HYPOTHESIS TESTING: FREQUENTIST APPROACH.

Size: px
Start display at page:

Download "HYPOTHESIS TESTING: FREQUENTIST APPROACH."

Transcription

1 HYPOTHESIS TESTING: FREQUENTIST APPROACH. These notes summarize the lectures on (the frequentist approach to) hypothesis testing. You should be familiar with the standard hypothesis testing from previous stats classes. Here, we will explain where this approach comes from and develop new ideas (all within the context of the parametric set-up). 1. Set-Up The basic setup of (the Neyman-Pearson approach to) hypothesis testing is as follows. There are two hypotheses you are trying to decide between: the null (H 0 ) and the alternative (H A ). If a hypothesis fully determines the behaviour (pdf/pmf or other) of the random variables then it is called simple, otherwise it is known as composite. The hypothesis test will reject H 0 in favour of H A if a test statistic T = T (X) falls into a rejection region (RR). We therefore have that: accept H 0 reject H 0 H 0 true Type I error H A true Type II error The probability of a type I error is denoted as α and is also known as the significance level of a test. The probability of a type II error is denoted as β. The power of a test is 1 β; the probability of doing the correct thing under H A. Note that if H A is composite then both β and power depend on the particular member of H A which holds. In this case we will often plot the power function. Ideally we would have that α = β = 0. However, in practice we most often have that decreasing α drives up β and vice versa. 2. Neyman-Pearson The main idea behind Neyman-Pearson is to fix α in advance (choose α to be small) and then to find a test which yields a small value of β. The Neyman-Pearson lemma tells us that the in such a set-up, the likelihood ratio test (LRT) is the most powerful of all the possible tests. This only works for two simple hypotheses. Date: November 25,

2 2 HYPOTHESIS TESTING: FREQUENTIST APPROACH. Thus, assume that H 0 and H A are both simple, and let f 0 (x) denote the pdf/pmf (likelihood) of the data under H 0 (and f A (x) under H A ). The LRT is the test which rejects if f 0 (x) f A (x) < c, where c is chosen in such a way so that P (reject) = α. Lemma 2.1 (Neyman-Pearson Lemma). Any other test with significance level α α has power less than or equal to that of the likelihood ratio test. First of all note that this is a very sensible thing to do (we reject H 0 if the data has a bigger likelihood under H A ). Thus, the basic idea is similar to that of maximum likelihood estimation. We next need to take the LRT test and translate it into something easier to handle. Example. ESP example (Bernoulli, sample size is 10). We have that P (T otal 6 H 0 ) = 0.02, and that P (T otal 5 H 0 ) = (for sample size 10), we therefore cannot choose α = 0.05 exactly. We will choose the rejection region to be {6, 7, 8, 9, 10}. In this case, the power function is given in Figure 1. The code used to generate Figure 1 in R was: x<-rep(0,250) for(i in 1:250) { x[i]<-1-pbinom(5,10,0.25+i/250*0.75) } plot(x) What happens if we do n independent tests at the same time? Example. Population = exponential. Example. Population = normal, variance known. 3. P-values Performing an α-level test is not very informative as to the amount of information for/against the alternative hypothesis. The quantity that does allow us to measure this is the p-value. The p-value is defined as the smallest value of α for which the null hypothesis will be rejected. Typically it is calculated as the probability of obtaining a test statistic as or more extreme than what was actually observed. Extreme is dictated by the form of the rejection region. For a specific example, in the ESP Bernoulli case, if

3 HYPOTHESIS TESTING: FREQUENTIST APPROACH. 3 x Index Figure 1. Power function in ESP example. we observed 5 total successes, then since we reject for T = total large, the p-value is calculated as P (T 5 H 0 ). Imho, you should always report the p-value for a hypothesis test in your research. Example. Under the null hypothesis show that the distribution of the p-value is Uniform[0, 1].

