EC2001 Econometrics 1 Dr. Jose Olmo Room D309

Size: px
Start display at page:

Download "EC2001 Econometrics 1 Dr. Jose Olmo Room D309"

Transcription

1 EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population: either (a real collection of things, such as UK citizens aged 18 to 25, or (b abstract notion such as all possible realizations of... etc. Example.- Possible cases in two tosses of a coin. 1

2 1.2 Differences between statistics, estimators and parameters A parameter is a property of the population. Not of the sample! Examples.- average wage in UK, life expectancy in Greenland, standard deviation in life expectancy in Greenland... A statistic is a function of observations. Example: sample mean, difference between sample and actual mean, number of people over 90 in a sample of ten individuals of Greenland... An estimator is a particular case of statistic. The difference lies in that an estimator gives us a guess of the value of a parameter of the population that is calculated from the information provided by the available data (sample. Note that an estimator is an statistic but not necessarily the opposite!!! 2

3 2 Sampling Distributions for the sample mean and sample variance The sample mean is determined by a sample of the population. The population mean is a parameter (constant. We will consider the estimators x n = 1 n n i=1 x i, denoting the sample mean, and Sn 2 = 1 n n 1 i=1 (x i x n 2, denoting the sample variance. Consider an independent and identically distributed (iid sample x 1,..., x n drawn from a N(µ, σ 2 distribution. By the properties of the normal distribution ( n xn µ σ N(0, 1, and (n 1S 2 n σ 2 χ 2 n 1. 3

4 3 Interval estimation for x n and S 2 n If the distribution of the estimator is known we can compute confidence intervals for the value of the parameter. We need to determine a significance level (α. Goal: Construct a confidence level s.t. the parameter is contained in it 100(1 α% of the times. In interval estimation we construct two functions g 1 (x 1,..., x n and g 2 (x 1,..., x n that depend on the sample observations such that P {g 1 (x 1,..., x n µ g 2 (x 1,..., x n } = 1 α. We operate on the equation P { g 1 > µ > g 2 } = 1 α. We do further computations in the expression to obtain P { ( x n g 1 ( σ > n xn µ σ > ( n xn g 2 } = 1 α. σ 4

5 Why do we do this??? Let us write it from left to right... P { ( x n g 2 ( σ < n xn µ σ < ( n xn g 1 σ } = 1 α. (1 The statistic ( x n µ σ follows a standard normal distribution. This means N(0, 1. This is our goal; find an standardized statistic. In this case the values of the distribution are tabulated and are in turn known. In fact we know from these tables the critical values z α/2 and z 1 α/2. Example: For α = 0.05, z α/2 = 1.96 and z 1 α/2 = P {z α/2 < ( x n µ σ < z1 α/2 } = 1 α. (2 Equating in equations (1 and (2 we have z α/2 = ( x n g 2 σ and z 1 α/2 = ( x n g 1 σ. Then g 2 = x n σ z α/2 and g 1 = x n σ z 1 α/2. 5

6 Furthermore given the normal distribution is symmetric we have z α/2 = z 1 α/2. Then g 2 = x n σ z α/2 and g 1 = x n + σ z α/2. The confidence interval at a level 1 α is given by [ x n σ z α/2, x n + σ z α/2 ]. Note that z α/2 < 0. Exercise 1.- The confidence interval for σ 2. Hint: P {χ α/2 < (n 1S2 n < χ σ 2 1 α/2 } = 1 α. (1 Solution:... P { (n 1S2 n χ 1 α/2 < σ 2 < (n 1S2 n } = 1 α. χ α/2 (2 6

7 4 Testing of Hypotheses An estimate gives us information about a parameter provided by the sample at hand. Our desire is to know the value of the parameter. We can use the value of the estimate and the uncertainty of the estimation to test different hypotheses. 1. Point hypothesis: H 0 : µ = 0 vs H 0 : µ Interval hypothesis: H 0 : 1 µ 1 vs H 1 : µ does not belong to [ 1, 1]. A test is a procedure that answers the question of whether the observed difference between the sample value and the population value hypothesized is real or due to uncertainty surrounding the estimation (sample variability. In order to see this we need the distribution function of a statistic relating the estimate and the parameter. 7

