Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities

Save this PDF as:

Size: px
Start display at page:

Download "Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities"

Transcription

1 Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify me if any of them are incorrect. Quiz #7 begins Wed after class and ends Friday. Quiz #8 begins Wed before Thanksgiving and ends on Monday after Thanksgiving. That is the last quiz. Jason Kramer will give the lecture this Friday. REVIEW OF HYPOTHESIS TEST FOR ONE PROPORTION ONE-SIDED TEST, WITH PICTURE H : p = p versus Ha: p > p Reject H if p-value <.5. For what values of z does that happen? Notation: level of significance = α (alpha, usually.5) p-value <.5 corresponds with z >.645 Normal, Mean=, StDev= HOMEWORK (due Fri, Nov 9): Chapter 2: #62, 83, AN ILLUSTRATION OF WHAT HAPPENS WHEN Ha IS TRUE A Specific Example Suppose the truth for the population proportion p is one standard deviation above the null value p. Then the mean for the standardized scores will be instead of. How often would we (correctly) reject the null hypothesis in that case? Answer (purple region) is Picture when the true population proportion is s.d. above null value Normal, Mean=, StDev= Section 2.2, Lesson 3 What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities Example: Case Study.6, aspirin and heart attacks. Found statistically significant relationship; p-value was <.. Possible errors: Type error (false positive) occurs when: Null hypothesis is actually true, but Conclusion of test is to Reject H and accept H a Type 2 error (false negative) occurs when: Alternative hypothesis is actually true, but Conclusion is that we cannot reject H

2 Heart attack and aspirin example: Null hypothesis: The proportion of men who would have heart attacks if taking aspirin = the proportion of men who would have heart attacks if taking placebo. Alternative hypothesis: The heart attack proportion is lower if men were to take aspirin than if they were not to take aspirin. Type error (false positive): Occurs if there really is no relationship between taking aspirin and heart attack prevention, but we conclude that there is a relationship. Consequence: Good for aspirin companies! Possible bad side effects from aspirin, with no redeeming value. Type 2 error (false negative): Occurs if there really is a relationship but study failed to find it. Consequence: Miss out on recommending something that could save lives! Which type of error is more serious? Probably all agree that Type 2 is more serious. Which could have occurred? Type error could have occurred. Type 2 could not have occurred, because we did find a significant relationship. Aspirin Example: Consequences of the decisions Decision: Reject H Don t conclude, Truth: Conclude aspirin aspirin works works H : Aspirin doesn t work Ha: Aspirin works Which error can occur? OK Type 2 error: Aspirin could save lives but we don t recognize benefits Type 2 can only occur if H is not Type error: People take aspirin needlessly; may suffer side effects OK Type can only occur if H is rejected. Which error can occur: Type error can only occur if aspirin doesn t work. Type 2 error can only occur if aspirin does work. rejected. Note that because H was rejected in this study, we could only have made a Type error, not a Type 2 error. Some analogies to hypothesis testing: Analogy : Courtroom: Null hypothesis: Defendant is innocent. Alternative hypothesis: Defendant is guilty Note that the two possible conclusions are not guilty and guilty. The conclusion not guilty is equivalent to don t reject H. We don t say defendant is innocent just like we don t accept H in hypothesis testing. Type error is when defendant is innocent but gets convicted Type 2 error is when defendant is guilty but does not get convicted. Which one is more serious??

3 Analogy 2: Medical test Null hypothesis: You do not have the disease Alternative hypothesis: You have the disease Type error: You don't have disease, but test says you do; a "false positive" Type 2 error: You do have disease, but test says you do not; a "false negative" Notes and Definitions: Probability related to Type error: The conditional probability of making a Type error, given that H is true, is the level of significance α. In most cases, this is.5. However, it should be adjusted to be lower (. is common) if a Type error is more serious than a Type 2 error. In probability notation: P(Reject H H is true) = α, usually.5. Normal, Mean=, StDev= Which is more serious?? Probability related to Type 2 error and Power: The conditional probability of making a correct decision, given that H a is true is called the power of the test. This can only be calculated if a specific value in Ha is specified. Conditional probability of making a Type 2 error = power. P(Reject H Ha is true) = power. P(Do not reject H Ha is true) = power = β =P(Type 2 error). Picture when the true population proportion is s.d. above null value Normal, Mean=, StDev= How can we increase power and decrease P(Type 2 error)? Power increases if: Sample size is increased (because having more evidence makes it easier to show that the alternative hypothesis is true, if it really is) The level of significance α is increased (because it s easier to reject H when the cutoff point for the p-value is larger) The actual difference between the sample estimate and the null value increases (because it s easier to detect a true difference if it s large) We have no control over this one! Probability of type 2 error = = = power Trade-off must be taken into account when choosing α. If α is small it s harder to reject H. If α is large it s easier to reject H : If Type error is more serious, use smaller α. If Type 2 error is more serious, use larger α.

