Testing Hypotheses About Proportions

Size: px
Start display at page:

Download "Testing Hypotheses About Proportions"

Transcription

1 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 the null and alternative hypotheses. 2. Verify necessary data conditions, and if met, 3. Assuming the null hypothesis is true, find the p-value. 4. Decide whether or not the result is statistically significant based on the p-value. 5. Report the conclusion in the context of the situation. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc Formulating Hypothesis Statements Does a majority of the population favor a new legal standard for the blood alcohol level that constitutes drunk driving? Hypothesis 1: The population proportion favoring the new standard is not a majority. More on Formulating Hypotheses Do female students study, on average, more than male students do? Hypothesis 1: On average, women do not study more than men do. Hypothesis 2: On average, women do study more than men do. Hypothesis 2: The population proportion favoring the new standard is a Copyright 2006 Brooks/Cole, a division majority. of Thomson Learning, Inc. 3 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 Terminology for the Two Choices Null hypothesis: Represented by H 0, is a statement that there is nothing happening. Generally thought of as the status quo, or no relationship, or no difference. Usually the researcher hopes to disprove or reject the null hypothesis. Alternative hypothesis: Represented by H a, is a statement that something is happening. Generally it is what the researcher hopes to prove. It may be a statement that the assumed status quo is false, or that there is a relationship, or that there is a difference. Examples of H 0 and H a Null hypothesis examples: There is no extrasensory perception. There is no difference between the mean pulse rates of men and women. There is no relationship between exercise intensity and the resulting aerobic benefit. Alternative hypothesis examples: There is extrasensory perception. Men have lower mean pulse rates than women do. Increasing exercise intensity increases the resulting aerobic benefit. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 6 1

2 Example 11.1 Are Side Effects Experienced by Fewer than 20% of Patients? Pharmaceutical company wants to claim that the proportion of patients who experience side effects is less than 20%. Null: 20% (or more) of users will experience side effects. Alternative: Fewer than 20% of users will experience side effects. Notice that the claim that the company hopes to prove is used as the alternative hypothesis. The alternative is one-sided. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 7 Example 11.2 Does a Majority Favor the Proposed Blood Alcohol Limit? Legislator s plan is to vote for the proposal if there is conclusive evidence that a majority of her constituents favor the proposal. H 0 : p.5 H a : p >.5 (not a majority) (a majority) Note: p = the proportion of her constituents that favors the proposal. The alternative is one-sided. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc Logic of Hypothesis Testing What if the Null is True? Similar to presumed innocent until proven guilty logic. We assume the null hypothesis is a possible truth until the sample data conclusively demonstrate otherwise. The Probability Question on Which Hypothesis Testing is Based If the null hypothesis is true about the population, what is the probability of observing sample data like that observed? Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 9 Example 11.3 Psychic Powers Cartoon: Two characters playing a coin-flipping game. Character 1: correctly guesses outcome of 100 flips. Character 2: just a coincidence Null: Alternative: Character 1 does not have Psychic Powers (is just guessing) Character 1 has Psychic Powers Q: If character only guessing, how likely is correctly guessing 100 consecutive fair coin tosses? A: (½) 100 => extraordinarily small. We reject the null hypothesis because the sample results are extremely inconsistent with it. We conclude character was using psychic powers. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc Reaching a Conclusion About the Two Hypotheses Data summary used to evaluate the two hypotheses is called the test statistic. Likelihood of observing a test statistic as extreme as what we did, or something even more extreme, if the null hypothesis is true is called the p-value. Decision: reject H 0 if the p-value is smaller than a designated level of significance, denoted by α (usually 0.05, sometimes 0.10 or 0.01). In this case the result is statistically significant. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 11 Stating the Two Possible Conclusions When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative hypothesis. Small is defined as a p-value α, where α = level of significance (usually 0.05). When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis. Not small is defined as a p-value > α,where α = level of significance (usually 0.05). Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 12 2

