6.0 Lesson Plan. Answer Questions. Regression. Transformation. Extrapolation. Residuals
|
|
- Hilary Wells
- 5 years ago
- Views:
Transcription
1 6.0 Lesson Plan Answer Questions Regression Transformation Extrapolation Residuals 1
2 Information about TAs Lab grader: Pontus, Hwk grader: Rachel, Quiz (Tuesday): Matt, Quiz (Thursday): Blake, All the grades are uploaded by Kai, The students can pick up the graded hwk, lab report and quiz in the filing cabinet in the SECC (Old chem, Rm 211). 2
3 6.1 More About Regression Recall the regression assumptions: 1. Each point (X i, Y i ) in the scatterplot satisfies: Y i = a + bx i + ɛ i where the ɛ i have a normal distribution with mean zero and (usually) unknown standard deviation. 2. The errors ɛ i have nothing to do with one another. A large error does not tend to be followed by another large error, for example. 3. The X i values are measured without error. (Thus all the error occurs in the vertical direction, and we do not need to minimize perpendicular distance to the line.) 3
4 A biologist wants to predict brain weight from body weight, based on a sample of 62 mammals. A portion of the data are shown below: bodywt brainwt log(bodywt) log(brainwt) arctic fox owl monkey cow grey wolf roe deer vervet Will this give an ecological correlation? 4
5 5
6 The regression equation is Y = X The correlation is the square root of.87266, or.9344, but it is heavily influenced by a few outliers. (Note: The correlation is the positive square root because the line has positive slope.) The standard deviation of the residuals is Residue is the typical distance of a point to the line (in the vertical direction). Under the Parameter Estimates portion of the printout, the last column tells whether the intercept and slope are significantly different from zero. Small numbers indicate significant differences; values less than.05 are usually taken to indicate real differences from zero, as opposed to chance errors. 6
7 The root mean square (RMSE) is the standard deviation of the vertical distances between each point and the estimated line. It is an estimate of the standard deviation of the vertical distances between the observations and the true line. Formally, where RMSE = 1 n [(Y 1 (â + ˆbX 1 )) (Y n (â + ˆbX n )) 2 ] â = Ȳ ˆb 1 n X; ˆb = n i=1 X iy i XȲ n i=1 X2 i X. 2 1 n Note that â + ˆbX i is the mean of the Y -value at X i 7
8 The regression line predicts the average value for the Y values at a given X. In practice, one wants to predict the individual value for a particular value of X. For example, if Beavis weighs 50 kilograms, then how much would his brain weigh? The prediction (in grams, without transformation) is Y = â + ˆbX = = But this is just the average for all mammals who weigh as much as Beavis does. 8
9 The individual value is less exact than the average value. To predict the average value, the only source of uncertainty is the exact location of the regression line (i.e., â and ˆb are estimates of the true intercept and slope). In order to predict Beavis s brainweight, the uncertainty about Beavis s deviation from the average is added to the uncertainty about the location of the line. For example, if Beavis weighs 50 kilograms, then his brain should weigh grams + ɛ. Assuming the regression model is correct, then ɛ have a normal distribution with mean zero and standard deviation
10 6.2 Transformations The scatterplot of the brainweight against bodyweight showed the the line was probably controlled by a few large values. (These are sometimes called high-leverage points.) Even worse, the scatterplot did not resemble the cigar-shaped point cloud that supports the regression assumptions listed before. In cases like this, one can consider making a transformation of the response variable or the explanatory variable or both. For this data, consider taking the logarithm (base 10) of the brainweight and body weight. The following scatterplot is much better. 10
11 11
12 Taking the log shows that the outliers are not surprising. The regression equation is now: log Y = log X Now 91.23% of the variation in brain weight is explained by body weight. Both the intercept and the slope are highly significant. The estimated standard deviation of ɛ is.317; this is the typical vertical distance between a point and the line. Making transformations is an art. Here the analysis suggests that log Y = log X.763 = Y = 8.1 X.763. So there is a power-law relationship between brain mass and body mass. (Note: 8.1 = ) 12
13 6.3 Extrapolation Predicting Y values for X values outside the range of X values observed in the data is extrapolation. This is risky, because you have no evidence that the linear relationship you have seen in the scatterplot continues to hold in the new X region. Extrapolated values can be entirely wrong. For example, it is unreliable to predict the brain weight of a blue whale. 13
14 6.4 Residuals Estimate the regression line (using JMP software or by calculating â and ˆb by hand). Then find the difference between each observed Y i and the predicted value Ŷi using the fitted line. These differences are called the residuals. Plot each difference against the corresponding X i value. This plot is called a residual plot. 14
