Simple Linear Regression Inference


 Francis Richardson
 5 years ago
 Views:
Transcription
1 Simple Linear Regression Inference 1
2 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 The interpretation of the Confidence Interval (CI) is fixed to a level α. If we sample more than a set of observations, each set would probably give a di_erent OLS estimate of β and therefore different CI. (1 α)% of these intervals would include β, and only (α)% of the sets would deviate from β by more than a specified Δ. 2
3 Standardize a Gaussian variable As we have previously seen: We can standardize such statement such that: 3
4 The tdistribution However σ 2 in unknown, and it must be estimated by s 2. Where t ν is a t distribution with ν= n 2 degrees of freedom. 4
5 CI limits Hence the CI for β at a (1 α)% can be calculated as follows: The CI specifies a range of values within which the true parameter is credible to lie. The level of likelihood is specified by the choice of α. 5
6 Hypothesis Testing Step 1. Fix a significance level α Step 2. Specify the null hypothesis Step 3. Specify the alternative hypothesis Step 4. Calculate the test statistic Step 5. Determine the acceptance/rejection region 6
7 Null and Alternative Hypothesis Given a starting conjecture (null hypothesis), considered true as working hypothesis, we may verify the Hypothesis evaluating the discrepancy between the sample observations and what one would expect from the null hypothesis. If, given a specific level of significance α, the discrepancy is significant, the null hypothesis is rejected. Otherwise it cannot be statistically rejected. It is important to notice that you may never conclusively accept the null hypothesis (Falsification Theory). 7
8 Hypothesis Testing: β1 = 0 We want to test whether there is a linear relationship between X and Y at a generic α = 0.05 level of significance. The hypothesis will then be: H 0 : β1 = 0 H A : β1 0 8
9 Hypothesis Testing: β1 = 0 The test statistic as before defined: Hence we reject the null hypothesis if: 9
10 Hypothesis Testing: β1 = 0 The Golden Rule When n is large, the tdistribution tends to a normal distribution, hence if we choose a 5% significance level as above we can use an approximate rule of thumb and reject the hypothesis whenever the test statistics is greater the 2. 10
11 Hypothesis Testing: b1 = 0 Pvalue and Significant levels The pvalue is a probability. The rule is: we reject the null hypothesis if pvalue< α. 11
12 Pvalue example Suppose you fix α = 0.5, therefore α/2 = 0.025, and the pvalue associated to the coefficient β 1 is It means that effectively there is a 1% chance that the relationship emerged randomly and a 99% chance that the relationship is real. Graphically we have: 12
13 Hypothesis Testing: β1 = c Similarly if we are testing for a specific constant value named c: H 0 : β1 = c H A : β1 c Hence the rejection criterion is: Keeping in mind the possibility of a Normal approximation for large samples. 13
14 The meaning of β 1 The parameter β 1 expresses how much change in Y follows a unitary change in X on average. if β 1 > 0 an increase in X is proportional to the increase in Y (direct relationship) if β 1 < 0 an increase in X is inversely proportional to the increase in Y (indirect relationship) 14
15 Regression and Correlation 15
16 Linear relationship The Σx i y i function measures the intensity of the linear relationship between X and Y. When this amount is expressed as mean observation contribution it is called covariance. 16
17 Linear relationship The covariance is a measure that depends on the measurement scalar. An alternative that is a relative index is the Pearson correlation coefficient. 17
18 Relationship and Dependence One must not however mistake the Pearson coefficient with the regression coefficient β 1 of a simple linear regression, as they have different formulas and different meanings. They can be linked by the following formula: Clearly the meaning also differs. The Pearson coefficient (r) measures the linear bidirectional relationship between X and Y (i.e. Y and X). 18
19 Linear relationship The regression coefficient β 1 measures the linear dependence of Y from X. Given a linear relationship, the dependence may be from X Y or Y X. In regression analysis one must ex ante decide which dependence to investigate. Correlation analysis in not concerned with causal links. Regression analysis is based of causal links. 19
20 Linear relationship Correlation does not imply causality. Example: If there are many hospital recoveries (X) in zones with many doctors (Y), it does not imply that doctors cause recoveries, but it identifies a strong linear link. 20
21 Residuals squared 21
22 Residual properties 22
23 Sum of squares TSS = Total Sum of Squares RSS = Residual Sum of Squares ESS = Explained Sum of Squares 23
24 R 2 : coefficient of determination The coefficient of determination R 2 expresses the proportion of total deviance explained by the regression model of Y on X. As one cannot explain more then all the existing deviance we can write as follows: 24
25 R 2 properties R 2 = 0 implies that the explicative contribution of the model is null, hence the deviance is completely expressed by the random component. R 2 = 1 implies that all N observations lie on the regression line, hence the whole variability is explained by the model. 25
