Course Objective This course is designed to give you a basic understanding of how to run regressions in SPSS.

Size: px
Start display at page:

Download "Course Objective This course is designed to give you a basic understanding of how to run regressions in SPSS."

Transcription

1 SPSS Regressions Social Science Research Lab American University, Washington, D.C. Web. Tel. x Course Objective This course is designed to give you a basic understanding of how to run regressions in SPSS. Learning Outcomes Learn how to make a scatter plot Learn how to run and interpret simple regressions Learn how to run a multiple regression Note: This tutorial uses the World95.sav dataset, which can be found at our website. Simple Regressions and the Scatter Plot A Simple regression analyzes the relationship between two variables. Scatter plot The relationship between two variables can be portrayed visually using a scatter plot. To create a scatter plot Go to Graphs>Legacy Dialogues>Scatter/dot then choose simple. Plug in your variables. For your dependent variable choose Infant mortality (babymort), and for your independent variable choose Women s literacy (lit_fema). Tip: The dependent variable is displayed on the (Y) vertical axis and the independent variable on the (X) horizontal axis. 1

2 An SPSS Scatter plot displaying the relationship between infant mortality and women s literacy The relationship between the two variables is estimated as a linear or straight line relationship, defined by the equation: Y = ax + b b is the intercept or the constant and a, the slope. The line is mathematically calculated such that the sum of distances from each observation to the line is minimized. By definition, the slope indicates the change in Y as a result of aunit change in X. The straight line is also called the regression line or the fit line and a is referred to as the regression coefficient. Tip: The method of calculating the regression coefficient (the slope) is called ordinary least squares, or OLS. OLS estimates the slope by minimizing the sum of squared differences between each predicted (ax + b) and the actual value of Y. One reason for squaring these distances is to ensure that all distances are positive. A positive regression coefficient indicates a positive relationship between the variables; the fit line will be upward sloping (see the example on the previous 2

3 page). A negative regression coefficient indicates a negative relationship between the variables; the fit line will be downward sloping. Testing for Statistical Significance The test of significance of the regression slope is a key test of hypothesis regression analysis that tells us whether the slope a is statistically different from 0. To determine whether the slope equals zero, a t-test is performed (For more information on t-tests see the SPSS bivariate statistics tutorial). As a general rule when observations on the scatter plot lie closely around the fit line, the regression line is more likely to be statistically significant: in other words, it will be more likely that the two variables are positively or negatively related. Fortunately, SPSS calculates the slope the intercept standard error of the slope, and the level at which the slope is statically significant. Let s look at an example: determine whether gas mileage and horsepower are related. Go to Analyze>Regression>Linear SPSS Linear Regression Window Infant Mortality (babymort) is our dependent variable and Female Literacy (lit_fema), our independent variable. Click OK. SPSS will produce a series of 3

4 tables which tell us about the nature of the relationship between these two variables. In the first table, the R 2, also called the coefficient of determination is very useful. It measures the proportion of the total variation in Y about its mean explained by the regression of Y on X. In this case, our regression explains 70.8 % of the variation of infant mortality. Typically, values of R 2 below 0.2 are considered weak, between 0.2 and 0.4, moderate, and above 0.4, strong. In the second table, we will focus on the F-statistic. By computing this statistic, we test the hypothesis that none of the explanatory variables help explain variation in Y about its mean. The information to pay attention to here is the probability shown as Sig. in the table. If this probability is below 0.05, we conclude that the F-statistic is large enough so that we can reject the hypothesis that none of the explanatory variables help explain variation in Y. This test is like a test of significance of the R 2. 4

5 Finally, the last table will help us determine whether infant mortality and women s literacy are significantly related, and the direction and strength of their relationship. The first important thing to note is that the sign of the coefficient of Females who read (%) is negative. It confirms our assumption (infant mortality decreases as women s education increases) and our visual analysis of the scatter plot (see page one for the scatter plot). Furthermore, the probability reported in the right column is very low. This implies that the slope a is statistically significant. To be less abstract, let us recall what those coefficients mean: they are the slope and the intercept of the regression line, i.e. Y = X What does this mean? It means that when literacy increases by one unit (i.e. 1%), infant mortality on average fall by 1.13 per thousand. In sum, R 2 is high, probabilities are low: WE ARE HAPPY! Multiple Regressions A multiple regression takes what we ve just done and adds several more variables to the mix. It is the right tool whenever you think that your dependent variable is explained by more than one independent variable. In our empirical work, we reasonably assume that gas mileage (Y) is not only explained by horsepower (X1) but also the weight (X2) of the car and its engine displacement (X3). Therefore, we will test the following equation: Y = a0 + a1x1 + a2x2 + a3x3 + error Y is the dependent variable, a0 is the constant, X1, X2, and X3 are the independent variables and their respective coefficients a1, a2, and a3, and the error term reflects all other factors that are not in the model. Scatter Plot Matrix Before performing any statistical work, do not forget to draw scatter plots of every independent variable against the dependent variable. Go to Graphs>Legacy Dialogues>Scatter/dot then choose Matrix Scatter 5