3 11.4 Testing Hypotheses About a Proportion Possible null and alternative hypotheses: 1. H 0 : p = p 0 versus H a : p p 0 (two-sided) 2. H 0 : p p 0 versus H a : p < p 0 (one-sided) 3. H 0 : p p 0 versus H a : p > p 0 (one-sided) p 0 = specific value called the null value. Often H 0 for a one-sided test is written as H 0 : p = p 0. Remember a p-value is computed assuming H 0 is true, and p 0 is the value used for that computation. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 13 The z-test for a Proportion Determine the sampling distribution of possible sample proportions when the true population proportion is p 0 (called the null value), the value specified in H 0. Using properties of this sampling distribution, calculate a standardized score (z-score) for the observed sample proportion. If the standardized score has a large magnitude, conclude that the sample proportion would be unlikely if the null value p 0 is true, and reject the null hypothesis. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 14 Conditions for Conducting the z-test 1. The sample should be a random sample from the population. Not always practical most use test procedure as long as sample is representative of the population for the question of interest. 2. The quantities np 0 and n(1 p 0 ) should both be at least 10. A sample size requirement. Some authors say at least 5 instead of our conservative 10. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 15 Example 11.6 The Importance of Order Survey of n = 190 college students. About half (92) asked: Randomly pick a letter - S or Q. Other half (98) asked: Randomly pick a letter - Q or S. Is there a preference for picking the first? Step 1: Determine the null and alternative hypotheses. Let p = proportion of population that would pick first letter. Null hypothesis: statement of nothing happening. If no general preference for either first or second letter, p =.5 Alternative hypothesis: researcher s belief or speculation. A preference for first letter => p is greater than.5. H 0 : p = p 0 versus H a : p > p 0 (one-sided) Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 16 Step 2: Verify necessary data conditions, and if met, 1. The sample should be a random sample from the population. The sample is a convenience sample of students who were enrolled for a class. Does not seem this will bias results for this question, so will view the sample as a random sample. 2. The quantities np 0 and n(1 p 0 ) should both be at least 10. With n = 190 and p 0 =.5, both n p 0 and n(1 p 0 ) equal 95, a quantity larger than 10, so the sample size condition is met. Step 2: Verify necessary data conditions, and if met, Of 92 students asked S or Q, 61 picked S, the first choice. Of 98 students asked Q or S, 53 picked Q, the first choice. Overall: 114 students picked first choice => 114/190 =.60. The sample proportion,.60, is used to compute the z-test statistic, the standardized score for measuring the difference between the =.60, and the null hypothesis value, p 0 =.50. The z-statistic = 2.76 (formula comes later). Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 17 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 18 3

4 Step 3: Assuming the null hypothesis is true, find the p-value. If the true p is.5, what is the probability that, for a sample of 190 people, the sample proportion could be as large as.60 (or larger)? or equivalently If the null hypothesis is true, what is the probability that the z-statistic could be as large as 2.76 (or larger)? Using computer (or reading from print-out): p-value = Step 4: Decide whether or not the result is statistically significant based on the p-value. Convention used by most researchers is to declare statistical significance when the p-value is smaller than The p-value = so the results are statistically significant and we can reject the null hypothesis. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 19 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 20 Step 5: Report the conclusion in the context of the problem. Details for Calculating the z-statistic The z-statistic for the significance test is Statistical Conclusion = Reject the null hypothesis that p = 0.50 Context Conclusion = there is statistically significant evidence that the first letter presented is preferred. represents the sample estimate of the proportion p 0 represents the specific value in null hypothesis n is the sample size Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 21 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 22 Example 11.1 Fewer than 20%? (cont). Clinical Trial of n = 400 patients. 68 patients experienced side effects. Can the company claim that fewer than 20% will experience side effects? Hypothesis testing steps: Step 1: Determine the null and alternative hypotheses Step 2: Verify necessary data conditions, and if met, Step 3: Assuming the null hypothesis is true, find the p-value. Step 4: Decide whether or not the result is statistically significant based on the p-value. Step 5: Report the conclusion in the context of the problem. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc

5 Using Minitab: To test hypotheses about a proportion, use Stat>Basic Statistics>1 Proportion. If the raw data are in a column of the worksheet, specify the column. If not, enter the summarized data. Click on Options, select confidence level, alternative, and check Use test and box. Check Perform hypothesis test and enter p 0. Click OK and read off results. Example 11.7 Left and Right Foot Lengths Sample: 112 of 215 college students with unequal right and left foot measurements. Let p = population proportion with a longer right foot. Are Left and Right Foot Lengths Equal or Different? 25 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc Role of Sample Size in Statistical Significance Cautions about Sample Size and Statistical Significance If a small to moderate effect in the population, a small sample has little chance of being statistically significant. With a large sample, even a small and unimportant effect in the population may be statistically significance. Example 11.8 Same Sample Proportion Can Produce Different Conclusions Taste Test: Sample of people taste both drinks and record how many like taste of Drink A better than B. Let p = H 0 : p =.5 H a : p.5 proportion in population that would prefer Drink A. (no preference) (preference for one or other) Results based on two sample sizes: n = 60 and n = 960 and the sample proportion for both is Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 27 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 28 Example 11.8 Different Conclusions (cont) Results when n = of the 60 preferred Drink A; Results when n = or the 960 preferred Drink A; Why more significant for larger n? The z-value changes because the sample size affects the standard error. When n =60, the null standard error =.065. When n = 960, the null standard error =.016. Increasing n decreases null standard error => an absolute difference between the sample proportion and null value is more significant Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 29 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 30 5

6 11.6 Real Importance versus Statistical Significance The p-value does not provide information about the magnitude of the effect. The magnitude of a statistically significant effect can be so small that the practical effect is not important. If sample size large enough, almost any null hypothesis can be rejected. Example 11.9 Birth Month and Height Headline: Spring Birthday Confers Height Advantage Austrian study of heights of 507,125 military recruits. Men born in spring were, on average, about 0.6 cm taller than men born in fall (Weber et al., Nature, 1998, 391: ). A small difference: 0.6 cm = about 1/4 inch. Sample size so large that even a very small difference was statistically significant. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 31 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 32 Case Study 11.1 Internet and Loneliness greater use of the Internet was associated with declines in participants communication with family members in the household, declines in size of their social circle, and increases in their depression and loneliness (Kraut et al., 1998, p. 1017) A closer look: actual effects were quite small. one hour a week on the Internet was associated, on average, with an increase of 0.03, or 1 percent on the depression scale (Harman, 30 August 1998, p. A3) What Can Go Wrong? A type 1 error can only occur when the null hypothesis is actually true. The error occurs by concluding that the alternative hypothesis is true. A type 2 error can only occur when the alternative hypothesis is actually true. The error occurs by concluding that the null hypothesis cannot be rejected. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 33 Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 34 Example Medical Analogy Null hypothesis: You do not have the disease. Alternative hypothesis: You do have the disease. Type 1 Error: You are told you have the disease, but you actually don t. The test result was a false positive. Consequence: You will be unnecessarily concerned about your health and you may receive unnecessary treatment. Type 2 Error : You are told that you do not have the disease, but you actually do. The test result was a false negative. Consequence: You do not receive treatment for a disease that you have. If this is a contagious disease, you may infect others. Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. 35 6

Mind on Statistics. Chapter 12

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

More information

Introduction to Hypothesis Testing

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

More information

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

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

More information

Chapter 8. Hypothesis Testing

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

More information

Module 7: Hypothesis Testing I Statistics (OA3102)

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

More information

Homework 6 Solutions

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

More information

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

More information

Lecture 8 Hypothesis 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

More information

Chapter 8 Introduction to Hypothesis Testing

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

More information

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE 1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

More information

Module 5 Hypotheses Tests: Comparing Two Groups

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

More information

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

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

More information

p-values and significance levels (false positive or false alarm rates)

p-values and significance levels (false positive or false alarm rates) p-values and significance levels (false positive or false alarm rates) Let's say 123 people in the class toss a coin. Call it "Coin A." There are 65 heads. Then they toss another coin. Call it "Coin B."

More information

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 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

More information

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

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

More information

How to Conduct a Hypothesis Test

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

More information

Online 12 - Sections 9.1 and 9.2-Doug Ensley

Online 12 - Sections 9.1 and 9.2-Doug Ensley Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics - Ensley Assignment: Online 12 - Sections 9.1 and 9.2 1. Does a P-value of 0.001 give strong evidence or not especially strong

More information

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

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

More information

15.0 More Hypothesis Testing

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

More information

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) 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

More information

9.1 Basic Principles of Hypothesis Testing

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

More information

Statistical Inference: Hypothesis Testing

Statistical Inference: Hypothesis Testing Statistical Inference: Hypothesis Testing Scott Evans, Ph.D. 1 The Big Picture Populations and Samples Sample / Statistics x, s, s 2 Population Parameters μ, σ, σ 2 Scott Evans, Ph.D. 2 Statistical Inference

More information

AP Statistics 2010 Scoring Guidelines

AP Statistics 2010 Scoring Guidelines AP Statistics 2010 Scoring Guidelines 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 in

More information

Mind on Statistics. Chapter 15

Mind on Statistics. Chapter 15 Mind on Statistics Chapter 15 Section 15.1 1. A student survey was done to study the relationship between class standing (freshman, sophomore, junior, or senior) and major subject (English, Biology, French,

More information

Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

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

More information

22. HYPOTHESIS TESTING

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?