15 If the assumptions for linear regression hold, what should one see in the residual plot? If the pattern of the residuals around the horizontal line at zero is: curved, then the assumption of linearity is violated. fan-shaped, then the assumption of constant standard deviation is violated (heteroscedasticity). filled with many outliers, then again the assumption of constant standard deviation is violated. shows a pattern (e.g., positive, negative, positive, negative,...) then the assumption of independent errors is violated. 15
16 When the residuals have a histogram that looks normal and when the residual plot shows no pattern, then we can use the normal distribution to make inferences about individuals. Suppose a residual plot had suggested that no transformation of the brainweight data were necessary. Then what percentage of 20-kilogram mammals have brains that weigh more than 180 grams? The regression equation says that the mean brainweight for 20 kilogram animals is * 20 = The sd of the residuals is Under the regression assumptions, the 20-kilogram mammals have brainweights that are normally distributed with mean and standard deviation The z-transform is ( )/ =.208. From the table, the area under the curve to the right of.208 is ( )/2 = %. 16
Stat 101: Lecture 6. Summer 2006
Stat 101: Lecture 6 Summer 2006 Outline Review and Questions Example for regression Transformations, Extrapolations, and Residual Review Mathematical model for regression Each point (X i, Y i ) in the
More information7.0 Lesson Plan. Regression. Residuals
7.0 Lesson Plan Regression Residuals 1 7.1 More About Regression Recall the regression assumptions: 1. Each point (X i, Y i ) in the scatterplot satisfies: Y i = ax i + b + ɛ i where the ɛ i have a normal
More information18.0 Multiple and Nonlinear Regression
18.0 Multiple and Nonlinear Regression 1 Answer Questions Multiple Regression Nonlinear Regression 18.1 Multiple Regression Recall the regression assumptions: 1. Each point (X i,y i ) in the scatterplot
More informationUnit 6 - Introduction to linear regression
Unit 6 - Introduction to linear regression Suggested reading: OpenIntro Statistics, Chapter 7 Suggested exercises: Part 1 - Relationship between two numerical variables: 7.7, 7.9, 7.11, 7.13, 7.15, 7.25,
More informationUnit 6 - Simple linear regression
Sta 101: Data Analysis and Statistical Inference Dr. Çetinkaya-Rundel Unit 6 - Simple linear regression LO 1. Define the explanatory variable as the independent variable (predictor), and the response variable
More informationAMS 7 Correlation and Regression Lecture 8
AMS 7 Correlation and Regression Lecture 8 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Suumer 2014 1 / 18 Correlation pairs of continuous observations. Correlation
More informationStat 529 (Winter 2011) A simple linear regression (SLR) case study. Mammals brain weights and body weights
Stat 529 (Winter 2011) A simple linear regression (SLR) case study Reading: Sections 8.1 8.4, 8.6, 8.7 Mammals brain weights and body weights Questions of interest Scatterplots of the data Log transforming
More informationLinear Regression and Correlation. February 11, 2009
Linear Regression and Correlation February 11, 2009 The Big Ideas To understand a set of data, start with a graph or graphs. The Big Ideas To understand a set of data, start with a graph or graphs. If
More informationAnnouncements. Lecture 10: Relationship between Measurement Variables. Poverty vs. HS graduate rate. Response vs. explanatory
Announcements Announcements Lecture : Relationship between Measurement Variables Statistics Colin Rundel February, 20 In class Quiz #2 at the end of class Midterm #1 on Friday, in class review Wednesday
More informationAnnouncements. Lecture 18: Simple Linear Regression. Poverty vs. HS graduate rate
Announcements Announcements Lecture : Simple Linear Regression Statistics 1 Mine Çetinkaya-Rundel March 29, 2 Midterm 2 - same regrade request policy: On a separate sheet write up your request, describing
More informationSociology 6Z03 Review I
Sociology 6Z03 Review I John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review I Fall 2016 1 / 19 Outline: Review I Introduction Displaying Distributions Describing
More informationChi-square tests. Unit 6: Simple Linear Regression Lecture 1: Introduction to SLR. Statistics 101. Poverty vs. HS graduate rate
Review and Comments Chi-square tests Unit : Simple Linear Regression Lecture 1: Introduction to SLR Statistics 1 Monika Jingchen Hu June, 20 Chi-square test of GOF k χ 2 (O E) 2 = E i=1 where k = total
More information3.2: Least Squares Regressions
3.2: Least Squares Regressions Section 3.2 Least-Squares Regression After this section, you should be able to INTERPRET a regression line CALCULATE the equation of the least-squares regression line CALCULATE
More informationSummarizing Data: Paired Quantitative Data
Summarizing Data: Paired Quantitative Data regression line (or least-squares line) a straight line model for the relationship between explanatory (x) and response (y) variables, often used to produce a
More informationLinear Regression. Linear Regression. Linear Regression. Did You Mean Association Or Correlation?