6 SPSS Scatter plot Matrix Window Lets plug in these variables: babymort, lit_fema and gdp_cap This will produce a matrix of scatter plots which looks like this: 6

7 A scatter plot like this is helpful for visually detecting relationships between your variables. For instance, you might observe a non-linear relationship between two variables, in which case, you should use different techniques (GDP per capita in general exhibits non-linear relationships with other variables). Correlation Matrix Before starting our multiple regression analysis, it is important to compute the correlation matrix. Go to Analyze>Correlate>Bivariate We select the three independent variables. Then click OK. The following table will show up in the output window. This preliminary step is important because independent variables should NOT be correlated with one another. If independent variables are correlated, this might affect the robustness of our results. In the fascinating world of statistics, this is referred to as the issue of multicollinearity. When we use multiple regression analysis, we attach a weight to any of the independent variables in order to explain the variation in Y. If two independent variables are strongly correlated, it becomes very hard to attach a weight to those variables because they basically convey the same information. As a result, the validity of our empirical work will be greatly affected. In general, 7

8 assuming two of the independent variables are correlated, the easiest solution is to ignore one of them variables one and to use simply the other one. In this particular case, all variables are strongly correlated with one another. I would recommend you should use only the women s literacy (which we did in the previous section) and not use GDP per capita or daily calorie intake. Let s conduct a multiple regression to explain its basic philosophy, even though our results will be sloppy. Running your Regression To run this regression, go to Analyze>Regression>Linear Select babymort as the dependent variable and calories, lit_fema, and gdp_cap as the independent variables. Click OK. The following tables show up in the output window. The R 2 for our new model is slightly greater than we obtained earlier (.711 vs..811). However, the variation is too large large considering that we added two new variables. Clearly, this is due to the fact that the new independent variables are strongly correlated and ultimately, do not bring much extra information. One of the flaw of the R 2 is that it is sensitive to the number of included independent variables. Specifically, addition of additional independent variables can only increase the R 2. In contrast, the Adjusted R 2 accounts for the number of independent variables. It may rise or fall with the addition of more variables. The Adjusted R 2 is greater than the one obtained in the former section. Therefore, the extra information brought by the new variables is greater than the penalty of adding variables (assuming that we did not encounter the issue of multicollinearity). 8

9 Finally, as expected from our correlation matrix, we find out that our independent variables are negatively correlated with infant mortality. If we look to the right column we find that our puzzling and strange result for GDP per capita is not statistically significant. This result is an illustration of what happens when there is extensive multicolinearity amongst your independent variables 9

SPSS Guide: Regression Analysis

SPSS Guide: Regression Analysis SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar

More information

Relationships Between Two Variables: Scatterplots and Correlation

Relationships Between Two Variables: Scatterplots and Correlation Relationships Between Two Variables: Scatterplots and Correlation Example: Consider the population of cars manufactured in the U.S. What is the relationship (1) between engine size and horsepower? (2)

More information

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

More information

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

More information

2. Simple Linear Regression

2. Simple Linear Regression Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

More information

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r), Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

More information

Introduction to Regression and Data Analysis

Introduction to Regression and Data Analysis Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

More information

Module 3: Correlation and Covariance

Module 3: Correlation and Covariance Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

More information

Module 5: Multiple Regression Analysis

Module 5: Multiple Regression Analysis Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

More information

Simple Predictive Analytics Curtis Seare

Simple Predictive Analytics Curtis Seare Using Excel to Solve Business Problems: Simple Predictive Analytics Curtis Seare Copyright: Vault Analytics July 2010 Contents Section I: Background Information Why use Predictive Analytics? How to use

More information

Linear Models in STATA and ANOVA

Linear Models in STATA and ANOVA Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 4-2 A Note on Non-Linear Relationships 4-4 Multiple Linear Regression 4-5 Removal of Variables 4-8 Independent Samples

More information

2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program. Statistics Module Part 3 2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

More information

Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation

Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage

More information

Premaster Statistics Tutorial 4 Full solutions

Premaster Statistics Tutorial 4 Full solutions Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

