test
statistic.
42. 4-42
Triple A Construction
Step 4.Step 4.
Reject the null hypothesis if the test statistic
is greater than the F-value in Appendix D.
df1 = k = 1
df2 = n – k – 1 = 6 – 1 – 1 = 4
The value of F associated with a 5% level of
significance and with degrees of freedom 1
and 4 is found in Appendix D.
F0.05,1,4 = 7.71
Fcalculated = 9.09
Reject H0 because 9.09 > 7.71
43. 4-43
F = 7.71
0.05
9.09
Triple A Construction
Figure 4.5
We can conclude there is a
statistically significant
relationship between X and
Y.
The r2
value of 0.69 means
about 69% of the variability
in sales (Y) is explained by
local payroll (X).
44. 4-44
Analysis of Variance (ANOVA) Table
When software is used to develop a regression
model, 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.
DF SS MS F SIGNIFICANCE
Regression k SSR MSR = SSR/k MSR/MSE P(F >
MSR/MSE)
Residual n - k - 1 SSE MSE =
SSE/(n - k - 1)
Total n - 1 SST
Table 4.4
45. 4-45
ANOVA for Triple A Construction
Because this probability is less than 0.05, we reject
the null hypothesis of no linear relationship and
conclude there is a linear relationship between X
and Y.
Program 4.1C
(partial)
P(F > 9.0909) = 0.0394
46. 4-46
Multiple Regression Analysis
Multiple regression modelsMultiple regression models are
extensions to the simple linear model
and allow the creation of models with
more than one independent variable.
Y = β0 + β1X1 + β2X2 + … + βkXk + ε
where
Y = dependent variable (response variable)
Xi = ith
independent variable (predictor or
explanatory variable)
β0 = intercept (value of Y when all Xi = 0)
βi = coefficient of the ith
independent variable
k = number of independent variables
ε = random error
47. 4-47
Multiple Regression Analysis
To estimate these values, a sample is taken the
following equation developed
kk XbXbXbbY ++++= ...ˆ 22110
where
= predicted value of Y
b0 = sample intercept (and is an estimate of
β0)
bi = sample coefficient of the ith variable
(and is an estimate of βi)
Yˆ
48. 4-48
Jenny Wilson Realty
Jenny Wilson wants to develop a model to determine
the suggested listing price for houses based on the
size and age of the house.
22110
ˆ XbXbbY ++=
where
= predicted value of dependent
variable (selling price)
b0 = Y intercept
X1 and X2 = value of the two independent
variables (square footage and age)
respectively
b1 and b2 = slopes for X1 and X2
respectively
Yˆ
She selects a sample of houses that have sold
recently and records the data shown in Table 4.5
49. 4-49
Jenny Wilson Real Estate Data
SELLING
PRICE ($)
SQUARE
FOOTAGE
AGE CONDITION
95,000 1,926 30 Good
119,000 2,069 40 Excellent
124,800 1,720 30 Excellent
135,000 1,396 15 Good
142,000 1,706 32 Mint
145,000 1,847 38 Mint
159,000 1,950 27 Mint
165,000 2,323 30 Excellent
182,000 2,285 26 Mint
183,000 3,752 35 Good
200,000 2,300 18 Good
211,000 2,525 17 Good
215,000 3,800 40 Excellent
219,000 1,740 12 MintTable 4.5
52. 4-52
Evaluating Multiple Regression Models
Evaluation is similar to simple linear
regression models.
The p-value for the F-test and r2
are
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.
53. 4-53
Evaluating Multiple Regression Models
To determine which independent
variables are significant, tests are
performed for each variable.
010 =β:H
011 ≠β:H
The test statistic is calculated and if the
p-value is lower than the level of
significance (α), the null hypothesis is
rejected.
54. 4-54
Jenny Wilson Realty
The model is statistically significant
The p-value for the F-test is 0.002.
r2
= 0.6719 so the model explains about 67% of
the variation in selling price (Y).
But the F-test is for the entire model and we can’t
tell if one or both of the independent variables are
significant.
By calculating the p-value of each variable, we can
assess the significance of the individual variables.
Since the p-value for X1 (square footage) and X2
(age) are both less than the significance level of
0.05, both null hypotheses can be rejected.
55. 4-55
Binary or Dummy Variables
BinaryBinary (or dummydummy or indicatorindicator) 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.
56. 4-56
Jenny Wilson Realty
Jenny believes a better model can be developed if
she includes information about the condition of
the property.
X3 = 1 if house is in excellent condition
= 0 otherwise
X4 = 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 X3 and X4 = 0, the house must be in good
condition.
59. 4-59
Model Building
The best model is a statistically significant
model with a high r2
and few variables.
As more variables are added to the model,
the r2
-value usually increases.
For this reason, the adjustedadjusted rr22
value is
often used to determine the usefulness of
an additional variable.
The adjusted r2
takes into account the
number of independent variables in the
model.
60. 4-60
Model Building
SST
SSE
SST
SSR
−== 12
r
The formula for r2
The formula for adjusted r2
)/(SST
)/(SSE
1
1
1Adjusted 2
−
−−
−=
n
kn
r
As the number of variables increases, the
adjusted r2
gets smaller unless the increase due to
the new variable is large enough to offset the
change in k.
61. 4-61
Model Building
In general, if a new variable increases the
adjusted r2
, it should probably be included in the
model.
In some cases, variables contain duplicate
information.
When two independent variables are correlated,
they are said to be collinear.collinear.
When more than two independent variables are
correlated, multicollinearitymulticollinearity exists.
When multicollinearity is present, hypothesis
tests for the individual coefficients are not valid
but the model may still be useful.
62. 4-62
Nonlinear Regression
In some situations, variables are not linear.
Transformations may be used to turn a
nonlinear model into a linear model.
*
* **
** *
* *
Linear relationship Nonlinear relationship
* *
** **
*
*
**
*
63. 4-63
Colonel Motors
Engineers at Colonel Motors want to use
regression analysis to improve fuel efficiency.
They have been asked to study the impact of
weight on miles per gallon (MPG).
MPG
WEIGHT (1,000
LBS.) MPG
WEIGHT (1,000
LBS.)
12 4.58 20 3.18
13 4.66 23 2.68
15 4.02 24 2.65
18 2.53 33 1.70
19 3.09 36 1.95
19 3.11 42 1.92
Table 4.6
65. 4-65
Colonel Motors
Program 4.4 This is a useful model with a
small F-test for significance
and a good r2
value.
Excel Output for Linear Regression Model with
MPG Data
67. 4-67
Colonel Motors
The nonlinear model is a quadratic model.
The easiest way to work with this model is to
develop a new variable.
2
2 weight)(=X
This gives us a model that can be solved with
linear regression software:
22110 XbXbbY ++=ˆ
68. 4-68
Colonel Motors
Program 4.5
A better model with a
smaller F-test for
significance and a larger
adjusted r2
value
21 43230879 XXY ...ˆ +−=
69. 4-69
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 X is questionable, as the
relationship may not hold outside the
sample data.
70. 4-70
Cautions and Pitfalls
A t-test for the intercept (b0) 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.
71. 4-71
Copyright
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