4 4 HYPOTHESIS TESTING: FREQUENTIST APPROACH. 4. Generalized Likelihood Ratio Test (GLRT) The LRT is optimal for testing a simple hypothesis against a simple hypothesis. However, often we wish to compare simple vs. composite or two composite hypotheses. As the name implies, the generalized LRT is a generalization of the LRT which allows us to handle composite hypotheses. Although no optimality results exist for the generalized version, we do have some nice asymptotic results, and it is easily motivated as a natural extension of the LRT. The set up for the GLRT is as follows. Let f(x θ) denote the pdf/pmf of the data if the parameter θ (possibly multivariate) is known. Notice that f(x θ) is actually the likelihood. The null hypothesis specifies that θ Θ 0 and the alternative says that θ Θ A. We let Θ denote Θ 0 Θ A. The GLRT rejects the null if Λ = max θ Θ 0 f(x θ) max θ ΘA f(x θ) is small. Indeed, this is very reasonable. In practice however it is often easier to work with Λ = max θ Θ 0 f(x θ) max θ Θ f(x θ) and reject H 0 if this is small. Since Λ = min(λ, 1) both versions actually do the same thing. We take the latter as our official definition of the GLRT. Example. Two-sided normal, unknown variance. Theorem 4.1. Under smoothness assumptions on the underlying pdf/pmf, the null distribution of 2 log Λ converges to a χ 2 distribution with degrees of freedom equal to dim Θ dim Θ 0 as the sample size tends to infinity. Since we reject for small values of Λ, we would reject for large values of 2 log Λ. Example. Compare this to the two-sided normal, unknown variance. 5. Power In the previous sections we have really avoided the issue of power. The LRT chooses the test with the highest power for a fixed significance level, but what if this isn t good enough? In practice it often is the case that increasing sample size increases power. The following examples are designed to illustrate this. Example. Normal, variance known. Example. Normal, variance unknown. (To calculate power, approximate using normal!)

5 HYPOTHESIS TESTING: FREQUENTIST APPROACH Duality of Confidence Intervals and Hypothesis Tests. A confidence interval (or set, in general) can be obtained by inverting a hypothesis test and vice versa. Example. Normal with known variance. Theorem 6.1. Suppose that for every value θ 0 in Θ there is a test at level α of the hypothesis H 0 : θ = θ 0. Denote the acceptance region of the test as A(θ 0 ). Then the set C(X) = {θ : X A(θ)} is a 100(1 α)% confidence region for θ. In words, a 100(1 α)% confidence region for θ consists of all those values of θ 0 for which the hypothesis that θ = θ 0 will not be rejected at level α. Theorem 6.2. Suppose that C(X) is a 100(1 α)% confidence region for θ: that is, for every θ 0 P (θ 0 C(X) θ = θ 0 ) = 1 α. Then an acceptance region for a test at level α of the hypothesis H 0 : θ = θ 0 is A(θ 0 ) = {X θ 0 C(X)}. In words, this says that the hypothesis that θ = θ 0 is accepted if θ 0 lies in the confidence region. This duality works exactly for the t-test and z-tests and associated confidence intervals. For other tests that are typically used (eg. testing a proportion or LRT for the Poisson, say) the typical tests do not invert exactly to the confidence interval and vice versa. This is not because duality fails in these cases, but because the test used is not an exact inversion of the confidence set. Example. Suppose a t-test rejects the two-sided hypothesis test for µ = 0 at the 5% level. Would the 90% CI contain zero? References [R] Rice, J.; Mathematical Statistics and Data Analysis Duxbury Press, 2nd Edition, Prepared by Hanna Jankowski Department of Statistics, University of Washington Box , Seattle, WA U.S.A.

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

PARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.

PARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation. PARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.. Beta Distribution We ll start by learning about the Beta distribution, since we end up using

More information

Lecture 21. Hypothesis Testing II

Lecture 21. Hypothesis Testing II Lecture 21. Hypothesis Testing II December 7, 2011 In the previous lecture, we dened a few key concepts of hypothesis testing and introduced the framework for parametric hypothesis testing. In the parametric

More information

Economics 520. Lecture Note 19: Hypothesis Testing via the Neyman-Pearson Lemma CB 8.1,

Economics 520. Lecture Note 19: Hypothesis Testing via the Neyman-Pearson Lemma CB 8.1, Economics 520 Lecture Note 9: Hypothesis Testing via the Neyman-Pearson Lemma CB 8., 8.3.-8.3.3 Uniformly Most Powerful Tests and the Neyman-Pearson Lemma Let s return to the hypothesis testing problem

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

Statistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation

Statistics - Lecture One. Outline. Charlotte Wickham  1. Basic ideas about estimation Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence

More information

Hypothesis Testing. BS2 Statistical Inference, Lecture 11 Michaelmas Term Steffen Lauritzen, University of Oxford; November 15, 2004