More information

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

More information

Correlation. What Is Correlation? Perfect Correlation. Perfect Correlation. Greg C Elvers

Correlation. What Is Correlation? Perfect Correlation. Perfect Correlation. Greg C Elvers Correlation Greg C Elvers What Is Correlation? Correlation is a descriptive statistic that tells you if two variables are related to each other E.g. Is your related to how much you study? When two variables

More information

Simple linear regression

Simple linear regression Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

More information

Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 13 Introduction to Linear Regression and Correlation Analysis Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

More information

Simple Linear Regression, Scatterplots, and Bivariate Correlation

Simple Linear Regression, Scatterplots, and Bivariate Correlation 1 Simple Linear Regression, Scatterplots, and Bivariate Correlation This section covers procedures for testing the association between two continuous variables using the SPSS Regression and Correlate analyses.

More information

Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

More information

The Dummy s Guide to Data Analysis Using SPSS

The Dummy s Guide to Data Analysis Using SPSS The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests

More information

We are often interested in the relationship between two variables. Do people with more years of full-time education earn higher salaries?

We are often interested in the relationship between two variables. Do people with more years of full-time education earn higher salaries? Statistics: Correlation Richard Buxton. 2008. 1 Introduction We are often interested in the relationship between two variables. Do people with more years of full-time education earn higher salaries? Do

More information

X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1) CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

More information

Regression and Correlation

Regression and Correlation Regression and Correlation Topics Covered: Dependent and independent variables. Scatter diagram. Correlation coefficient. Linear Regression line. by Dr.I.Namestnikova 1 Introduction Regression analysis

More information

Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Section 14 Simple Linear Regression: Introduction to Least Squares Regression Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

More information

PLOTTING DATA AND INTERPRETING GRAPHS

PLOTTING DATA AND INTERPRETING GRAPHS PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they

More information

Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96 1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

More information

Univariate Regression

Univariate Regression Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

More information

Pearson s Correlation

Pearson s Correlation Pearson s Correlation Correlation the degree to which two variables are associated (co-vary). Covariance may be either positive or negative. Its magnitude depends on the units of measurement. Assumes the

More information

Elements of a graph. Click on the links below to jump directly to the relevant section

Elements of a graph. Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

More information

Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

More information

Correlation key concepts:

Correlation key concepts: CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

More information

Factors affecting online sales

Factors affecting online sales Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4

More information

Chapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS

Chapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Chapter Seven Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Section : An introduction to multiple regression WHAT IS MULTIPLE REGRESSION? Multiple

More information

Example: Boats and Manatees

Example: Boats and Manatees Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant

More information

Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

More information

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

More information

Copyright 2007 by Laura Schultz. All rights reserved. Page 1 of 5

Copyright 2007 by Laura Schultz. All rights reserved. Page 1 of 5 Using Your TI-83/84 Calculator: Linear Correlation and Regression Elementary Statistics Dr. Laura Schultz This handout describes how to use your calculator for various linear correlation and regression

More information

POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.

POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression

More information

WEB APPENDIX. Calculating Beta Coefficients. b Beta Rise Run Y 7.1 1 8.92 X 10.0 0.0 16.0 10.0 1.6

WEB APPENDIX. Calculating Beta Coefficients. b Beta Rise Run Y 7.1 1 8.92 X 10.0 0.0 16.0 10.0 1.6 WEB APPENDIX 8A Calculating Beta Coefficients The CAPM is an ex ante model, which means that all of the variables represent before-thefact, expected values. In particular, the beta coefficient used in

More information

Getting Correct Results from PROC REG

Getting Correct Results from PROC REG Getting Correct Results from PROC REG Nathaniel Derby, Statis Pro Data Analytics, Seattle, WA ABSTRACT PROC REG, SAS s implementation of linear regression, is often used to fit a line without checking

More information

Chapter 23. Inferences for Regression

Chapter 23. Inferences for Regression Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

More information

Module 5: Statistical Analysis

Module 5: Statistical Analysis Module 5: Statistical Analysis To answer more complex questions using your data, or in statistical terms, to test your hypothesis, you need to use more advanced statistical tests. This module reviews the

More information

Scatter Plots with Error Bars

Scatter Plots with Error Bars Chapter 165 Scatter Plots with Error Bars Introduction The procedure extends the capability of the basic scatter plot by allowing you to plot the variability in Y and X corresponding to each point. Each

More information

SPSS Explore procedure

SPSS Explore procedure SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,

More information

Scatter Plot, Correlation, and Regression on the TI-83/84

Scatter Plot, Correlation, and Regression on the TI-83/84 Scatter Plot, Correlation, and Regression on the TI-83/84 Summary: When you have a set of (x,y) data points and want to find the best equation to describe them, you are performing a regression. This page