26 R 2 properties All intermediate cases imply that the higher/lower the value of R 2 the more/less of variability is expressed by the choice of the model. A model with an R 2 = 0.80 implies that 80% of the variability is explained by the chosen model. The R 2 is said to be a fitting index as it measures the adaptiveness of the chosen model to the specified dataset. 26
27 R 2 and r Hence the coefficient of determination is equal to the coefficient of correlation squared. Another equally simple and efficient relationship is can be obtained by: 27
28 ANalysis Of VAriance The total variability decomposition shows the error component and the model component. We also know that: 28
29 ANalysis Of VAriance As the square of a Normal distribution is a Chisquared distribution: Omitting here the proof for: The ratio of χ 2 divided by the d.f. is known to be an F distribution, as follows: 29
30 ANalysis Of VAriance One may then verify the null hypothesis H 0 : β 1 = 0 with respect to H A : β 1 0 with the following test statistic: A strong linear relationship between X and Y will determine a high valued test statistic, which supports the model, and reject the null hypothesis. Therefore: * F F 1; n 2 H0 : 1 0 is rejected empirical value theoretical value 30
31 Anova table 31
32 Forecast Often the estimated regression model is used to predict the expected Y value (i.e ˆ Y ) which corresponds to a specific value of the X variable in this case indicated with Xκ. The standard error of such value is equal to: 32
33 Forecast The 1 α confidence intervals for such value are then specified as: We may notice that the value of the s.e. increases proportionally to the distance between Xκ and X, therefore taking distant forecasts will worsen the quality of the estimate. It can also happen that the linear relation breaks down for very high or low Xκ values, as it is limited to the observe scatter plot. In this case taking a forecast outside the range of the observed values can be misleading. 33
34 Residual analysis The residual analysis is a graphical approach used to evaluate if the assumptions of the model are valid and thus to determine whether the regression model selected is appropriate. The assumption that can be graphically evaluated are: Linearity; Independence of errors; Normality of errors; Equal variance. 34
35 Linearity To evaluate linearity, plot the residuals on the vertical axis against the corresponding Xi values on the horizontal axis. If the linear model is appropriate fro the data, there will be no apparent pattern in this plot. 35
36 Independence For data collected over time, residuals can be plotted versus the time dimension to search for a relationship between consecutive residuals. Normality It can be evaluated with a normal probability plot that compare the actual versus the theoretical values of the residuals. 36
37 Equal variance Plot residuals versus X i. 37
38 Business Exercises Managerial decisions are often based on the relationship between two or more variables. For example, after considering the relationship between advertising expenditures and sales, a marketing manager might attempt to predict sales for a given level of advertising expenditure. A public utility might use the relationship between daily temperature and the demand for electricity to predict electricity consumption on next months temperature forecasts. 38
39 Business Exercises A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this the company randomly chooses six small cities and offers the candy bar at different prices. Using the candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below: 39
40 Business Exercises: ˆ ˆ 0, 1 40
41 Business Exercises: TSS 41
42 Business Exercises: TSS 42
43 Business Exercises: ESS 43
44 Numerical Example 44
45 45
46 46
47 Therefore one can conclude that 95 times out of 100 the true value β 1 lies between 0.25 and
48 The linear relationship is very strong, 97%. As an alternative to the Ftest one can present the ttest. Which implies that the variable of circulating vehicles is significant in the regression. 48
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 informationChapter 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 informationMultiple Linear Regression
Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is
More informationElements of statistics (MATH04871)
Elements of statistics (MATH04871) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis 
More informationPart 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 information1. 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 1propZint 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" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More information17. 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 informationOutline. Topic 4  Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares
Topic 4  Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test  Fall 2013 R 2 and the coefficient of correlation
More informationFactors 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 informationUnit 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 informationChapter 7 Notes  Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:
Chapter 7 Notes  Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a
More informationCHAPTER 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 information2. 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 informationPOLYNOMIAL 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 informationCOMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.