More information

Statistics 2014 Scoring Guidelines

Statistics 2014 Scoring Guidelines AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

More information

Name: Date: Use the following to answer questions 3-4:

Name: Date: Use the following to answer questions 3-4: Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

More information

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

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

More information

6: Introduction to Hypothesis Testing

6: Introduction to Hypothesis Testing 6: Introduction to Hypothesis Testing Significance testing is used to help make a judgment about a claim by addressing the question, Can the observed difference be attributed to chance? We break up significance

More information

Mind on Statistics. Chapter 13

Mind on Statistics. Chapter 13 Mind on Statistics Chapter 13 Sections 13.1-13.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question

More information

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

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

More information

Chapter 7 Part 2. Hypothesis testing Power

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

More information

Extending Hypothesis Testing. p-values & confidence intervals

Extending Hypothesis Testing. p-values & confidence intervals Extending Hypothesis Testing p-values & confidence intervals So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by

More information

Introduction to Hypothesis Testing OPRE 6301

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

More information

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

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

More information

Chapter 8 Section 1. Homework A

Chapter 8 Section 1. Homework A Chapter 8 Section 1 Homework A 8.7 Can we use the large-sample confidence interval? In each of the following circumstances state whether you would use the large-sample confidence interval. The variable

More information

1 Hypothesis Testing. H 0 : population parameter = hypothesized value:

1 Hypothesis Testing. H 0 : population parameter = hypothesized value: 1 Hypothesis Testing In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data

More information

5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives

5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives C H 8A P T E R Outline 8 1 Steps in Traditional Method 8 2 z Test for a Mean 8 3 t Test for a Mean 8 4 z Test for a Proportion 8 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc.

More information

Step 1: Set up hypotheses that ask a question about the population by setting up two opposite statements about the possible value of the parameters.

Step 1: Set up hypotheses that ask a question about the population by setting up two opposite statements about the possible value of the parameters. HYPOTHESIS TEST CLASS NOTES Hypothesis Test: Procedure that allows us to ask a question about an unknown population parameter Uses sample data to draw a conclusion about the unknown population parameter.

More information

Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

More information

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 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

More information

Homework 5 Solutions

Homework 5 Solutions Math 130 Assignment Chapter 18: 6, 10, 38 Chapter 19: 4, 6, 8, 10, 14, 16, 40 Chapter 20: 2, 4, 9 Chapter 18 Homework 5 Solutions 18.6] M&M s. The candy company claims that 10% of the M&M s it produces

More information

Lecture 13 More on hypothesis testing

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

More information

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

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

More information

1-3 id id no. of respondents 101-300 4 respon 1 responsible for maintenance? 1 = no, 2 = yes, 9 = blank

1-3 id id no. of respondents 101-300 4 respon 1 responsible for maintenance? 1 = no, 2 = yes, 9 = blank Basic Data Analysis Graziadio School of Business and Management Data Preparation & Entry Editing: Inspection & Correction Field Edit: Immediate follow-up (complete? legible? comprehensible? consistent?

More information

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and

More information

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. 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

More information

3.4 Statistical inference for 2 populations based on two samples

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

More information

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

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

More information

Hypothesis Testing. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006

Hypothesis Testing. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006 Hypothesis Testing Lecture 4 Hypothesis Testing Hypothesis testing is about making decisions Is a hypothesis true or false? Are women paid less, on average, than men? Principles of Hypothesis Testing The

More information

6. Statistical Inference: Significance Tests

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

More information

AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

More information

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

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

More information

Hypothesis Testing: Significance

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

More information

About Hypothesis Testing

About Hypothesis Testing 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

More information

Hypothesis Testing and Confidence Interval Estimation

Hypothesis Testing and Confidence Interval Estimation Biostatistics for Health Care Researchers: A Short Course Hypothesis Testing and Confidence Interval Estimation Presented ed by: Susan M. Perkins, Ph.D. Division of Biostatistics Indiana University School