8 In the case of the mean, t = ( x n µ S tν with ν denoting the degrees of freedom of a Student s t-distribution. In this example if the sample has n observations ν = n Criteria to reject H 0 The case of hypothesis testing is similar to confidence intervals (sometimes is the same thing. We reject at a significance level (α. This means we assign an α probability to be wrong in our conclusion. This α determines a critical level such that if the statistic exceeds this value we consider the null hypothesis is false. 8

9 Hypothesis tests can be one sided or twosided: 1. Two-sided: H 0 : µ = 4 vs H 0 : µ 4 2. One-sided: H 0 : µ = 4 vs H 0 : µ < 4 or H 0 : µ = 4 vs H 0 : µ > 4. Example.- (a two sided case H 0 : µ = 4 vs H 0 : µ 4 and consider α = 0.05 and n = 100. Critical level: t 99,1 α/2, given it is a two sided test. Rejection criteria: RH 0 if ( x n 4 S < t99,α/2 or ( n xn 4 S > t99,1 α/2. We can use alternatively the absolute value to write the rejection criteria more compactly, ( x n 4 S > t99,1 α. An alternative viewpoint: measure the probability in the tail (p-values. 9

10 P { ( x n 4 S > tn 1 } = p value or P {t n 1 > ( x n 4 S } = p value depending on the context. Rejection criteria: p value < α/2. For the absolute value, P {t n 1 > ( x n 4 S } = p value. Rejection criteria: p value < α. Exercise 2.- Do this example for a x n = 4.23 and S = 2.5, b x n = 4.05 and S = 0.1. Does this agree with what your intuition would say? Example.- (a one-sided case H 0 : µ = 4 vs H 0 : µ < 4 and consider α = 0.05 and n =

11 Critical level: t 99,α, given it is a onesided (left test. Rejection criteria (for a left-sided test: < t99,α. ( xn 4 S An alternative viewpoint: measure the probability in the tail (p-values. For the left tail this is } = p value. P {t n 1 ( x n 4 S Rejection criteria: p value < α for a one-sided test. 11

12 Exercise 3.- Do this example for a x n = 4.5 and S = 0.5, b x n = and S = Does this agree with what your intuition would say? 4.2 Type of errors in hypothesis testing We can commit two types of errors in hypothesis testing. Type I error: Rejecting the null hypothesis when it is true. Type II error: Accepting the null hypothesis when it is false. This is called β error. Thus α = P {RH 0 H 0 is true}, β = P {AH 0 H 0 is false}. Exercise 4.- Why do you think type I error is also called α error? 12

Hypothesis Testing One Sample Tests

Hypothesis Testing One Sample Tests STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal

More information

Introductory Econometrics. Review of statistics (Part II: Inference)

Introductory Econometrics. Review of statistics (Part II: Inference) Introductory Econometrics Review of statistics (Part II: Inference) Jun Ma School of Economics Renmin University of China October 1, 2018 1/16 Null and alternative hypotheses Usually, we have two competing

More information

Summary: the confidence interval for the mean (σ 2 known) with gaussian assumption

Summary: the confidence interval for the mean (σ 2 known) with gaussian assumption Summary: the confidence interval for the mean (σ known) with gaussian assumption on X Let X be a Gaussian r.v. with mean µ and variance σ. If X 1, X,..., X n is a random sample drawn from X then the confidence

More information

INTERVAL ESTIMATION AND HYPOTHESES TESTING

INTERVAL ESTIMATION AND HYPOTHESES TESTING INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,

More information

exp{ (x i) 2 i=1 n i=1 (x i a) 2 (x i ) 2 = exp{ i=1 n i=1 n 2ax i a 2 i=1

exp{ (x i) 2 i=1 n i=1 (x i a) 2 (x i ) 2 = exp{ i=1 n i=1 n 2ax i a 2 i=1 4 Hypothesis testing 4. Simple hypotheses A computer tries to distinguish between two sources of signals. Both sources emit independent signals with normally distributed intensity, the signals of the first