4 Ways to picture the errors: Truth, decisions, consequences, conditional (row) probabilities: Decision: Truth: Don t reject H Reject H Error can occur: H Correct Type error Type error can only α α occur if H true H a Type 2 error Correct Type 2 error can only β = power power occur if H a is true. Error can occur: H not rejected H rejected SECTION 2.4: Test for difference in 2 proportions Reminder from when we started Chapter 9 Five situations we will cover for the rest of this quarter: Parameter name and description Population parameter Sample statistic For Categorical Variables: One population proportion (or probability) p p Difference in two population proportions p p 2 p For Quantitative Variables: One population mean µ x Population mean of paired differences (dependent samples, paired) µ d d Difference in two population means µ (independent samples) µ 2 x x2 For each situation will we: Learn about the sampling distribution for the sample statistic Learn how to find a confidence interval for the true value of the parameter Test hypotheses about the true value of the parameter Comparing two proportions from independent samples Reminder on how we get independent samples (Lecture 9): Random samples taken separately from two populations and same response variable is recorded. Example: Compare proportions who think global warming is a problem, in two different years. One random sample taken and a variable recorded, but units are categorized to form two populations. Example: Compare 2 and over with under 2 for proportion who drink alcohol. Participants randomly assigned to one of two treatment conditions, and same response variable is recorded. Example: Compare aspirin and placebo groups for proportions who had heart attacks. Hypothesis Test for Difference in Two Proportions Example: (Source: Poll taken in June 26, just after the May release of An Inconvenient Truth Asked 5 people In your view, is global warming a very serious problem, somewhat serious, not too serious, or not a problem? Results: 65/5 =.4 or 4% answered Very serious Poll taken again with different 5 people in October, 29. Results: 525/5 =.35 or 35% answered Very serious. Question: Did the population proportion that thinks it s very serious go down from 26 to 29, or is it chance fluctuation?

5 Notation and numbers for the Example: Population parameter of interest is p p 2 where: p = p 2 = proportion of all US adults in May 26 who thought global warming was a serious problem. proportion of all US adults in Oct 29 who thought global warming was a serious problem. p = sample estimate from May 26 = X /n = 65/5 =.4 p 2 = sample estimate from Oct 29 = X 2 /n 2 = 525/5 =.35 Sample statistic is p =.4.35 =. 6 Five steps to hypothesis testing for difference in 2 proportions: See Summary Box on pages STEP : Determine the null and alternative hypotheses. Null hypothesis is H : p p 2 = (or p = p 2 ); null value = Alternative hypothesis is one of these, based on context: H a : p p 2 (or p p 2 ) H a : p p 2 > (or p > p 2 ) H a : p p 2 < (or p < p 2 ) EXAMPLE: Did the population proportion who think global warming is very serious drop from 26 to 29? This is the alternative hypothesis. (Note that it s a one-sided test.) H : p p 2 = (no actual change in population proportions) H a : p p 2 > (or p > p 2 ; 26 proportion > 29 proportion) STEP 2: Verify data conditions. If met, summarize data into test statistic. For Difference in Two Proportions: Data conditions: n p and n( p ) are both at least for both samples. Test statistic: sample statistic null value z = (null) standard error Sample statistic = p Null value = Null standard error: Computed assuming null hypothesis is true. If null hypothesis is true, then p = p 2 We get an estimate for the common value of p using both samples, then use that in the standard error formula. Details on next page. X + X p = n + n 2 = 2 combined successes combined sample sizes ( p) p2 ( ) Null standard error = estimate of n n2 combined estimate p in place of both p and p 2. So the test statistic is: z = ( p ) = p( p) p( p) + n n 2 p + ( p ) p( p)( + ) n n 2, using