More information

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 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

More information

Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008 Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

More information

research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other

research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other 1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric

More information

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

More information

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance Chapter 8 Hypothesis Testing 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: σ Known 8 5 Testing a Claim About a Mean: σ

More information

Hypothesis testing - Steps

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 =

More information

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

More information

Sample Size Planning, Calculation, and Justification

Sample Size Planning, Calculation, and Justification Sample Size Planning, Calculation, and Justification Theresa A Scott, MS Vanderbilt University Department of Biostatistics theresa.scott@vanderbilt.edu http://biostat.mc.vanderbilt.edu/theresascott Theresa

More information

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so: Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

More information

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto

More information

Power and Sample Size Determination

Power and Sample Size Determination Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

More information

HYPOTHESIS TESTING WITH SPSS:

HYPOTHESIS TESTING WITH SPSS: HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER

More information

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

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

More information

Practice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.

Practice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems. Practice problems for Homework 1 - confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

Introduction to Hypothesis Testing

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

More information

Statistical Foundations:

Statistical Foundations: Statistical Foundations: Hypothesis Testing Psychology 790 Lecture #9 9/19/2006 Today sclass Hypothesis Testing. General terms and philosophy. Specific Examples Hypothesis Testing Rules of the NHST Game

More information

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

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,

More information

Chapter 2. Hypothesis testing in one population

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

More information

STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS

STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could

More information

Mind on Statistics. Chapter 8

Mind on Statistics. Chapter 8 Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable

More information

7 Hypothesis testing - one sample tests

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

More information

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

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

More information

Tests for Two Proportions

Tests for Two Proportions Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics

More information

Inferential Statistics

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

More information

MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

More information

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

More information

WISE Power Tutorial All Exercises

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

More information

The Chi-Square Test. STAT E-50 Introduction to Statistics

The Chi-Square Test. STAT E-50 Introduction to Statistics STAT -50 Introduction to Statistics The Chi-Square Test The Chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed

More information

Unit 27: Comparing Two Means

Unit 27: Comparing Two Means Unit 27: Comparing Two Means Prerequisites Students should have experience with one-sample t-procedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One

More information

FAT-FREE OR REGULAR PRINGLES: CAN TASTERS TELL THE DIFFERENCE?

FAT-FREE OR REGULAR PRINGLES: CAN TASTERS TELL THE DIFFERENCE? CHAPTER 10 Hypothesis Tests Involving a Sample Mean or Proportion FAT-FREE OR REGULAR PRINGLES: CAN TASTERS TELL THE DIFFERENCE? Michael Newman/PhotoEdit When the makers of Pringles potato chips came out

More information

Correlational Research

Correlational Research Correlational Research Chapter Fifteen Correlational Research Chapter Fifteen Bring folder of readings The Nature of Correlational Research Correlational Research is also known as Associational Research.

More information

Hypothesis Testing. Bluman Chapter 8

Hypothesis Testing. Bluman Chapter 8 CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

More information

MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010 MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

More information

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation Leslie Chandrakantha lchandra@jjay.cuny.edu Department of Mathematics & Computer Science John Jay College of

More information

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

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

More information

Testing a claim about a population mean

Testing a claim about a population mean Introductory Statistics Lectures Testing a claim about a population mean One sample hypothesis test of the mean Department of Mathematics Pima Community College Redistribution of this material is prohibited

More information

Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing

Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing 1) Hypothesis testing and confidence interval estimation are essentially two totally different statistical procedures

More information

Tests for Two Survival Curves Using Cox s Proportional Hazards Model

Tests for Two Survival Curves Using Cox s Proportional Hazards Model Chapter 730 Tests for Two Survival Curves Using Cox s Proportional Hazards Model Introduction A clinical trial is often employed to test the equality of survival distributions of two treatment groups.

More information

Statistical Inference and t-tests

Statistical Inference and t-tests 1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

More information

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

More information

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

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.

More information

Confidence Interval: pˆ = E = Indicated decision: < p <

Confidence Interval: pˆ = E = Indicated decision: < p < Hypothesis (Significance) Tests About a Proportion Example 1 The standard treatment for a disease works in 0.675 of all patients. A new treatment is proposed. Is it better? (The scientists who created

More information