More information

Chapter 5: HYPOTHESIS TESTING

Chapter 5: HYPOTHESIS TESTING MATH411: Applied Statistics Dr. YU, Chi Wai Chapter 5: HYPOTHESIS TESTING 1 WHAT IS HYPOTHESIS TESTING? As its name indicates, it is about a test of hypothesis. To be more precise, we would first translate

More information

Lectures 5 & 6: Hypothesis Testing

Lectures 5 & 6: Hypothesis Testing Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across

More information

Chapter 9 Inferences from Two Samples

Chapter 9 Inferences from Two Samples Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review

More information

STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots. March 8, 2015

STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots. March 8, 2015 STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots March 8, 2015 The duality between CI and hypothesis testing The duality between CI and hypothesis

More information

LECTURE 6. Introduction to Econometrics. Hypothesis testing & Goodness of fit

LECTURE 6. Introduction to Econometrics. Hypothesis testing & Goodness of fit LECTURE 6 Introduction to Econometrics Hypothesis testing & Goodness of fit October 25, 2016 1 / 23 ON TODAY S LECTURE We will explain how multiple hypotheses are tested in a regression model We will define

More information

Confidence Intervals and Hypothesis Tests

Confidence Intervals and Hypothesis Tests Confidence Intervals and Hypothesis Tests STA 281 Fall 2011 1 Background The central limit theorem provides a very powerful tool for determining the distribution of sample means for large sample sizes.

More information

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses. 1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately

More information

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire

More information

Section 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples

Section 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means

More information

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology

More information

Chapter 5 Confidence Intervals

Chapter 5 Confidence Intervals Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is

More information

Preliminary Statistics Lecture 5: Hypothesis Testing (Outline)

Preliminary Statistics Lecture 5: Hypothesis Testing (Outline) 1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 5: Hypothesis Testing (Outline) Gujarati D. Basic Econometrics, Appendix A.8 Barrow M. Statistics

More information

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................

More information

Probabilities & Statistics Revision

Probabilities & Statistics Revision Probabilities & Statistics Revision Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 January 6, 2017 Christopher Ting QF

More information

i=1 X i/n i=1 (X i X) 2 /(n 1). Find the constant c so that the statistic c(x X n+1 )/S has a t-distribution. If n = 8, determine k such that

i=1 X i/n i=1 (X i X) 2 /(n 1). Find the constant c so that the statistic c(x X n+1 )/S has a t-distribution. If n = 8, determine k such that Math 47 Homework Assignment 4 Problem 411 Let X 1, X,, X n, X n+1 be a random sample of size n + 1, n > 1, from a distribution that is N(µ, σ ) Let X = n i=1 X i/n and S = n i=1 (X i X) /(n 1) Find the

More information

LECTURE 5. Introduction to Econometrics. Hypothesis testing

LECTURE 5. Introduction to Econometrics. Hypothesis testing LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will

More information

Chapter 8 of Devore , H 1 :

Chapter 8 of Devore , H 1 : Chapter 8 of Devore TESTING A STATISTICAL HYPOTHESIS Maghsoodloo A statistical hypothesis is an assumption about the frequency function(s) (i.e., PDF or pdf) of one or more random variables. Stated in

More information

Estimating the accuracy of a hypothesis Setting. Assume a binary classification setting

Estimating the accuracy of a hypothesis Setting. Assume a binary classification setting Estimating the accuracy of a hypothesis Setting Assume a binary classification setting Assume input/output pairs (x, y) are sampled from an unknown probability distribution D = p(x, y) Train a binary classifier

More information

STAT 513 fa 2018 Lec 02

STAT 513 fa 2018 Lec 02 STAT 513 fa 2018 Lec 02 Inference about the mean and variance of a Normal population Karl B. Gregory Fall 2018 Inference about the mean and variance of a Normal population Here we consider the case in