More information

Part 1: Background - Graphing

Part 1: Background - Graphing Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and

More information

Linear Regression. Chapter 5. Prediction via Regression Line Number of new birds and Percent returning. Least Squares

Linear Regression. Chapter 5. Prediction via Regression Line Number of new birds and Percent returning. Least Squares Linear Regression Chapter 5 Regression Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects

More information

Absorbance Spectrophotometry: Analysis of FD&C Red Food Dye #40 Calibration Curve Procedure

Absorbance Spectrophotometry: Analysis of FD&C Red Food Dye #40 Calibration Curve Procedure Absorbance Spectrophotometry: Analysis of FD&C Red Food Dye #40 Calibration Curve Procedure Note: there is a second document that goes with this one! 2046 - Absorbance Spectrophotometry. Make sure you

More information

Summary of important mathematical operations and formulas (from first tutorial):

Summary of important mathematical operations and formulas (from first tutorial): EXCEL Intermediate Tutorial Summary of important mathematical operations and formulas (from first tutorial): Operation Key Addition + Subtraction - Multiplication * Division / Exponential ^ To enter a

More information

Exercise 1.12 (Pg. 22-23)

Exercise 1.12 (Pg. 22-23) Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

with functions, expressions and equations which follow in units 3 and 4.

with functions, expressions and equations which follow in units 3 and 4. Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a

More information

Session 9 Case 3: Utilizing Available Software Statistical Analysis

Session 9 Case 3: Utilizing Available Software Statistical Analysis Session 9 Case 3: Utilizing Available Software Statistical Analysis Michelle Phillips Economist, PURC michelle.phillips@warrington.ufl.edu With material from Ted Kury Session Overview With Data from Cases

More information

Introduction to Data Analysis in Hierarchical Linear Models

Introduction to Data Analysis in Hierarchical Linear Models Introduction to Data Analysis in Hierarchical Linear Models April 20, 2007 Noah Shamosh & Frank Farach Social Sciences StatLab Yale University Scope & Prerequisites Strong applied emphasis Focus on HLM

More information

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

More information

How To Run Statistical Tests in Excel

How To Run Statistical Tests in Excel How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting

More information

Mixed 2 x 3 ANOVA. Notes

Mixed 2 x 3 ANOVA. Notes Mixed 2 x 3 ANOVA This section explains how to perform an ANOVA when one of the variables takes the form of repeated measures and the other variable is between-subjects that is, independent groups of participants

More information

Curve Fitting in Microsoft Excel By William Lee

Curve Fitting in Microsoft Excel By William Lee Curve Fitting in Microsoft Excel By William Lee This document is here to guide you through the steps needed to do curve fitting in Microsoft Excel using the least-squares method. In mathematical equations

More information

This chapter will demonstrate how to perform multiple linear regression with IBM SPSS

This chapter will demonstrate how to perform multiple linear regression with IBM SPSS CHAPTER 7B Multiple Regression: Statistical Methods Using IBM SPSS This chapter will demonstrate how to perform multiple linear regression with IBM SPSS first using the standard method and then using the

More information

2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or

2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or Simple and Multiple Regression Analysis Example: Explore the relationships among Month, Adv.$ and Sales $: 1. Prepare a scatter plot of these data. The scatter plots for Adv.$ versus Sales, and Month versus

More information

Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.

Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line. Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Module 7 Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. You are given information about a straight line. Use two points to graph the equation.

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Dealing with Data in Excel 2010

Dealing with Data in Excel 2010 Dealing with Data in Excel 2010 Excel provides the ability to do computations and graphing of data. Here we provide the basics and some advanced capabilities available in Excel that are useful for dealing

More information

Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases:

Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases: Profile Analysis Introduction Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases: ) Comparing the same dependent variables

More information

Causal Forecasting Models

Causal Forecasting Models CTL.SC1x -Supply Chain & Logistics Fundamentals Causal Forecasting Models MIT Center for Transportation & Logistics Causal Models Used when demand is correlated with some known and measurable environmental

More information

Directions for using SPSS

Directions for using SPSS Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...

More information

Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade

Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Section 3 Part 1. Relationships between two numerical variables

Section 3 Part 1. Relationships between two numerical variables Section 3 Part 1 Relationships between two numerical variables 1 Relationship between two variables The summary statistics covered in the previous lessons are appropriate for describing a single variable.