Did You Mean Association Or Correlation? AP Statistics Chapter 8 Be careful not to use the word correlation when you really mean association. Often times people will incorrectly use the word correlation
More informationAnnouncements. Unit 6: Simple Linear Regression Lecture : Introduction to SLR. Poverty vs. HS graduate rate. Modeling numerical variables
Announcements Announcements Unit : Simple Linear Regression Lecture : Introduction to SLR Statistics 1 Mine Çetinkaya-Rundel April 2, 2013 Statistics 1 (Mine Çetinkaya-Rundel) U - L1: Introduction to SLR
More informationChapter 7. Scatterplots, Association, and Correlation
Chapter 7 Scatterplots, Association, and Correlation Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 29 Objective In this chapter, we study relationships! Instead, we investigate
More informationSection 3.3. How Can We Predict the Outcome of a Variable? Agresti/Franklin Statistics, 1of 18
Section 3.3 How Can We Predict the Outcome of a Variable? Agresti/Franklin Statistics, 1of 18 Regression Line Predicts the value for the response variable, y, as a straight-line function of the value of
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More informationBivariate data analysis
Bivariate data analysis Categorical data - creating data set Upload the following data set to R Commander sex female male male male male female female male female female eye black black blue green green
More informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationLecture 18: Simple Linear Regression
Lecture 18: Simple Linear Regression BIOS 553 Department of Biostatistics University of Michigan Fall 2004 The Correlation Coefficient: r The correlation coefficient (r) is a number that measures the strength
More informationChapter 5 Friday, May 21st
Chapter 5 Friday, May 21 st Overview In this Chapter we will see three different methods we can use to describe a relationship between two quantitative variables. These methods are: Scatterplot Correlation
More informationChapter 7. Linear Regression (Pt. 1) 7.1 Introduction. 7.2 The Least-Squares Regression Line
Chapter 7 Linear Regression (Pt. 1) 7.1 Introduction Recall that r, the correlation coefficient, measures the linear association between two quantitative variables. Linear regression is the method of fitting
More informationLecture 16: Again on Regression
Lecture 16: Again on Regression S. Massa, Department of Statistics, University of Oxford 10 February 2016 The Normality Assumption Body weights (Kg) and brain weights (Kg) of 62 mammals. Species Body weight
More informationSimple Linear Regression for the MPG Data
Simple Linear Regression for the MPG Data 2000 2500 3000 3500 15 20 25 30 35 40 45 Wgt MPG What do we do with the data? y i = MPG of i th car x i = Weight of i th car i =1,...,n n = Sample Size Exploratory
More informationCorrelation and Regression
Correlation and Regression October 25, 2017 STAT 151 Class 9 Slide 1 Outline of Topics 1 Associations 2 Scatter plot 3 Correlation 4 Regression 5 Testing and estimation 6 Goodness-of-fit STAT 151 Class
More informationCan you tell the relationship between students SAT scores and their college grades?