More information

Stats Review Chapter 14. Mary Stangler Center for Academic Success Revised 8/16

Stats Review Chapter 14. Mary Stangler Center for Academic Success Revised 8/16 Stats Review Chapter 14 Revised 8/16 Note: This review is meant to highlight basic concepts from the course. It does not cover all concepts presented by your instructor. Refer back to your notes, unit

More information

Performance Evaluation and Comparison

Performance Evaluation and Comparison Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Cross Validation and Resampling 3 Interval Estimation

More information

CH.9 Tests of Hypotheses for a Single Sample

CH.9 Tests of Hypotheses for a Single Sample CH.9 Tests of Hypotheses for a Single Sample Hypotheses testing Tests on the mean of a normal distributionvariance known Tests on the mean of a normal distributionvariance unknown Tests on the variance

More information

BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings

BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings Yujin Chung October 4th, 2016 Fall 2016 Yujin Chung Lec6: Statistical hypothesis testings Fall 2016 1/30 Previous Two types of statistical

More information

Inference in Regression Analysis

Inference in Regression Analysis Inference in Regression Analysis Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 4, Slide 1 Today: Normal Error Regression Model Y i = β 0 + β 1 X i + ǫ i Y i value

More information

ME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV

ME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV Theory of Engineering Experimentation Chapter IV. Decision Making for a Single Sample Chapter IV 1 4 1 Statistical Inference The field of statistical inference consists of those methods used to make decisions

More information

Math 152. Rumbos Fall Solutions to Exam #2

Math 152. Rumbos Fall Solutions to Exam #2 Math 152. Rumbos Fall 2009 1 Solutions to Exam #2 1. Define the following terms: (a) Significance level of a hypothesis test. Answer: The significance level, α, of a hypothesis test is the largest probability

More information

STAT 4385 Topic 01: Introduction & Review

STAT 4385 Topic 01: Introduction & Review STAT 4385 Topic 01: Introduction & Review Xiaogang Su, Ph.D. Department of Mathematical Science University of Texas at El Paso xsu@utep.edu Spring, 2016 Outline Welcome What is Regression Analysis? Basics

More information

CHAPTER 8. Test Procedures is a rule, based on sample data, for deciding whether to reject H 0 and contains:

CHAPTER 8. Test Procedures is a rule, based on sample data, for deciding whether to reject H 0 and contains: CHAPTER 8 Test of Hypotheses Based on a Single Sample Hypothesis testing is the method that decide which of two contradictory claims about the parameter is correct. Here the parameters of interest are

More information

Distribution-Free Procedures (Devore Chapter Fifteen)

Distribution-Free Procedures (Devore Chapter Fifteen) Distribution-Free Procedures (Devore Chapter Fifteen) MATH-5-01: Probability and Statistics II Spring 018 Contents 1 Nonparametric Hypothesis Tests 1 1.1 The Wilcoxon Rank Sum Test........... 1 1. Normal

More information

HYPOTHESIS TESTING. Hypothesis Testing

HYPOTHESIS TESTING. Hypothesis Testing MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.

More information

Inference and Regression

Inference and Regression Inference and Regression Assignment 4 - Solutions Department of IOMS Professor William Greene Phone: 212.998.0876 Office: KMC 7-90 Home page:www.stern.nyu.edu/~wgreene Email: wgreene@stern.nyu.edu Course

More information

Bias Variance Trade-off

Bias Variance Trade-off Bias Variance Trade-off The mean squared error of an estimator MSE(ˆθ) = E([ˆθ θ] 2 ) Can be re-expressed MSE(ˆθ) = Var(ˆθ) + (B(ˆθ) 2 ) MSE = VAR + BIAS 2 Proof MSE(ˆθ) = E((ˆθ θ) 2 ) = E(([ˆθ E(ˆθ)]

More information

Quantitative Methods for Economics, Finance and Management (A86050 F86050)

Quantitative Methods for Economics, Finance and Management (A86050 F86050) Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge

More information

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only

More information

AMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015

AMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015 AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking

More information

(a) (3 points) Construct a 95% confidence interval for β 2 in Equation 1.

