Hypothesis Testing – Introduction

Slides:



Advertisements
Similar presentations
Introduction to Hypothesis Testing
Advertisements

Introduction to Hypothesis Testing
Hypothesis testing Another judgment method of sampling data.
Anthony Greene1 Simple Hypothesis Testing Detecting Statistical Differences In The Simplest Case:  and  are both known I The Logic of Hypothesis Testing:
Lecture XXIII.  In general there are two kinds of hypotheses: one concerns the form of the probability distribution (i.e. is the random variable normally.
Hypothesis Testing making decisions using sample data.
Decision Errors and Power
1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.
Likelihood ratio tests
Statistical Significance What is Statistical Significance? What is Statistical Significance? How Do We Know Whether a Result is Statistically Significant?
HYPOTHESIS TESTING Four Steps Statistical Significance Outcomes Sampling Distributions.
Statistical Significance What is Statistical Significance? How Do We Know Whether a Result is Statistically Significant? How Do We Know Whether a Result.
Hypothesis Testing: Type II Error and Power.
1 Statistical Inference Note: Only worry about pages 295 through 299 of Chapter 12.
Fall 2006 – Fundamentals of Business Statistics 1 Chapter 8 Introduction to Hypothesis Testing.
Chapter 3 Hypothesis Testing. Curriculum Object Specified the problem based the form of hypothesis Student can arrange for hypothesis step Analyze a problem.
PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 6 Chicago School of Professional Psychology.
Introduction to Testing a Hypothesis Testing a treatment Descriptive statistics cannot determine if differences are due to chance. A sampling error occurs.
The Neymann-Pearson Lemma Suppose that the data x 1, …, x n has joint density function f(x 1, …, x n ;  ) where  is either  1 or  2. Let g(x 1, …,
Business Statistics - QBM117 Introduction to hypothesis testing.
Hypothesis Testing – Introduction
1 © Lecture note 3 Hypothesis Testing MAKE HYPOTHESIS ©
Presented by Mohammad Adil Khan
Hypothesis Testing.
8 - 1 © 2003 Pearson Prentice Hall Chi-Square (  2 ) Test of Variance.
Sections 8-1 and 8-2 Review and Preview and Basics of Hypothesis Testing.
Chapter 8 Introduction to Hypothesis Testing
1 Virtual COMSATS Inferential Statistics Lecture-17 Ossam Chohan Assistant Professor CIIT Abbottabad.
1 1 Slide © 2005 Thomson/South-Western Chapter 9, Part B Hypothesis Tests Population Proportion Population Proportion Hypothesis Testing and Decision Making.
7 Elementary Statistics Hypothesis Testing. Introduction to Hypothesis Testing Section 7.1.
Overview Basics of Hypothesis Testing
1 Power and Sample Size in Testing One Mean. 2 Type I & Type II Error Type I Error: reject the null hypothesis when it is true. The probability of a Type.
1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.
STA Statistical Inference
AP STATISTICS LESSON 10 – 4 ( DAY 1 ) INFERENCE AS DECISION.
IE241: Introduction to Hypothesis Testing. We said before that estimation of parameters was one of the two major areas of statistics. Now let’s turn to.
Chapter 8 Introduction to Hypothesis Testing ©. Chapter 8 - Chapter Outcomes After studying the material in this chapter, you should be able to: 4 Formulate.
Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 8 th Edition Chapter 9 Hypothesis Testing: Single.
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-1 σ σ.
Introduction to Hypothesis Testing
Lecture 6 inferential statistics  Research hypotheses  Statistical hypotheses  Acceptable risks  ‘Real world model’  Decision rules  Experiment report.
Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example,
Introduction to Testing a Hypothesis Testing a treatment Descriptive statistics cannot determine if differences are due to chance. Sampling error means.
Inference as Design Target Goal: I can calculate and interpret a type I and type II error. 9.1c h.w: pg 547: 15, 19, 21.
Sampling Distributions Statistics Introduction Let’s assume that the IQ in the population has a mean (  ) of 100 and a standard deviation (  )
Power of a test. power The power of a test (against a specific alternative value) Is In practice, we carry out the test in hope of showing that the null.
Hypothesis Testing Steps for the Rejection Region Method State H 1 and State H 0 State the Test Statistic and its sampling distribution (normal or t) Determine.
Today: Hypothesis testing. Example: Am I Cheating? If each of you pick a card from the four, and I make a guess of the card that you picked. What proportion.
Chapter Ten McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. One-Sample Tests of Hypothesis.
Chapter 9: Hypothesis Tests for One Population Mean 9.2 Terms, Errors, and Hypotheses.
Example The strength of concrete depends, to some extent on the method used for drying it. Two different drying methods were tested independently on specimens.
Slide 20-1 Copyright © 2004 Pearson Education, Inc.
Lec. 19 – Hypothesis Testing: The Null and Types of Error.
Statistical Inference for the Mean Objectives: (Chapter 8&9, DeCoursey) -To understand the terms variance and standard error of a sample mean, Null Hypothesis,
Introduction to Hypothesis Test – Part 2
Hypothesis Testing I The One-sample Case
Review and Preview and Basics of Hypothesis Testing
Tests of Significance The reasoning of significance tests
Hypothesis Testing: Hypotheses
Hypothesis Testing Summer 2017 Summer Institutes.
CONCEPTS OF HYPOTHESIS TESTING
Introduction to Inference
Chapter 9 Hypothesis Testing.
P-value Approach for Test Conclusion
Hypothesis Testing – Introduction
Chapter 9: Hypothesis Testing
Introduction to Inference
AP Statistics: Chapter 21
Hypothesis Testing – Introduction
Inference as Decision Section 10.4.
Presentation transcript:

Hypothesis Testing – Introduction Hypothesis: A conjecture about the distribution of some random variables. A hypothesis can be simple or composite. A simple hypothesis completely specifies the distribution. A composite does not. There are two types of hypotheses: The null hypothesis, H0, is the current belief. The alternative hypothesis, Ha, is your belief; it is what you want to show. week 8

Testing Process Hypothesis testing is a proof by contradiction. The testing process has four steps: Step 1: Assume H0 is true. Step 2: Use statistical theory to make a statistic (function of the data) that includes H0. This statistic is called the test statistic. Step 3: Find the probability that the test statistic would take a value as extreme or more extreme than that actually observed. Think of this as: probability of getting our sample assuming H0 is true. Step 4: If the probability we calculated in step 3 is high it means that the sample is likely under H0 and so we have no evidence against H0. If the probability is low it means that the sample is unlikely under H0. This in turn means one of two things; either H0 is false or we are unlucky and H0 is true. week 8

Example week 8

Graphical Representation Let Sn be the set of all possible samples of size n from the population we are sampling from. Let C be the set of all samples for which we reject H0. It is called the critical region. is the set of all samples for which we fail to reject H0. It is called the acceptance region. week 8

Example week 8

Decision Errors When we perform a statistical test we hope that our decision will be correct, but sometimes it will be wrong. There are two possible errors that can be made in hypothesis test. The error made by rejecting the null hypothesis H0 when in fact H0 is true is called a type I error. The error made by failing to reject the null hypothesis H0 when in fact H0 is false is called a type II error. week 8

Size of a Test The probability that defines the critical region is called the size of the test and is denoted by α. The size of the test is also the probability of type I error. Example... week 8

Power The probability that a fixed size  test will reject H0 when H0 is false is called the power of the test. Power is not about an error. We want high power. Example… week 8

Decision Rules A hypothesis test is a decision made where we attach a probability of type I error and fix it to be α. However, for any set up there are lots of decision rules with the same size. Example: week 8

Neyman Pearson Lemma - Introduction We start by picking an α. For any α there is infinite number of possible decision rules (infinite number of critical regions). Each critical region has a power. Neyman Pearson Lemma tells us how to find the critical region (i.e test) that has the highest power. week 8

Neyman Pearson Lemma If C is a critical region of size α and k is a constant such that inside C (i.e. reject H0) outside C (i.e. fail to reject H0) Then C is the most powerful test of H0: θ = θ0 versus Ha:θ = θ1. week 8

Translation of Lemma L0 is the probability of the sample under H0. L1 is the probability of the sample under Ha. If then θ0 is more likely, i.e., H0 is more likely true. If then θ1 is more likely, i.e., Ha is more likely true. But we need to ensure P(inside C | H0) = α. So we find k and C all at once by solving week 8

Examples week 8

Proof of Neyman Pearson Lemma week 8