6 Step 2 for the Example: Data conditions are met, since both sample sizes are 5. X + X combined successes n + n combined sample sizes p = = = = =.38 2 Pictures: On left: Sampling distribution of p when population proportions are equal and sample sizes are both 5, showing where the observed value of.6 falls. On right: The same picture, converted to z-scores. 5 5 Null standard error = (. 38)(.38)( + ) =. 77 Sampling distribution for phat - p2hat, when p=p2, n=n2=5 Normal, Mean=, StDev=.77 Standard normal distribution Normal, Mean=, StDev= Test statistic: sample statistic null value.6 z = = = 3.39 (null) standard error.77 Possible values of phat - p2hat.35.6 Values of z Note that area above p =.6 is so small you can t see it! STEP 3: Assuming the null hypothesis is true, find the p-value. General: p-value = the probability of a test statistic as extreme as the one observed or more so, in the direction of H a, if the null hypothesis is true. Difference in two proportions, same idea as one proportion. Depends on the alternative hypothesis. See pictures on p. 57 Alternative hypothesis: H a : p p 2 > (a one-sided hypothesis) H a : p p 2 < (a one-sided hypothesis) H a : p p 2 (a two-sided hypothesis) p-value is: Area above the test statistic z Area below the test statistic z 2 the area above z = area in tails beyond z and z Example: Alternative hypothesis is one-sided H a : p p 2 > p-value = Area above the test statistic z = 3.39 From Table A., p-value = area above 3.39 =.9997 =.3. STEP 4: Decide whether or not the result is statistically significant based on the p-value. Example: Use α of.5, as usual p-value =.3 <.5, so: Reject the null hypothesis. Accept the alternative hypothesis The result is statistically significant

7 Step 5: Report the conclusion in the context of the situation. Example: Conclusion: From May 26 to October 29 there was a statistically significant decrease in the proportion of US adults who think global warming is very serious. Interpretation of the p-value (for this one-sided test): It s a conditional probability. Conditional on the null hypothesis being true (equal population proportions), what is the probability that we would observe a sample difference as large as the one observed or larger just by chance? Specific to this example: If there really were no change in the proportion of the population who think global warming is very serious what is the probability of observing a sample proportion in 29 that is.6 (6%) or more lower than the sample proportion in 26? Answer: The probability is.3. Therefore, we reject the idea (the hypothesis) that there was no change in the population proportion.

HOMEWORK (due Fri, Nov 19): Chapter 12: #62, 83, 101

Today: Section 2.2, Lesson 3: What can go wrong with hyothesis testing Section 2.4: Hyothesis tests for difference in two roortions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify

Example for testing one population mean:

Today: Sections 13.1 to 13.3 ANNOUNCEMENTS: We will finish hypothesis testing for the 5 situations today. See pages 586-587 (end of Chapter 13) for a summary table. Quiz for week 8 starts Wed, ends Monday

HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

Homework 6 Solutions

Math 17, Section 2 Spring 2011 Assignment Chapter 20: 12, 14, 20, 24, 34 Chapter 21: 2, 8, 14, 16, 18 Chapter 20 20.12] Got Milk? The student made a number of mistakes here: Homework 6 Solutions 1. Null

Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.

Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample

Chapter 8 Introduction to Hypothesis Testing

Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm.

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm. Political Science 15 Lecture 12: Hypothesis Testing Sampling

Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

6. Statistical Inference: Significance Tests

6. Statistical Inference: Significance Tests Goal: Use statistical methods to check hypotheses such as Women's participation rates in elections in France is higher than in Germany. (an effect) Ethnic divisions

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of

How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

Chapter 21. More About Tests and Intervals. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 21 More About Tests and Intervals Copyright 2012, 2008, 2005 Pearson Education, Inc. Zero In on the Null Null hypotheses have special requirements. To perform a hypothesis test, the null must be

Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic.

Hypothesis Testing Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic. Population Characteristics are things like The mean of a population

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

Lecture 8 Hypothesis Testing

Lecture 8 Hypothesis Testing Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech Midterm 1 Score 46 students Highest score: 98 Lowest

Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine

Hypothesis testing. Power of a test. Alternative is greater than Null. Probability

Probability February 14, 2013 Debdeep Pati Hypothesis testing Power of a test 1. Assuming standard deviation is known. Calculate power based on one-sample z test. A new drug is proposed for people with

Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

Two-sample hypothesis testing, I 9.07 3/09/2004

Two-sample hypothesis testing, I 9.07 3/09/2004 But first, from last time More on the tradeoff between Type I and Type II errors The null and the alternative: Sampling distribution of the mean, m, given

CHAPTER 15: Tests of Significance: The Basics

CHAPTER 15: Tests of Significance: The Basics The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner Lecture PowerPoint Slides Chapter 15 Concepts 2 The Reasoning of Tests of Significance

22. HYPOTHESIS TESTING

22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?