More information

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

More information

Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

More information

Chapter 5 Estimating Demand Functions

Chapter 5 Estimating Demand Functions Chapter 5 Estimating Demand Functions 1 Why do you need statistics and regression analysis? Ability to read market research papers Analyze your own data in a simple way Assist you in pricing and marketing

More information

There are six different windows that can be opened when using SPSS. The following will give a description of each of them.

There are six different windows that can be opened when using SPSS. The following will give a description of each of them. SPSS Basics Tutorial 1: SPSS Windows There are six different windows that can be opened when using SPSS. The following will give a description of each of them. The Data Editor The Data Editor is a spreadsheet

More information

Stat 412/512 CASE INFLUENCE STATISTICS. Charlotte Wickham. stat512.cwick.co.nz. Feb 2 2015

Stat 412/512 CASE INFLUENCE STATISTICS. Charlotte Wickham. stat512.cwick.co.nz. Feb 2 2015 Stat 412/512 CASE INFLUENCE STATISTICS Feb 2 2015 Charlotte Wickham stat512.cwick.co.nz Regression in your field See website. You may complete this assignment in pairs. Find a journal article in your field

More information

FRICTION, WORK, AND THE INCLINED PLANE

FRICTION, WORK, AND THE INCLINED PLANE FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle

More information

Regression Analysis (Spring, 2000)

Regression Analysis (Spring, 2000) Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity

More information

Introduction to Quantitative Methods

Introduction to Quantitative Methods Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Firms that survive in the long run are usually those that A) remain small. B) strive for the largest

More information

Simple Regression Theory II 2010 Samuel L. Baker

Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Descriptive statistics consist of methods for organizing and summarizing data. It includes the construction of graphs, charts and tables, as well various descriptive measures such

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

Analysis of Variance ANOVA

Analysis of Variance ANOVA Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

More information

17. SIMPLE LINEAR REGRESSION II

17. SIMPLE LINEAR REGRESSION II 17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

More information

SPSS Resources. 1. See website (readings) for SPSS tutorial & Stats handout

SPSS Resources. 1. See website (readings) for SPSS tutorial & Stats handout Analyzing Data SPSS Resources 1. See website (readings) for SPSS tutorial & Stats handout Don t have your own copy of SPSS? 1. Use the libraries to analyze your data 2. Download a trial version of SPSS

More information

Nonlinear Regression Functions. SW Ch 8 1/54/

Nonlinear Regression Functions. SW Ch 8 1/54/ Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General

More information

Chapter 7: Modeling Relationships of Multiple Variables with Linear Regression

Chapter 7: Modeling Relationships of Multiple Variables with Linear Regression Chapter 7: Modeling Relationships of Multiple Variables with Linear Regression Overview Chapters 5 and 6 examined methods to test relationships between two variables. Many research projects, however, require

More information

CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there

CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there is a relationship between variables, To find out the

More information

An analysis appropriate for a quantitative outcome and a single quantitative explanatory. 9.1 The model behind linear regression

An analysis appropriate for a quantitative outcome and a single quantitative explanatory. 9.1 The model behind linear regression Chapter 9 Simple Linear Regression An analysis appropriate for a quantitative outcome and a single quantitative explanatory variable. 9.1 The model behind linear regression When we are examining the relationship

More information

Financial Risk Management Exam Sample Questions/Answers

Financial Risk Management Exam Sample Questions/Answers Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM-99, Question 4 Random walk assumes that returns from one time period

More information

Panel Data Analysis in Stata

Panel Data Analysis in Stata Panel Data Analysis in Stata Anton Parlow Lab session Econ710 UWM Econ Department??/??/2010 or in a S-Bahn in Berlin, you never know.. Our plan Introduction to Panel data Fixed vs. Random effects Testing

More information

containing Kendall correlations; and the OUTH = option will create a data set containing Hoeffding statistics.

containing Kendall correlations; and the OUTH = option will create a data set containing Hoeffding statistics. Getting Correlations Using PROC CORR Correlation analysis provides a method to measure the strength of a linear relationship between two numeric variables. PROC CORR can be used to compute Pearson product-moment

More information

Formula for linear models. Prediction, extrapolation, significance test against zero slope.

Formula for linear models. Prediction, extrapolation, significance test against zero slope. Formula for linear models. Prediction, extrapolation, significance test against zero slope. Last time, we looked the linear regression formula. It s the line that fits the data best. The Pearson correlation

More information

Moderation. Moderation

Moderation. Moderation Stats - Moderation Moderation A moderator is a variable that specifies conditions under which a given predictor is related to an outcome. The moderator explains when a DV and IV are related. Moderation

More information