Correlation One Challenge Can you tell the relationship between students SAT scores and their college grades? A: The higher SAT scores are, the better GPA may be. B: The higher SAT scores are, the lower
More informationappstats8.notebook October 11, 2016
Chapter 8 Linear Regression Objective: Students will construct and analyze a linear model for a given set of data. Fat Versus Protein: An Example pg 168 The following is a scatterplot of total fat versus
More informationStatistics for EES 7. Linear regression and linear models
Statistics for EES 7. Linear regression and linear models Dirk Metzler http://www.zi.biologie.uni-muenchen.de/evol/statgen.html 26. May 2009 Contents 1 Univariate linear regression: how and why? 2 t-test
More informationReview. Midterm Exam. Midterm Review. May 6th, 2015 AMS-UCSC. Spring Session 1 (Midterm Review) AMS-5 May 6th, / 24
Midterm Exam Midterm Review AMS-UCSC May 6th, 2015 Spring 2015. Session 1 (Midterm Review) AMS-5 May 6th, 2015 1 / 24 Topics Topics We will talk about... 1 Review Spring 2015. Session 1 (Midterm Review)
More informationNature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals. Regression Output. Conditions for inference.
Understanding regression output from software Nature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals In 1966 Cyril Burt published a paper called The genetic determination of differences
More informationMATH 2560 C F03 Elementary Statistics I LECTURE 9: Least-Squares Regression Line and Equation
MATH 2560 C F03 Elementary Statistics I LECTURE 9: Least-Squares Regression Line and Equation 1 Outline least-squares regresion line (LSRL); equation of the LSRL; interpreting the LSRL; correlation and
More informationStatistical View of Least Squares
May 23, 2006 Purpose of Regression Some Examples Least Squares Purpose of Regression Purpose of Regression Some Examples Least Squares Suppose we have two variables x and y Purpose of Regression Some Examples
More informationChapter 2: Looking at Data Relationships (Part 3)
Chapter 2: Looking at Data Relationships (Part 3) Dr. Nahid Sultana Chapter 2: Looking at Data Relationships 2.1: Scatterplots 2.2: Correlation 2.3: Least-Squares Regression 2.5: Data Analysis for Two-Way
More information9. Linear Regression and Correlation
9. Linear Regression and Correlation Data: y a quantitative response variable x a quantitative explanatory variable (Chap. 8: Recall that both variables were categorical) For example, y = annual income,
More informationDetermine is the equation of the LSRL. Determine is the equation of the LSRL of Customers in line and seconds to check out.. Chapter 3, Section 2
3.2c Computer Output, Regression to the Mean, & AP Formulas Be sure you can locate: the slope, the y intercept and determine the equation of the LSRL. Slope is always in context and context is x value.
More informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More information2. Outliers and inference for regression
Unit6: Introductiontolinearregression 2. Outliers and inference for regression Sta 101 - Spring 2016 Duke University, Department of Statistical Science Dr. Çetinkaya-Rundel Slides posted at http://bit.ly/sta101_s16
More informationScatterplots and Correlation
Bivariate Data Page 1 Scatterplots and Correlation Essential Question: What is the correlation coefficient and what does it tell you? Most statistical studies examine data on more than one variable. Fortunately,
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter Seven Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Seven Notes Spring 2011 1 / 42 Outline
More informationUNIT 12 ~ More About Regression
***SECTION 15.1*** The Regression Model When a scatterplot shows a relationship between a variable x and a y, we can use the fitted to the data to predict y for a given value of x. Now we want to do tests
More informationChapter 8. Linear Regression. Copyright 2010 Pearson Education, Inc.
Chapter 8 Linear Regression Copyright 2010 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright
More information1. Use Scenario 3-1. In this study, the response variable is
Chapter 8 Bell Work Scenario 3-1 The height (in feet) and volume (in cubic feet) of usable lumber of 32 cherry trees are measured by a researcher. The goal is to determine if volume of usable lumber can
More informationCorrelation and Regression Notes. Categorical / Categorical Relationship (Chi-Squared Independence Test)
Relationship Hypothesis Tests Correlation and Regression Notes Categorical / Categorical Relationship (Chi-Squared Independence Test) Ho: Categorical Variables are independent (show distribution of conditional
More informationRegression Modelling. Dr. Michael Schulzer. Centre for Clinical Epidemiology and Evaluation
Regression Modelling Dr. Michael Schulzer Centre for Clinical Epidemiology and Evaluation REGRESSION Origins: a historical note. Sir Francis Galton: Natural inheritance (1889) Macmillan, London. Regression
More information15.0 Linear Regression
15.0 Linear Regression 1 Answer Questions Lines Correlation Regression 15.1 Lines The algebraic equation for a line is Y = β 0 + β 1 X 2 The use of coordinate axes to show functional relationships was
More informationSection I: Multiple Choice Select the best answer for each question.