15.0 More Hypothesis Testing

15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis

Reasoning with Uncertainty More about Hypothesis Testing. P-values, types of errors, power of a test

Reasoning with Uncertainty More about Hypothesis Testing P-values, types of errors, power of a test P-Values and Decisions Your conclusion about any null hypothesis should be accompanied by the P-value

Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

Terminology. 2 There is no mathematical difference between the errors, however. The bottom line is that we choose one type

Hypothesis Testing 10.2.1 Terminology The null hypothesis H 0 is a nothing hypothesis, whose interpretation could be that nothing has changed, there is no difference, there is nothing special taking place,

Hypothesis Testing: Two Means, Paired Data, Two Proportions

Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this

9.1 Basic Principles of Hypothesis Testing

9. Basic Principles of Hypothesis Testing Basic Idea Through an Example: On the very first day of class I gave the example of tossing a coin times, and what you might conclude about the fairness of the

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing.

This is Dr. Chumney. The focus of this lecture is hypothesis testing both what it is, how hypothesis tests are used, and how to conduct hypothesis tests. 1 In this lecture, we will talk about both theoretical

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

2 Precision-based sample size calculations

Statistics: An introduction to sample size calculations Rosie Cornish. 2006. 1 Introduction One crucial aspect of study design is deciding how big your sample should be. If you increase your sample size

Hypothesis Testing --- One Mean

Hypothesis Testing --- One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis

Hypothesis Testing. Concept of Hypothesis Testing

Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population

Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide

Two Related Samples t Test

Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

Introduction to Hypothesis Testing

Introduction to Hypothesis Testing A Hypothesis Test for μ Heuristic Hypothesis testing works a lot like our legal system. In the legal system, the accused is innocent until proven guilty. After examining

Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections - we are still here Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.5 The t Test 9.6 Comparing the

Hypothesis Testing. Jungmo Yoon CMC, Claremont McKeena College. Jungmo Yoon (CMC) Testing CMC, / 7

Hypothesis Testing Jungmo Yoon Claremont McKeena College CMC, 2009 Jungmo Yoon (CMC) Testing CMC, 2009 1 / 7 Concepts of Hypothesis Testing A criminal trial as an example of hypothesis testing. A jury

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide

Hypothesis testing S2

Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to

Introduction to Hypothesis Testing

Introduction to Hypothesis Testing A Hypothesis Test for Heuristic Hypothesis testing works a lot like our legal system. In the legal system, the accused is innocent until proven guilty. After examining

Lecture 13 More on hypothesis testing

Lecture 13 More on hypothesis testing Thais Paiva STA 111 - Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111 - Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error

About Hypothesis Testing TABLE OF CONTENTS About Hypothesis Testing... 1 What is a HYPOTHESIS TEST?... 1 Hypothesis Testing... 1 Hypothesis Testing... 1 Steps in Hypothesis Testing... 2 Steps in Hypothesis

Chapter 7. Section Introduction to Hypothesis Testing

Section 7.1 - Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

Review for Exam 2. H 0 : p 1 p 2 = 0 H A : p 1 p 2 0

Review for Exam 2 1 Time in the shower The distribution of the amount of time spent in the shower (in minutes) of all Americans is right-skewed with mean of 8 minutes and a standard deviation of 10 minutes.

Module 7: Hypothesis Testing I Statistics (OA3102)

Module 7: Hypothesis Testing I Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 10.1-10.5 Revision: 2-12 1 Goals for this Module

Mind on Statistics. Chapter 12

Mind on Statistics Chapter 12 Sections 12.1 Questions 1 to 6: For each statement, determine if the statement is a typical null hypothesis (H 0 ) or alternative hypothesis (H a ). 1. There is no difference

STA102 Class Notes Chapter Hypothesis Testing

10. Hypothesis Testing In this chapter, we continue to study methods for making inference about a population using our sample. In chapter 9, we used CIs, essentially, to answer questions of the type, What

Ch. 8 Hypothesis Testing

Ch. 8 Hypothesis Testing 8.1 Foundations of Hypothesis Testing Definitions In statistics, a hypothesis is a claim about a property of a population. A hypothesis test is a standard procedure for testing

AP Statistics 2009 Scoring Guidelines Form B

AP Statistics 2009 Scoring Guidelines Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

Hypothesis. Testing Examples and Case Studies. Chapter 23. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Hypothesis Chapter 23 Testing Examples and Case Studies Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis

WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

8-2 Basics of Hypothesis Testing. Definitions. Rare Event Rule for Inferential Statistics. Null Hypothesis

8-2 Basics of Hypothesis Testing Definitions This section presents individual components of a hypothesis test. We should know and understand the following: How to identify the null hypothesis and alternative

Chapter 16 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 16 Multiple Choice Questions (The answers are provided after the last question.) 1. Which of the following symbols represents a population parameter? a. SD b. σ c. r d. 0 2. If you drew all possible

HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

A Trial Analogy for Statistical. Hypothesis Testing. Legal Trial Begin with claim: Statistical Significance Test Hypotheses (statements)

A Trial Analogy for Statistical Slide 1 Hypothesis Testing Legal Trial Begin with claim: Smith is not guilty If this is rejected, we accept Smith is guilty reasonable doubt Present evidence (facts) Evaluate

Statistical inference provides methods for drawing conclusions about a population from sample data.