Hypothesis Testing. BS2 Statistical Inference, Lecture 11 Michaelmas Term Steffen Lauritzen, University of Oxford; November 15, 2004 Hypothesis Testing BS2 Statistical Inference, Lecture 11 Michaelmas Term 2004 Steffen Lauritzen, University of Oxford; November 15, 2004 Hypothesis testing We consider a family of densities F = {f(x; θ),

More information

Institute of Actuaries of India

Institute of Actuaries of India Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the

More information

The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80

The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80 The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80 71. Decide in each case whether the hypothesis is simple

More information

LECTURE 10: NEYMAN-PEARSON LEMMA AND ASYMPTOTIC TESTING. The last equality is provided so this can look like a more familiar parametric test.

LECTURE 10: NEYMAN-PEARSON LEMMA AND ASYMPTOTIC TESTING. The last equality is provided so this can look like a more familiar parametric test. Economics 52 Econometrics Professor N.M. Kiefer LECTURE 1: NEYMAN-PEARSON LEMMA AND ASYMPTOTIC TESTING NEYMAN-PEARSON LEMMA: Lesson: Good tests are based on the likelihood ratio. The proof is easy in the

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

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

Hypothesis Testing Chap 10p460

Hypothesis Testing Chap 10p460 Hypothesis Testing Chap 1p46 Elements of a statistical test p462 - Null hypothesis - Alternative hypothesis - Test Statistic - Rejection region Rejection Region p462 The rejection region (RR) specifies

More information

Summary of Chapters 7-9

Summary of Chapters 7-9 Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two

More information

Recall that in order to prove Theorem 8.8, we argued that under certain regularity conditions, the following facts are true under H 0 : 1 n

Recall that in order to prove Theorem 8.8, we argued that under certain regularity conditions, the following facts are true under H 0 : 1 n Chapter 9 Hypothesis Testing 9.1 Wald, Rao, and Likelihood Ratio Tests Suppose we wish to test H 0 : θ = θ 0 against H 1 : θ θ 0. The likelihood-based results of Chapter 8 give rise to several possible

More information

Math 494: Mathematical Statistics

Math 494: Mathematical Statistics Math 494: Mathematical Statistics Instructor: Jimin Ding jmding@wustl.edu Department of Mathematics Washington University in St. Louis Class materials are available on course website (www.math.wustl.edu/

More information

STATS 200: Introduction to Statistical Inference. Lecture 29: Course review

STATS 200: Introduction to Statistical Inference. Lecture 29: Course review STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout

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

Testing Hypothesis. Maura Mezzetti. Department of Economics and Finance Università Tor Vergata

Testing Hypothesis. Maura Mezzetti. Department of Economics and Finance Università Tor Vergata Maura Department of Economics and Finance Università Tor Vergata Hypothesis Testing Outline It is a mistake to confound strangeness with mystery Sherlock Holmes A Study in Scarlet Outline 1 The Power Function

More information

A Very Brief Summary of Statistical Inference, and Examples

A Very Brief Summary of Statistical Inference, and Examples A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2009 Prof. Gesine Reinert Our standard situation is that we have data x = x 1, x 2,..., x n, which we view as realisations of random

More information

Review. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda

Review. DS GA 1002 Statistical and Mathematical Models.   Carlos Fernandez-Granda Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Probability and statistics Probability: Framework for dealing with

More information

Lecture Testing Hypotheses: The Neyman-Pearson Paradigm

Lecture Testing Hypotheses: The Neyman-Pearson Paradigm Math 408 - Mathematical Statistics Lecture 29-30. Testing Hypotheses: The Neyman-Pearson Paradigm April 12-15, 2013 Konstantin Zuev (USC) Math 408, Lecture 29-30 April 12-15, 2013 1 / 12 Agenda Example:

More information

4.5.1 The use of 2 log Λ when θ is scalar

4.5.1 The use of 2 log Λ when θ is scalar 4.5. ASYMPTOTIC FORM OF THE G.L.R.T. 97 4.5.1 The use of 2 log Λ when θ is scalar Suppose we wish to test the hypothesis NH : θ = θ where θ is a given value against the alternative AH : θ θ on the basis

More information

Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses 9.2 Testing Simple Hypotheses 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.6 Comparing the Means of Two

More information

Definition 3.1 A statistical hypothesis is a statement about the unknown values of the parameters of the population distribution.