(a) (3 points) Construct a 95% confidence interval for β 2 in Equation 1. Problem 1 (21 points) An economist runs the regression y i = β 0 + x 1i β 1 + x 2i β 2 + x 3i β 3 + ε i (1) The results are summarized in the following table: Equation 1. Variable Coefficient Std. Error

More information

Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009 there were participants

Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009 there were participants 18.650 Statistics for Applications Chapter 5: Parametric hypothesis testing 1/37 Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009

More information

M(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1

M(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1 Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)

More information

Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.

Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc. Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,

More information

Class 24. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 24. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700 Class 4 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 9. and 9.3 Lecture Chapter 10.1-10.3 Review Exam 6 Problem Solving

More information

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation

More information

Part 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2

Part 1: You are given the following system of two equations: x + 2y = 16 3x 4y = 2 Solving Systems of Equations Algebraically Teacher Notes Comment: As students solve equations throughout this task, have them continue to explain each step using properties of operations or properties

More information

Chapter 10: Inferences based on two samples

Chapter 10: Inferences based on two samples November 16 th, 2017 Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 1: Descriptive statistics Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter 8: Confidence

More information

Probability and Statistics Notes

Probability 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 information

Visual interpretation with normal approximation

Visual interpretation with normal approximation Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation

More information

VTU Edusat Programme 16

VTU Edusat Programme 16 VTU Edusat Programme 16 Subject : Engineering Mathematics Sub Code: 10MAT41 UNIT 8: Sampling Theory Dr. K.S.Basavarajappa Professor & Head Department of Mathematics Bapuji Institute of Engineering and

More information

Statistical inference

Statistical inference Statistical inference Contents 1. Main definitions 2. Estimation 3. Testing L. Trapani MSc Induction - Statistical inference 1 1 Introduction: definition and preliminary theory In this chapter, we shall

More information

Problem Set 4 - Solutions

Problem Set 4 - Solutions Problem Set 4 - Solutions Econ-310, Spring 004 8. a. If we wish to test the research hypothesis that the mean GHQ score for all unemployed men exceeds 10, we test: H 0 : µ 10 H a : µ > 10 This is a one-tailed

More information

Chapter 12: Inference about One Population

Chapter 12: Inference about One Population Chapter 1: Inference about One Population 1.1 Introduction In this chapter, we presented the statistical inference methods used when the problem objective is to describe a single population. Sections 1.

More information

Topic 15: Simple Hypotheses

Topic 15: Simple Hypotheses Topic 15: November 10, 2009 In the simplest set-up for a statistical hypothesis, we consider two values θ 0, θ 1 in the parameter space. We write the test as H 0 : θ = θ 0 versus H 1 : θ = θ 1. H 0 is

More information

Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.

Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6. Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized

More information

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1) Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ

More information

Statistics Part IV Confidence Limits and Hypothesis Testing. Joe Nahas University of Notre Dame

Statistics Part IV Confidence Limits and Hypothesis Testing. Joe Nahas University of Notre Dame Statistics Part IV Confidence Limits and Hypothesis Testing Joe Nahas University of Notre Dame Statistic Outline (cont.) 3. Graphical Display of Data A. Histogram B. Box Plot C. Normal Probability Plot

More information

STA Module 10 Comparing Two Proportions

STA Module 10 Comparing Two Proportions STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare

More information

7 Estimation. 7.1 Population and Sample (P.91-92)

7 Estimation. 7.1 Population and Sample (P.91-92) 7 Estimation MATH1015 Biostatistics Week 7 7.1 Population and Sample (P.91-92) Suppose that we wish to study a particular health problem in Australia, for example, the average serum cholesterol level for

More information

280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE Tests of Statistical Hypotheses

280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE Tests of Statistical Hypotheses 280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE 9-1.2 Tests of Statistical Hypotheses To illustrate the general concepts, consider the propellant burning rate problem introduced earlier. The null

More information

CBA4 is live in practice mode this week exam mode from Saturday!