Chapter 15 Tests of Significance: The Basics Statistical inference provides methods for drawing conclusions about a population from sample data. Two of the most common types of statistical inference: 1)

Probability & Statistics

Probability & Statistics BITS Pilani K K Birla Goa Campus Dr. Jajati Keshari Sahoo Department of Mathematics TEST OF HYPOTHESIS There are many problems in which, rather then estimating the value of a parameter,

Hypothesis Testing. Mark Huiskes, LIACS Probability and Statistics, Mark Huiskes, LIACS, Lecture 10

Hypothesis Testing Mark Huiskes, LIACS mark.huiskes@liacs.nl Estimating sample standard deviation Suppose we have a sample x_1,, x_n Average: xbar = (x_1 + + x_n) / n Standard deviation estimate s? Two

Decisions under uncertainty: A choice between two conflicting theories

HYPOTHESIS TESTING The logic of hypothesis testing One-sample vs. two-sample problem Null vs. alternative hypothesis Type I and type II error Significance level α Power 1-β One-sided vs. two-sided test

STA 2023H Solutions for Practice Test 4

1. Which statement is not true about confidence intervals? A. A confidence interval is an interval of values computed from sample data that is likely to include the true population value. B. An approximate

An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

Basic Statistics Self Assessment Test

Basic Statistics Self Assessment Test Professor Douglas H. Jones PAGE 1 A soda-dispensing machine fills 12-ounce cans of soda using a normal distribution with a mean of 12.1 ounces and a standard deviation

Hypothesis Testing. April 21, 2009

Hypothesis Testing April 21, 2009 Your Claim is Just a Hypothesis I ve never made a mistake. Once I thought I did, but I was wrong. Your Claim is Just a Hypothesis Confidence intervals quantify how sure

Section: 101 (10am-11am) 102 (11am-12pm) 103 (1pm-2pm) 104 (1pm-2pm)

Stat 0 Midterm Exam Instructor: Tessa Childers-Day 1 May 014 Please write your name and student ID below, and circle your section. With your signature, you certify that you have not observed poor or dishonest

Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there

Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

Chris Slaughter, DrPH. GI Research Conference June 19, 2008

Chris Slaughter, DrPH Assistant Professor, Department of Biostatistics Vanderbilt University School of Medicine GI Research Conference June 19, 2008 Outline 1 2 3 Factors that Impact Power 4 5 6 Conclusions

reductio ad absurdum null hypothesis, alternate hypothesis

Chapter 10 s Using a Single Sample 10.1: Hypotheses & Test Procedures Basics: In statistics, a hypothesis is a statement about a population characteristic. s are based on an reductio ad absurdum form of

Versions 1a Page 1 of 17

Note to Students: This practice exam is intended to give you an idea of the type of questions the instructor asks and the approximate length of the exam. It does NOT indicate the exact questions or the

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

Hypothesis Testing: Significance

STAT 101 Dr. Kari Lock Morgan Hypothesis Testing: Significance SECTION 4.3, 4.5 Significance level (4.3) Statistical conclusions (4.3) Type I and II errors (4.3) Statistical versus practical significance

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

8-1 8-2 8-3 8-4 8-5 8-6

8-1 Review and Preview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-4 Testing a Claim About a Mean: s Known 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing a Claim

Sociology 6Z03 Topic 15: Statistical Inference for Means

Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall 2016 1 / 41 Outline: Statistical

T-test in SPSS Hypothesis tests of proportions Confidence Intervals (End of chapter 6 material)

T-test in SPSS Hypothesis tests of proportions Confidence Intervals (End of chapter 6 material) Definition of p-value: The probability of getting evidence as strong as you did assuming that the null hypothesis

Problem Set 2 - Solutions

Ecn 102 - Analysis of Economic Data University of California - Davis January 13, 2010 John Parman Problem Set 2 - Solutions This problem set will be not be graded and does not need to be turned in. However,

9.1 Hypothesis Testing

9.1 Hypothesis Testing Define: 1. Null Hypothesis 2. Alternative Hypothesis Null Hypothesis: H 0, statement that the population proportion, or population mean is EQUAL TO a number population proportion

Chapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited

Chapter 8 Professor April 20, 2014 In Chapter 8, we continue our study of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the