Chapter 3 AP Statistics Practice Test (TPS- 4 p200) Section I: Multiple Choice Select the best answer for each question. 1. A school guidance counselor examines the number of extracurricular activities
More informationThe response variable depends on the explanatory variable.
A response variable measures an outcome of study. > dependent variables An explanatory variable attempts to explain the observed outcomes. > independent variables The response variable depends on the explanatory
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationDr. Allen Back. Sep. 23, 2016
Dr. Allen Back Sep. 23, 2016 Look at All the Data Graphically A Famous Example: The Challenger Tragedy Look at All the Data Graphically A Famous Example: The Challenger Tragedy Type of Data Looked at the
More informationStat 101 L: Laboratory 5
Stat 101 L: Laboratory 5 The first activity revisits the labeling of Fun Size bags of M&Ms by looking distributions of Total Weight of Fun Size bags and regular size bags (which have a label weight) of
More informationChapter 14. Statistical versus Deterministic Relationships. Distance versus Speed. Describing Relationships: Scatterplots and Correlation
Chapter 14 Describing Relationships: Scatterplots and Correlation Chapter 14 1 Statistical versus Deterministic Relationships Distance versus Speed (when travel time is constant). Income (in millions of
More informationChapter 6: Exploring Data: Relationships Lesson Plan
Chapter 6: Exploring Data: Relationships Lesson Plan For All Practical Purposes Displaying Relationships: Scatterplots Mathematical Literacy in Today s World, 9th ed. Making Predictions: Regression Line
More informationWarm-up Using the given data Create a scatterplot Find the regression line
Time at the lunch table Caloric intake 21.4 472 30.8 498 37.7 335 32.8 423 39.5 437 22.8 508 34.1 431 33.9 479 43.8 454 42.4 450 43.1 410 29.2 504 31.3 437 28.6 489 32.9 436 30.6 480 35.1 439 33.0 444
More informationLecture 16 - Correlation and Regression
Lecture 16 - Correlation and Regression Statistics 102 Colin Rundel April 1, 2013 Modeling numerical variables Modeling numerical variables So far we have worked with single numerical and categorical variables,
More informationMrs. Poyner/Mr. Page Chapter 3 page 1
Name: Date: Period: Chapter 2: Take Home TEST Bivariate Data Part 1: Multiple Choice. (2.5 points each) Hand write the letter corresponding to the best answer in space provided on page 6. 1. In a statistics
More informationChapter 10. Regression. Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania
Chapter 10 Regression Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Scatter Diagrams A graph in which pairs of points, (x, y), are
More informationINFERENCE FOR REGRESSION
CHAPTER 3 INFERENCE FOR REGRESSION OVERVIEW In Chapter 5 of the textbook, we first encountered regression. The assumptions that describe the regression model we use in this chapter are the following. We
More informationChapter 8. Linear Regression /71
Chapter 8 Linear Regression 1 /71 Homework p192 1, 2, 3, 5, 7, 13, 15, 21, 27, 28, 29, 32, 35, 37 2 /71 3 /71 Objectives Determine Least Squares Regression Line (LSRL) describing the association of two
More informationSTA Module 5 Regression and Correlation. Learning Objectives. Learning Objectives (Cont.) Upon completing this module, you should be able to:
STA 2023 Module 5 Regression and Correlation Learning Objectives Upon completing this module, you should be able to: 1. Define and apply the concepts related to linear equations with one independent variable.
More informationChapter 16. Simple Linear Regression and dcorrelation
Chapter 16 Simple Linear Regression and dcorrelation 16.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationExample: Can an increase in non-exercise activity (e.g. fidgeting) help people gain less weight?