Definition 3.1 A statistical hypothesis is a statement about the unknown values of the parameters of the population distribution. Hypothesis Testing Definition 3.1 A statistical hypothesis is a statement about the unknown values of the parameters of the population distribution. Suppose the family of population distributions is indexed

More information

Chapters 10. Hypothesis Testing

Chapters 10. Hypothesis Testing Chapters 10. Hypothesis Testing Some examples of hypothesis testing 1. Toss a coin 100 times and get 62 heads. Is this coin a fair coin? 2. Is the new treatment on blood pressure more effective than the

More information

Topic 10: Hypothesis Testing

Topic 10: Hypothesis Testing Topic 10: Hypothesis Testing Course 003, 2016 Page 0 The Problem of Hypothesis Testing A statistical hypothesis is an assertion or conjecture about the probability distribution of one or more random variables.

More information

Lecture 10: Generalized likelihood ratio test

Lecture 10: Generalized likelihood ratio test Stat 200: Introduction to Statistical Inference Autumn 2018/19 Lecture 10: Generalized likelihood ratio test Lecturer: Art B. Owen October 25 Disclaimer: These notes have not been subjected to the usual

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

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

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: The Generalized Likelihood Ratio Test

Hypothesis Testing: The Generalized Likelihood Ratio Test Hypothesis Testing: The Generalized Likelihood Ratio Test Consider testing the hypotheses H 0 : θ Θ 0 H 1 : θ Θ \ Θ 0 Definition: The Generalized Likelihood Ratio (GLR Let L(θ be a likelihood for a random

More information

One sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:

One sided tests. An example of a two sided alternative is what we ve been using for our two sample tests: One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get

More information

Lecture 21: October 19

Lecture 21: October 19 36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 21: October 19 21.1 Likelihood Ratio Test (LRT) To test composite versus composite hypotheses the general method is to use

More information

STAT 830 Hypothesis Testing

STAT 830 Hypothesis Testing STAT 830 Hypothesis Testing Richard Lockhart Simon Fraser University STAT 830 Fall 2018 Richard Lockhart (Simon Fraser University) STAT 830 Hypothesis Testing STAT 830 Fall 2018 1 / 30 Purposes of These

More information

Lecture 12 November 3

Lecture 12 November 3 STATS 300A: Theory of Statistics Fall 2015 Lecture 12 November 3 Lecturer: Lester Mackey Scribe: Jae Hyuck Park, Christian Fong Warning: These notes may contain factual and/or typographic errors. 12.1

More information

Topic 10: Hypothesis Testing

Topic 10: Hypothesis Testing Topic 10: Hypothesis Testing Course 003, 2017 Page 0 The Problem of Hypothesis Testing A statistical hypothesis is an assertion or conjecture about the probability distribution of one or more random variables.

More information

Robustness and Distribution Assumptions

Robustness and Distribution Assumptions Chapter 1 Robustness and Distribution Assumptions 1.1 Introduction In statistics, one often works with model assumptions, i.e., one assumes that data follow a certain model. Then one makes use of methodology

More information

Testing Statistical Hypotheses

Testing Statistical Hypotheses E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions

More information

STAT 830 Hypothesis Testing

STAT 830 Hypothesis Testing STAT 830 Hypothesis Testing Hypothesis testing is a statistical problem where you must choose, on the basis of data X, between two alternatives. We formalize this as the problem of choosing between two

More information

Two hours. To be supplied by the Examinations Office: Mathematical Formula Tables THE UNIVERSITY OF MANCHESTER. 21 June :45 11:45

Two hours. To be supplied by the Examinations Office: Mathematical Formula Tables THE UNIVERSITY OF MANCHESTER. 21 June :45 11:45 Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS 21 June 2010 9:45 11:45 Answer any FOUR of the questions. University-approved

More information

simple if it completely specifies the density of x

simple if it completely specifies the density of x 3. Hypothesis Testing Pure significance tests Data x = (x 1,..., x n ) from f(x, θ) Hypothesis H 0 : restricts f(x, θ) Are the data consistent with H 0? H 0 is called the null hypothesis simple if it completely

More information

8: Hypothesis Testing

8: Hypothesis Testing Some definitions 8: Hypothesis Testing. Simple, compound, null and alternative hypotheses In test theory one distinguishes between simple hypotheses and compound hypotheses. A simple hypothesis Examples:

More information

Hypothesis Test. The opposite of the null hypothesis, called an alternative hypothesis, becomes

Hypothesis Test. The opposite of the null hypothesis, called an alternative hypothesis, becomes Neyman-Pearson paradigm. Suppose that a researcher is interested in whether the new drug works. The process of determining whether the outcome of the experiment points to yes or no is called hypothesis

More information

Partitioning the Parameter Space. Topic 18 Composite Hypotheses

Partitioning the Parameter Space. Topic 18 Composite Hypotheses Topic 18 Composite Hypotheses Partitioning the Parameter Space 1 / 10 Outline Partitioning the Parameter Space 2 / 10 Partitioning the Parameter Space Simple hypotheses limit us to a decision between one

More information

F79SM STATISTICAL METHODS

F79SM STATISTICAL METHODS F79SM STATISTICAL METHODS SUMMARY NOTES 9 Hypothesis testing 9.1 Introduction As before we have a random sample x of size n of a population r.v. X with pdf/pf f(x;θ). The distribution we assign to X is

More information

7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between

7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between 7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation

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

Hypothesis Testing - Frequentist

Hypothesis Testing - Frequentist Frequentist Hypothesis Testing - Frequentist Compare two hypotheses to see which one better explains the data. Or, alternatively, what is the best way to separate events into two classes, those originating

More information

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

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

Hypothesis testing (cont d)

Hypothesis testing (cont d) Hypothesis testing (cont d) Ulrich Heintz Brown University 4/12/2016 Ulrich Heintz - PHYS 1560 Lecture 11 1 Hypothesis testing Is our hypothesis about the fundamental physics correct? We will not be able

More information

http://www.math.uah.edu/stat/hypothesis/.xhtml 1 of 5 7/29/2009 3:14 PM Virtual Laboratories > 9. Hy pothesis Testing > 1 2 3 4 5 6 7 1. The Basic Statistical Model As usual, our starting point is a random

More information

BTRY 4090: Spring 2009 Theory of Statistics

BTRY 4090: Spring 2009 Theory of Statistics BTRY 4090: Spring 2009 Theory of Statistics Guozhang Wang September 25, 2010 1 Review of Probability We begin with a real example of using probability to solve computationally intensive (or infeasible)

More information

Topic 3: Hypothesis Testing

Topic 3: Hypothesis Testing CS 8850: Advanced Machine Learning Fall 07 Topic 3: Hypothesis Testing Instructor: Daniel L. Pimentel-Alarcón c Copyright 07 3. Introduction One of the simplest inference problems is that of deciding between

More information

FYST17 Lecture 8 Statistics and hypothesis testing. Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons

FYST17 Lecture 8 Statistics and hypothesis testing. Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons FYST17 Lecture 8 Statistics and hypothesis testing Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons 1 Plan for today: Introduction to concepts The Gaussian distribution Likelihood functions Hypothesis

More information

STA 732: Inference. Notes 2. Neyman-Pearsonian Classical Hypothesis Testing B&D 4

STA 732: Inference. Notes 2. Neyman-Pearsonian Classical Hypothesis Testing B&D 4 STA 73: Inference Notes. Neyman-Pearsonian Classical Hypothesis Testing B&D 4 1 Testing as a rule Fisher s quantification of extremeness of observed evidence clearly lacked rigorous mathematical interpretation.

More information

Testing and Model Selection

Testing and Model Selection Testing and Model Selection This is another digression on general statistics: see PE App C.8.4. The EViews output for least squares, probit and logit includes some statistics relevant to testing hypotheses

More information

Testing Statistical Hypotheses

Testing Statistical Hypotheses E.L. Lehmann Joseph P. Romano, 02LEu1 ttd ~Lt~S Testing Statistical Hypotheses Third Edition With 6 Illustrations ~Springer 2 The Probability Background 28 2.1 Probability and Measure 28 2.2 Integration.........