CBA4 is live in practice mode this week exam mode from Saturday! Announcements CBA4 is live in practice mode this week exam mode from Saturday! Material covered: Confidence intervals (both cases) 1 sample hypothesis tests (both cases) Hypothesis tests for 2 means as

More information

Statistical Inference

Statistical Inference Statistical Inference Classical and Bayesian Methods Revision Class for Midterm Exam AMS-UCSC Th Feb 9, 2012 Winter 2012. Session 1 (Revision Class) AMS-132/206 Th Feb 9, 2012 1 / 23 Topics Topics We will

More information

Multiple Regression Analysis

Multiple Regression Analysis Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 +... β k x k + u 2. Inference 0 Assumptions of the Classical Linear Model (CLM)! So far, we know: 1. The mean and variance of the OLS estimators

More information

Single Sample Means. SOCY601 Alan Neustadtl

Single Sample Means. SOCY601 Alan Neustadtl Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size

More information

z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests

z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests z and t tests for the mean of a normal distribution Confidence intervals for the mean Binomial tests Chapters 3.5.1 3.5.2, 3.3.2 Prof. Tesler Math 283 Fall 2018 Prof. Tesler z and t tests for mean Math

More information

Advanced Experimental Design

Advanced Experimental Design Advanced Experimental Design Topic Four Hypothesis testing (z and t tests) & Power Agenda Hypothesis testing Sampling distributions/central limit theorem z test (σ known) One sample z & Confidence intervals

More information

(1) Introduction to Bayesian statistics

(1) Introduction to Bayesian statistics Spring, 2018 A motivating example Student 1 will write down a number and then flip a coin If the flip is heads, they will honestly tell student 2 if the number is even or odd If the flip is tails, they

More information

EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)

EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) 1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For

More information

Announcements. Unit 3: Foundations for inference Lecture 3: Decision errors, significance levels, sample size, and power.

Announcements. Unit 3: Foundations for inference Lecture 3: Decision errors, significance levels, sample size, and power. Announcements Announcements Unit 3: Foundations for inference Lecture 3:, significance levels, sample size, and power Statistics 101 Mine Çetinkaya-Rundel October 1, 2013 Project proposal due 5pm on Friday,

More information

STAT 135 Lab 7 Distributions derived from the normal distribution, and comparing independent samples.

STAT 135 Lab 7 Distributions derived from the normal distribution, and comparing independent samples. STAT 135 Lab 7 Distributions derived from the normal distribution, and comparing independent samples. Rebecca Barter March 16, 2015 The χ 2 distribution The χ 2 distribution We have seen several instances

More information

ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12

ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12 ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12 Winter 2012 Lecture 13 (Winter 2011) Estimation Lecture 13 1 / 33 Review of Main Concepts Sampling Distribution of Sample Mean

More information

Topic 19 Extensions on the Likelihood Ratio

Topic 19 Extensions on the Likelihood Ratio Topic 19 Extensions on the Likelihood Ratio Two-Sided Tests 1 / 12 Outline Overview Normal Observations Power Analysis 2 / 12 Overview The likelihood ratio test is a popular choice for composite hypothesis

More information

CIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8

CIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval

More information

Hypothesis Testing. 1 Definitions of test statistics. CB: chapter 8; section 10.3

Hypothesis Testing. 1 Definitions of test statistics. CB: chapter 8; section 10.3 Hypothesis Testing CB: chapter 8; section 0.3 Hypothesis: statement about an unknown population parameter Examples: The average age of males in Sweden is 7. (statement about population mean) The lowest

More information

Relating Graph to Matlab

Relating Graph to Matlab There are two related course documents on the web Probability and Statistics Review -should be read by people without statistics background and it is helpful as a review for those with prior statistics

More information

Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test

Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test 1 Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test Learning Objectives After this section, you should be able to DESCRIBE the relationship between the significance level of a test, P(Type

More information

Introduction to Statistics

Introduction to Statistics MTH4106 Introduction to Statistics Notes 15 Spring 2013 Testing hypotheses about the mean Earlier, we saw how to test hypotheses about a proportion, using properties of the Binomial distribution It is

More information

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008

2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Ch. 5 Hypothesis Testing

Ch. 5 Hypothesis Testing Ch. 5 Hypothesis Testing The current framework of hypothesis testing is largely due to the work of Neyman and Pearson in the late 1920s, early 30s, complementing Fisher s work on estimation. As in estimation,

More information

How do we compare the relative performance among competing models?