Example: Can an increase in non-exercise activity (e.g. fidgeting) help people gain less weight? 16 subjects overfed for 8 weeks Explanatory: change in energy use from non-exercise activity (calories)
More information11 Correlation and Regression
Chapter 11 Correlation and Regression August 21, 2017 1 11 Correlation and Regression When comparing two variables, sometimes one variable (the explanatory variable) can be used to help predict the value
More informationChapter 8: Transformations. October 15, For most practical problems, there is no theory to tell us the correct form for the mean function,
Chapter 8: Transformations October 15, 2018 For most practical problems, there is no theory to tell us the correct form for the mean function, and any parametric form we use is little more than an approximation
More informationContents. 1 Review of Residuals. 2 Detecting Outliers. 3 Influential Observations. 4 Multicollinearity and its Effects
Contents 1 Review of Residuals 2 Detecting Outliers 3 Influential Observations 4 Multicollinearity and its Effects W. Zhou (Colorado State University) STAT 540 July 6th, 2015 1 / 32 Model Diagnostics:
More informationIntroduction and Single Predictor Regression. Correlation
Introduction and Single Predictor Regression Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning Correlation A correlation
More informationUNIVERSITY OF MASSACHUSETTS. Department of Mathematics and Statistics. Basic Exam - Applied Statistics. Tuesday, January 17, 2017
UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam - Applied Statistics Tuesday, January 17, 2017 Work all problems 60 points are needed to pass at the Masters Level and 75
More informationSTA 302f16 Assignment Five 1
STA 30f16 Assignment Five 1 Except for Problem??, these problems are preparation for the quiz in tutorial on Thursday October 0th, and are not to be handed in As usual, at times you may be asked to prove
More informationChapter 14. Linear least squares
Serik Sagitov, Chalmers and GU, March 5, 2018 Chapter 14 Linear least squares 1 Simple linear regression model A linear model for the random response Y = Y (x) to an independent variable X = x For a given
More informationInfluencing Regression
Math Objectives Students will recognize that one point can influence the correlation coefficient and the least-squares regression line. Students will differentiate between an outlier and an influential
More informationChapter 7 Linear Regression
Chapter 7 Linear Regression 1 7.1 Least Squares: The Line of Best Fit 2 The Linear Model Fat and Protein at Burger King The correlation is 0.76. This indicates a strong linear fit, but what line? The line
More informationRegression Models - Introduction
Regression Models - Introduction In regression models there are two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent
More informationRegression and correlation. Correlation & Regression, I. Regression & correlation. Regression vs. correlation. Involve bivariate, paired data, X & Y
Regression and correlation Correlation & Regression, I 9.07 4/1/004 Involve bivariate, paired data, X & Y Height & weight measured for the same individual IQ & exam scores for each individual Height of
More informationSTAT 361 Fall Homework 1: Solution. It is customary to use a special sign Σ as an abbreviation for the sum of real numbers
STAT 361 Fall 2016 Homework 1: Solution It is customary to use a special sign Σ as an abbreviation for the sum of real numbers x 1, x 2,, x n : x i = x 1 + x 2 + + x n. If x 1,..., x n are real numbers,
More informationThe following formulas related to this topic are provided on the formula sheet:
Student Notes Prep Session Topic: Exploring Content The AP Statistics topic outline contains a long list of items in the category titled Exploring Data. Section D topics will be reviewed in this session.
More information10.1: Scatter Plots & Trend Lines. Essential Question: How can you describe the relationship between two variables and use it to make predictions?
10.1: Scatter Plots & Trend Lines Essential Question: How can you describe the relationship between two variables and use it to make predictions? Vocab Two-variable data: two data points, one individual/object.
More informationAP Statistics Bivariate Data Analysis Test Review. Multiple-Choice
Name Period AP Statistics Bivariate Data Analysis Test Review Multiple-Choice 1. The correlation coefficient measures: (a) Whether there is a relationship between two variables (b) The strength of the
More informationLooking at data: relationships
Looking at data: relationships Least-squares regression IPS chapter 2.3 2006 W. H. Freeman and Company Objectives (IPS chapter 2.3) Least-squares regression p p The regression line Making predictions:
More informationInference for Regression Inference about the Regression Model and Using the Regression Line, with Details. Section 10.1, 2, 3
Inference for Regression Inference about the Regression Model and Using the Regression Line, with Details Section 10.1, 2, 3 Basic components of regression setup Target of inference: linear dependency
More informationAny of 27 linear and nonlinear models may be fit. The output parallels that of the Simple Regression procedure.