277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies
More informationDATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More information2013 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 informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationChapter 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 informationIntroduction 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 informationAdditional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm
Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jintselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm
More information11. Analysis of Casecontrol Studies Logistic Regression
Research methods II 113 11. Analysis of Casecontrol Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:
More informationBivariate Statistics Session 2: Measuring Associations ChiSquare Test
Bivariate Statistics Session 2: Measuring Associations ChiSquare Test Features Of The ChiSquare Statistic The chisquare test is nonparametric. That is, it makes no assumptions about the distribution
More informationNCSS 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 informationWeek TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480
1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationPlease follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used
More informationClass 19: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationSection 13, Part 1 ANOVA. Analysis Of Variance
Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationTesting for Lack of Fit
Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit
More informationII. DISTRIBUTIONS distribution normal distribution. standard scores
Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,
More informationHow To Check For Differences In The One Way Anova
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationMULTIPLE 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 informationGeneralized Linear Models
Generalized Linear Models We have previously worked with regression models where the response variable is quantitative and normally distributed. Now we turn our attention to two types of models where the
More informationA Primer on Forecasting Business Performance
A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.
More information2. 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 information12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationConsider a study in which. How many subjects? The importance of sample size calculations. An insignificant effect: two possibilities.
Consider a study in which How many subjects? The importance of sample size calculations Office of Research Protections Brown Bag Series KB Boomer, Ph.D. Director, boomer@stat.psu.edu A researcher conducts
More information1 Simple Linear Regression I Least Squares Estimation
Simple Linear Regression I Least Squares Estimation Textbook Sections: 8. 8.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationUsing Excel for inferential statistics
FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied
More information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
More informationTime series Forecasting using HoltWinters Exponential Smoothing
Time series Forecasting using HoltWinters Exponential Smoothing Prajakta S. Kalekar(04329008) Kanwal Rekhi School of Information Technology Under the guidance of Prof. Bernard December 6, 2004 Abstract
More informationEconometrics 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 informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More information2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measuresoffit in multiple regression Assumptions
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationStatistical Models in R
Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 16233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationTwoSample TTests Allowing Unequal Variance (Enter Difference)
Chapter 45 TwoSample TTests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationSimple 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 informationCorrelation 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 information12: Analysis of Variance. Introduction
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
More informationPrinciples of Hypothesis Testing for Public Health
Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions
More informationCalculating PValues. Parkland College. Isela Guerra Parkland College. Recommended Citation
Parkland College A with Honors Projects Honors Program 2014 Calculating PValues Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating PValues" (2014). A with Honors Projects.
More informationChapter 3 Quantitative Demand Analysis
Managerial Economics & Business Strategy Chapter 3 uantitative Demand Analysis McGrawHill/Irwin Copyright 2010 by the McGrawHill Companies, Inc. All rights reserved. Overview I. The Elasticity Concept
More informationCausal 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 informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationBusiness Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.
Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGrawHill/Irwin, 2008, ISBN: 9780073319889. Required Computing
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationOverview Classes. 123 Logistic regression (5) 193 Building and applying logistic regression (6) 263 Generalizations of logistic regression (7)
Overview Classes 123 Logistic regression (5) 193 Building and applying logistic regression (6) 263 Generalizations of logistic regression (7) 24 Loglinear models (8) 54 1517 hrs; 5B02 Building and
More informationPoint Biserial Correlation Tests
Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the productmoment correlation calculated between a continuous random variable
More informationExample: Boats and Manatees
Figure 96 Example: Boats and Manatees Slide 1 Given the sample data in Table 91, find the value of the linear correlation coefficient r, then refer to Table A6 to determine whether there is a significant
More informationWHAT IS A JOURNAL CLUB?
WHAT IS A JOURNAL CLUB? With its September 2002 issue, the American Journal of Critical Care debuts a new feature, the AJCC Journal Club. Each issue of the journal will now feature an AJCC Journal Club
More informationRegression 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 information1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ
STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material
More informationIntroduction 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 informationSAS Software to Fit the Generalized Linear Model
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
More informationSection 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 informationSPSS TUTORIAL & EXERCISE BOOK
UNIVERSITY OF MISKOLC Faculty of Economics Institute of Business Information and Methods Department of Business Statistics and Economic Forecasting PETRA PETROVICS SPSS TUTORIAL & EXERCISE BOOK FOR BUSINESS
More informationLinear Models for Continuous Data
Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationInteraction between quantitative predictors
Interaction between quantitative predictors In a firstorder model like the ones we have discussed, the association between E(y) and a predictor x j does not depend on the value of the other predictors
More informationAugust 2012 EXAMINATIONS Solution Part I
August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,
More informationKSTAT MINIMANUAL. Decision Sciences 434 Kellogg Graduate School of Management
KSTAT MINIMANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To
More informationAnalysis of Variance. MINITAB User s Guide 2 31
3 Analysis of Variance Analysis of Variance Overview, 32 OneWay Analysis of Variance, 35 TwoWay Analysis of Variance, 311 Analysis of Means, 313 Overview of Balanced ANOVA and GLM, 318 Balanced
More informationDongfeng Li. Autumn 2010
Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis
More informationMultiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear.
Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.
More informationCourse Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics
Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGrawHill/Irwin, 2010, ISBN: 9780077384470 [This
More informationBasic Statistics and Data Analysis for Health Researchers from Foreign Countries
Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association
More informationPremaster 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 informationSTART Selected Topics in Assurance
START Selected Topics in Assurance Related Technologies Table of Contents Introduction Some Statistical Background Fitting a Normal Using the Anderson Darling GoF Test Fitting a Weibull Using the Anderson
More informationDescription. Textbook. Grading. Objective
EC151.02 Statistics for Business and Economics (MWF 8:008:50) Instructor: Chiu Yu Ko Office: 462D, 21 Campenalla Way Phone: 26093 Email: kocb@bc.edu Office Hours: by appointment Description This course
More informationThe Big Picture. Correlation. Scatter Plots. Data
The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered
More informationBowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition
Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology StepbyStep  Excel Microsoft Excel is a spreadsheet software application
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationSimple 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 informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationConfidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
More informationINTRODUCTORY STATISTICS
INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
More informationOnce saved, if the file was zipped you will need to unzip it. For the files that I will be posting you need to change the preferences.
1 Commands in JMP and Statcrunch Below are a set of commands in JMP and Statcrunch which facilitate a basic statistical analysis. The first part concerns commands in JMP, the second part is for analysis
More informationLets suppose we rolled a sixsided die 150 times and recorded the number of times each outcome (16) occured. The data is
In this lab we will look at how R can eliminate most of the annoying calculations involved in (a) using ChiSquared tests to check for homogeneity in twoway tables of catagorical data and (b) computing
More information