How do we compare the relative performance among competing models? How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model

More information

Chapter 27 Summary Inferences for Regression

Chapter 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 information

Part III: Unstructured Data

Part III: Unstructured Data Inf1-DA 2010 2011 III: 51 / 89 Part III Unstructured Data Data Retrieval: III.1 Unstructured data and data retrieval Statistical Analysis of Data: III.2 Data scales and summary statistics III.3 Hypothesis

More information

1 Probability theory. 2 Random variables and probability theory.

1 Probability theory. 2 Random variables and probability theory. Probability theory Here we summarize some of the probability theory we need. If this is totally unfamiliar to you, you should look at one of the sources given in the readings. In essence, for the major

More information

STAT 135 Lab 5 Bootstrapping and Hypothesis Testing

STAT 135 Lab 5 Bootstrapping and Hypothesis Testing STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,

More information

WISE Power Tutorial Answer Sheet

WISE Power Tutorial Answer Sheet ame Date Class WISE Power Tutorial Answer Sheet Power: The B.E.A.. Mnemonic Select true or false for each scenario: (Assuming no other changes) True False 1. As effect size increases, power decreases.

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness

More information

Statistics for IT Managers

Statistics for IT Managers Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample

More information

6.4 Type I and Type II Errors

6.4 Type I and Type II Errors 6.4 Type I and Type II Errors Ulrich Hoensch Friday, March 22, 2013 Null and Alternative Hypothesis Neyman-Pearson Approach to Statistical Inference: A statistical test (also known as a hypothesis test)

More information

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y

More information

Mathematical Statistics

Mathematical Statistics Mathematical Statistics MAS 713 Chapter 8 Previous lecture: 1 Bayesian Inference 2 Decision theory 3 Bayesian Vs. Frequentist 4 Loss functions 5 Conjugate priors Any questions? Mathematical Statistics

More information

Hypothesis tests for two means

Hypothesis tests for two means Chapter 3 Hypothesis tests for two means 3.1 Introduction Last week you were introduced to the concept of hypothesis testing in statistics, and we considered hypothesis tests for the mean if we have a

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT

More information

Probability theory and inference statistics! Dr. Paola Grosso! SNE research group!! (preferred!)!!

Probability theory and inference statistics! Dr. Paola Grosso! SNE research group!!  (preferred!)!! Probability theory and inference statistics Dr. Paola Grosso SNE research group p.grosso@uva.nl paola.grosso@os3.nl (preferred) Roadmap Lecture 1: Monday Sep. 22nd Collecting data Presenting data Descriptive

More information

Fundamental Probability and Statistics

Fundamental Probability and Statistics Fundamental Probability and Statistics "There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are

More information

Problem 1 (20) Log-normal. f(x) Cauchy

Problem 1 (20) Log-normal. f(x) Cauchy ORF 245. Rigollet Date: 11/21/2008 Problem 1 (20) f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 4 2 0 2 4 Normal (with mean -1) 4 2 0 2 4 Negative-exponential x x f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.5

More information

Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing

Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October

More information

Normal (Gaussian) distribution The normal distribution is often relevant because of the Central Limit Theorem (CLT):

Normal (Gaussian) distribution The normal distribution is often relevant because of the Central Limit Theorem (CLT): Lecture Three Normal theory null distributions Normal (Gaussian) distribution The normal distribution is often relevant because of the Central Limit Theorem (CLT): A random variable which is a sum of many

More information

14.30 Introduction to Statistical Methods in Economics Spring 2009

14.30 Introduction to Statistical Methods in Economics Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Hypothesis Testing hypothesis testing approach formulation of the test statistic

Hypothesis Testing hypothesis testing approach formulation of the test statistic Hypothesis Testing For the next few lectures, we re going to look at various test statistics that are formulated to allow us to test hypotheses in a variety of contexts: In all cases, the hypothesis testing

More information