STATGRAPHICS Rev. 9/13/213 Calibration Models Summary... 1 Data Input... 3 Analysis Summary... 5 Analysis Options... 7 Plot of Fitted Model... 9 Predicted Values... 1 Confidence Intervals... 11 Observed
More informationTHE ROYAL STATISTICAL SOCIETY 2008 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE (MODULAR FORMAT) MODULE 4 LINEAR MODELS
THE ROYAL STATISTICAL SOCIETY 008 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE (MODULAR FORMAT) MODULE 4 LINEAR MODELS The Society provides these solutions to assist candidates preparing for the examinations
More informationStatistical View of Least Squares
Basic Ideas Some Examples Least Squares May 22, 2007 Basic Ideas Simple Linear Regression Basic Ideas Some Examples Least Squares Suppose we have two variables x and y Basic Ideas Simple Linear Regression
More informationMath 1710 Class 20. V2u. Last Time. Graphs and Association. Correlation. Regression. Association, Correlation, Regression Dr. Back. Oct.
,, Dr. Back Oct. 14, 2009 Son s Heights from Their Fathers Galton s Original 1886 Data If you know a father s height, what can you say about his son s? Son s Heights from Their Fathers Galton s Original
More informationChapter 8. Linear Regression. The Linear Model. Fat Versus Protein: An Example. The Linear Model (cont.) Residuals
Chapter 8 Linear Regression Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8-1 Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Fat Versus
More information7. Do not estimate values for y using x-values outside the limits of the data given. This is called extrapolation and is not reliable.
AP Statistics 15 Inference for Regression I. Regression Review a. r à correlation coefficient or Pearson s coefficient: indicates strength and direction of the relationship between the explanatory variables
More informationLecture 19: Inference for SLR & Transformations
Lecture 19: Inference for SLR & Transformations Statistics 101 Mine Çetinkaya-Rundel April 3, 2012 Announcements Announcements HW 7 due Thursday. Correlation guessing game - ends on April 12 at noon. Winner
More informationASSIGNMENT 3 SIMPLE LINEAR REGRESSION. Old Faithful
ASSIGNMENT 3 SIMPLE LINEAR REGRESSION In the simple linear regression model, the mean of a response variable is a linear function of an explanatory variable. The model and associated inferential tools
More informationSimple Linear Regression Using Ordinary Least Squares
Simple Linear Regression Using Ordinary Least Squares Purpose: To approximate a linear relationship with a line. Reason: We want to be able to predict Y using X. Definition: The Least Squares Regression
More informationChapter 9. Correlation and Regression
Chapter 9 Correlation and Regression Lesson 9-1/9-2, Part 1 Correlation Registered Florida Pleasure Crafts and Watercraft Related Manatee Deaths 100 80 60 40 20 0 1991 1993 1995 1997 1999 Year Boats in
More informationKeller: Stats for Mgmt & Econ, 7th Ed July 17, 2006
Chapter 17 Simple Linear Regression and Correlation 17.1 Regression Analysis Our problem objective is to analyze the relationship between interval variables; regression analysis is the first tool we will
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 10 Correlation and Regression 10-1 Overview 10-2 Correlation 10-3 Regression 10-4
More informationChapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
More informationApplied Regression Analysis. Section 4: Diagnostics and Transformations
Applied Regression Analysis Section 4: Diagnostics and Transformations 1 Regression Model Assumptions Y i = β 0 + β 1 X i + ɛ Recall the key assumptions of our linear regression model: (i) The mean of
More informationProb/Stats Questions? /32
Prob/Stats 10.4 Questions? 1 /32 Prob/Stats 10.4 Homework Apply p551 Ex 10-4 p 551 7, 8, 9, 10, 12, 13, 28 2 /32 Prob/Stats 10.4 Objective Compute the equation of the least squares 3 /32 Regression A scatter
More informationBasic Business Statistics 6 th Edition
Basic Business Statistics 6 th Edition Chapter 12 Simple Linear Regression Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based
More informationAnalysis of Bivariate Data
Analysis of Bivariate Data Data Two Quantitative variables GPA and GAES Interest rates and indices Tax and fund allocation Population size and prison population Bivariate data (x,y) Case corr® 2 Independent
More informationChapter 3: Describing Relationships
Chapter 3: Describing Relationships Section 3.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 3 Describing Relationships 3.1 Scatterplots and Correlation 3.2 Section 3.2
More informationChapter 10 Correlation and Regression
Chapter 10 Correlation and Regression 10-1 Review and Preview 10-2 Correlation 10-3 Regression 10-4 Variation and Prediction Intervals 10-5 Multiple Regression 10-6 Modeling Copyright 2010, 2007, 2004
More information