9.2 Linear Regression Key Concepts: –Residuals –Least Squares Criterion –Regression Line –Using a Regression Equation to Make Predictions.

Slides:

Advertisements

Similar presentations

Chapter 12 Simple Linear Regression

Advertisements

Linear Regression.  The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu:  The model won’t be perfect, regardless.

1 Functions and Applications

Regression Analysis Module 3. Regression Regression is the attempt to explain the variation in a dependent variable using the variation in independent.

Simple Linear Regression. Start by exploring the data Construct a scatterplot  Does a linear relationship between variables exist?  Is the relationship.

Chapter 8 Linear Regression © 2010 Pearson Education 1.

CHAPTER 8: LINEAR REGRESSION

Regression Regression: Mathematical method for determining the best equation that reproduces a data set Linear Regression: Regression method applied with.

LINEAR REGRESSION: What it Is and How it Works Overview What is Bivariate Linear Regression? The Regression Equation How It’s Based on r.

LINEAR REGRESSION: Evaluating Regression Models. Overview Standard Error of the Estimate Goodness of Fit Coefficient of Determination Regression Coefficients.

LINEAR REGRESSION: What it Is and How it Works. Overview What is Bivariate Linear Regression? The Regression Equation How It’s Based on r.

Chapter 12 Simple Regression

ESTIMATING THE REGRESSION COEFFICIENTS FOR SIMPLE LINEAR REGRESSION.

AP Statistics Chapter 8: Linear Regression

Linear Regression Larson/Farber 4th ed. 1 Section 9.2.

Business Statistics - QBM117 Least squares regression.

Introduction to Regression Analysis, Chapter 13,

Simple Linear Regression Analysis

Chapter 2 – Simple Linear Regression - How. Here is a perfect scenario of what we want reality to look like for simple linear regression. Our two variables.

Regression, Residuals, and Coefficient of Determination Section 3.2.

1 1 Slide Simple Linear Regression Chapter 14 BA 303 – Spring 2011.

Linear Regression Analysis

Adapted from Walch Education A linear equation describes a situation where there is a near- constant rate of change. An exponential equation describes.

Linear Regression.

Section 9.2 Linear Regression © 2012 Pearson Education, Inc. All rights reserved.

Introduction to Linear Regression and Correlation Analysis

Simple Linear Regression

Biostatistics Unit 9 – Regression and Correlation.

1 FORECASTING Regression Analysis Aslı Sencer Graduate Program in Business Information Systems.

Chapter 6 & 7 Linear Regression & Correlation

Section 4.2 Regression Equations and Predictions.

Least-Squares Regression Section 3.3. Why Create a Model? There are two reasons to create a mathematical model for a set of bivariate data. To predict.

Linear Regression Least Squares Method: the Meaning of r 2.

Section 5.2: Linear Regression: Fitting a Line to Bivariate Data.

Statistical Methods Statistical Methods Descriptive Inferential

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 8 Linear Regression.

Simple Linear Regression. Deterministic Relationship If the value of y (dependent) is completely determined by the value of x (Independent variable) (Like.

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 4 Section 2 – Slide 1 of 20 Chapter 4 Section 2 Least-Squares Regression.

Regression Regression relationship = trend + scatter

PS 225 Lecture 20 Linear Regression Equation and Prediction.

Section 9.2 Linear Regression. Section 9.2 Objectives Find the equation of a regression line Predict y-values using a regression equation.

Chapter 8 Linear Regression *The Linear Model *Residuals *Best Fit Line *Correlation and the Line *Predicated Values *Regression.

Chapter 6 (cont.) Difference Estimation. Recall the Regression Estimation Procedure 2.

Creating a Residual Plot and Investigating the Correlation Coefficient.

7-3 Line of Best Fit Objectives

AP STATISTICS Section 3.2 Least Squares Regression.

Linear Regression 1 Section 9.2. Section 9.2 Objectives 2 Find the equation of a regression line Predict y-values using a regression equation.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Correlation and Regression 9.

CHAPTER 8 Linear Regression. Residuals Slide  The model won’t be perfect, regardless of the line we draw.  Some points will be above the line.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.3 Predicting the Outcome.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1.

^ y = a + bx Stats Chapter 5 - Least Squares Regression

LEAST-SQUARES REGRESSION 3.2 Least Squares Regression Line and Residuals.

Linear Prediction Correlation can be used to make predictions – Values on X can be used to predict values on Y – Stronger relationships between X and Y.

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Simple Linear Regression Analysis Chapter 13.

1 Simple Linear Regression and Correlation Least Squares Method The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES.

1 1 Slide The Simple Linear Regression Model n Simple Linear Regression Model y =  0 +  1 x +  n Simple Linear Regression Equation E( y ) =  0 + 

Simple Linear Regression The Coefficients of Correlation and Determination Two Quantitative Variables x variable – independent variable or explanatory.

Chapter 14 Introduction to Regression Analysis. Objectives Regression Analysis Uses of Regression Analysis Method of Least Squares Difference between.

Correlation and Regression Ch 4. Why Regression and Correlation We need to be able to analyze the relationship between two variables (up to now we have.

Chapter Correlation and Regression 1 of 84 9 © 2012 Pearson Education, Inc. All rights reserved.

Statistics 350 Lecture 2. Today Last Day: Section Today: Section 1.6 Homework #1: Chapter 1 Problems (page 33-38): 2, 5, 6, 7, 22, 26, 33, 34,

1 Objective Given two linearly correlated variables (x and y), find the linear function (equation) that best describes the trend. Section 10.3 Regression.

Chapter 5 LSRL. Bivariate data x – variable: is the independent or explanatory variable y- variable: is the dependent or response variable Use x to predict.

CHAPTER 3 Describing Relationships

Chapter 5 LSRL.

Linear Regression.

Linear regression Fitting a straight line to observations.

Lesson 2.2 Linear Regression.

Presentation transcript:

9.2 Linear Regression Key Concepts: –Residuals –Least Squares Criterion –Regression Line –Using a Regression Equation to Make Predictions

9.2 Linear Regression In this section, we will assume a significant linear correlation exists between the two variables of interest. –We would like to find the equation of the line that best fits the data. How do we decide whether one line is a better fit than another? We start by measuring the residuals (or amount of error). Each residual represents the difference between what our proposed line predicts for a given vale of x and the actual observed value (see p. 486). A positive residual tells us our line over-estimated the observed value. A negative residual tells us our line under- estimated the observed value. If the residual = 0, we have an exact match.

9.2 Linear Regression The next step is to add all the residuals together. –The line that best fits the data should have the smallest sum of residuals. Unfortunately, whenever we add all the residuals of a data set, we end up with a sum equal to zero. We can avoid this problem by squaring the residuals and then adding. –According to the Least Squares Criterion, the line of best fit (called the regression line) will have the smallest sum of squared residuals.

9.2 Linear Regression The equation of a regression line for an independent variable x and a dependent variable y is: where is the predicted y-value for a given x-value. –We use the Least Squares Criterion and Calculus to derive the slope m and y-intercept b of the regression line.

9.2 Linear Regression The slope m and y-intercept b of a regression line are given by:

9.2 Linear Regression It is best to use a regression line to make predictions for x-vales over (or close to) the range of the original data. Extrapolation (using a regression line to make predictions for x-values well beyond the range of the original data) can lead to highly inaccurate results. Practice building a regression equation and using it to make predictions: #20 p. 491 (Wins and Earned Run Averages) #24 p. 492 (High-Fiber Cereals: Caloric and Sugar Content)