More information

ST495: Survival Analysis: Hypothesis testing and confidence intervals

ST495: Survival Analysis: Hypothesis testing and confidence intervals ST495: Survival Analysis: Hypothesis testing and confidence intervals Eric B. Laber Department of Statistics, North Carolina State University April 3, 2014 I remember that one fateful day when Coach took

More information

STAT 801: Mathematical Statistics. Hypothesis Testing

STAT 801: Mathematical Statistics. Hypothesis Testing STAT 801: Mathematical Statistics Hypothesis Testing Hypothesis testing: a statistical problem where you must choose, on the basis o data X, between two alternatives. We ormalize this as the problem o

More information

Political Science 236 Hypothesis Testing: Review and Bootstrapping

Political Science 236 Hypothesis Testing: Review and Bootstrapping Political Science 236 Hypothesis Testing: Review and Bootstrapping Rocío Titiunik Fall 2007 1 Hypothesis Testing Definition 1.1 Hypothesis. A hypothesis is a statement about a population parameter The

More information

Hypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =

Hypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n = Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,

More information

Stat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS

Stat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS Stat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS 1a. Under the null hypothesis X has the binomial (100,.5) distribution with E(X) = 50 and SE(X) = 5. So P ( X 50 > 10) is (approximately) two tails

More information

Lecture 8: Information Theory and Statistics

Lecture 8: Information Theory and Statistics Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 23, 2015 1 / 50 I-Hsiang

More information

Statistics for Particle Physics. Kyle Cranmer. New York University. Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009

Statistics for Particle Physics. Kyle Cranmer. New York University. Kyle Cranmer (NYU) CERN Academic Training, Feb 2-5, 2009 Statistics for Particle Physics Kyle Cranmer New York University 91 Remaining Lectures Lecture 3:! Compound hypotheses, nuisance parameters, & similar tests! The Neyman-Construction (illustrated)! Inverted

More information

Lecture 28: Asymptotic confidence sets

Lecture 28: Asymptotic confidence sets Lecture 28: Asymptotic confidence sets 1 α asymptotic confidence sets Similar to testing hypotheses, in many situations it is difficult to find a confidence set with a given confidence coefficient or level

More information

STA 2101/442 Assignment 3 1

STA 2101/442 Assignment 3 1 STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance

More information

Composite Hypotheses and Generalized Likelihood Ratio Tests

Composite Hypotheses and Generalized Likelihood Ratio Tests Composite Hypotheses and Generalized Likelihood Ratio Tests Rebecca Willett, 06 In many real world problems, it is difficult to precisely specify probability distributions. Our models for data may involve

More information

TUTORIAL 8 SOLUTIONS #

TUTORIAL 8 SOLUTIONS # TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level

More information

4 Hypothesis testing. 4.1 Types of hypothesis and types of error 4 HYPOTHESIS TESTING 49

4 Hypothesis testing. 4.1 Types of hypothesis and types of error 4 HYPOTHESIS TESTING 49 4 HYPOTHESIS TESTING 49 4 Hypothesis testing In sections 2 and 3 we considered the problem of estimating a single parameter of interest, θ. In this section we consider the related problem of testing whether

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

Introduction 1. STA442/2101 Fall See last slide for copyright information. 1 / 33

Introduction 1. STA442/2101 Fall See last slide for copyright information. 1 / 33 Introduction 1 STA442/2101 Fall 2016 1 See last slide for copyright information. 1 / 33 Background Reading Optional Chapter 1 of Linear models with R Chapter 1 of Davison s Statistical models: Data, and

More information

H 2 : otherwise. that is simply the proportion of the sample points below level x. For any fixed point x the law of large numbers gives that

H 2 : otherwise. that is simply the proportion of the sample points below level x. For any fixed point x the law of large numbers gives that Lecture 28 28.1 Kolmogorov-Smirnov test. Suppose that we have an i.i.d. sample X 1,..., X n with some unknown distribution and we would like to test the hypothesis that is equal to a particular distribution

More information

LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2

LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept

More information

2. What are the tradeoffs among different measures of error (e.g. probability of false alarm, probability of miss, etc.)?

2. What are the tradeoffs among different measures of error (e.g. probability of false alarm, probability of miss, etc.)? ECE 830 / CS 76 Spring 06 Instructors: R. Willett & R. Nowak Lecture 3: Likelihood ratio tests, Neyman-Pearson detectors, ROC curves, and sufficient statistics Executive summary In the last lecture we

More information

Master s Written Examination

Master s Written Examination Master s Written Examination Option: Statistics and Probability Spring 016 Full points may be obtained for correct answers to eight questions. Each numbered question which may have several parts is worth

More information

Direction: This test is worth 250 points and each problem worth points. DO ANY SIX

Direction: This test is worth 250 points and each problem worth points. DO ANY SIX Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes

More information

Lecture 8: Information Theory and Statistics

Lecture 8: Information Theory and Statistics Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and Estimation I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 22, 2015

More information

Interval Estimation. Chapter 9

Interval Estimation. Chapter 9 Chapter 9 Interval Estimation 9.1 Introduction Definition 9.1.1 An interval estimate of a real-values parameter θ is any pair of functions, L(x 1,..., x n ) and U(x 1,..., x n ), of a sample that satisfy

More information

Basic Concepts of Inference

Basic Concepts of Inference Basic Concepts of Inference Corresponds to Chapter 6 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT) with some slides by Jacqueline Telford (Johns Hopkins University) and Roy Welsch (MIT).

More information

Homework 7: Solutions. P3.1 from Lehmann, Romano, Testing Statistical Hypotheses.

Homework 7: Solutions. P3.1 from Lehmann, Romano, Testing Statistical Hypotheses. Stat 300A Theory of Statistics Homework 7: Solutions Nikos Ignatiadis Due on November 28, 208 Solutions should be complete and concisely written. Please, use a separate sheet or set of sheets for each

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

Using R in Undergraduate and Graduate Probability and Mathematical Statistics Courses*

Using R in Undergraduate and Graduate Probability and Mathematical Statistics Courses* Using R in Undergraduate and Graduate Probability and Mathematical Statistics Courses* Amy G. Froelich Michael D. Larsen Iowa State University *The work presented in this talk was partially supported by

More information

Topic 17: Simple Hypotheses

Topic 17: Simple Hypotheses Topic 17: November, 2011 1 Overview and Terminology Statistical hypothesis testing is designed to address the question: Do the data provide sufficient evidence to conclude that we must depart from our

More information

Review. December 4 th, Review

Review. December 4 th, Review December 4 th, 2017 Att. Final exam: Course evaluation Friday, 12/14/2018, 10:30am 12:30pm Gore Hall 115 Overview Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 6: Statistics and Sampling Distributions Chapter

More information

Econometrics. 4) Statistical inference

Econometrics. 4) Statistical inference 30C00200 Econometrics 4) Statistical inference Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Confidence intervals of parameter estimates Student s t-distribution

More information

STAT Chapter 8: Hypothesis Tests

STAT Chapter 8: Hypothesis Tests STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular

More information

Tests and Their Power

Tests and Their Power Tests and Their Power Ling Kiong Doong Department of Mathematics National University of Singapore 1. Introduction In Statistical Inference, the two main areas of study are estimation and testing of hypotheses.

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

Psychology 282 Lecture #4 Outline Inferences in SLR

Psychology 282 Lecture #4 Outline Inferences in SLR Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations

More information

Stat 5101 Lecture Notes

Stat 5101 Lecture Notes Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random

More information

Parameter Estimation and Fitting to Data

Parameter Estimation and Fitting to Data Parameter Estimation and Fitting to Data Parameter estimation Maximum likelihood Least squares Goodness-of-fit Examples Elton S. Smith, Jefferson Lab 1 Parameter estimation Properties of estimators 3 An

More information

Lecture 4: Testing Stuff

Lecture 4: Testing Stuff Lecture 4: esting Stuff. esting Hypotheses usually has three steps a. First specify a Null Hypothesis, usually denoted, which describes a model of H 0 interest. Usually, we express H 0 as a restricted

More information

Central Limit Theorem ( 5.3)

Central Limit Theorem ( 5.3) Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately

More information

Econ 325: Introduction to Empirical Economics

Econ 325: Introduction to Empirical Economics Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population

More information

Statistical Theory MT 2007 Problems 4: Solution sketches

Statistical Theory MT 2007 Problems 4: Solution sketches Statistical Theory MT 007 Problems 4: Solution sketches 1. Consider a 1-parameter exponential family model with density f(x θ) = f(x)g(θ)exp{cφ(θ)h(x)}, x X. Suppose that the prior distribution has the

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

One-sample categorical data: approximate inference

One-sample categorical data: approximate inference One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution

More information

LECTURE 5 HYPOTHESIS TESTING

LECTURE 5 HYPOTHESIS TESTING October 25, 2016 LECTURE 5 HYPOTHESIS TESTING Basic concepts In this lecture we continue to discuss the normal classical linear regression defined by Assumptions A1-A5. Let θ Θ R d be a parameter of interest.

More information