Hypothesis Testing Introduction

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Hypothesis Testing Introduction"

Transcription

1 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, H 0, is the current belief. The alternative hypothesis, H a, is your belief; it is what you want to show. STA286 week 10 1

2 Examples Each of the following situations requires a significance test about a population mean μ. State the appropriate null hypothesis H 0 and alternative hypothesis H a in each case. (a) The mean area of the several thousand apartments in a new development is advertised to be 1250 square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion. (b) Larry's car consume on average 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased. (c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target. STA286 week 10 2

3 Test Statistic The test is based on a statistic that estimate the parameter that appears in the hypotheses. Usually this is the same estimate we would use in a confidence interval for the parameter. When H 0 is true, we expect the estimate to take a value near the parameter value specified in H 0. Values of the estimate far from the parameter value specified by H 0 give evidence against H 0. The alternative hypothesis determines which directions count against H 0. A test statistic measures compatibility between the null hypothesis and the data. We use it for the probability calculation that we need for our test of significance It is a random variable with a distribution that we know. STA286 week 10 3

4 Example An air freight company wishes to test whether or not the mean weight of parcels shipped on a particular root exceeds 10 pounds. A random sample of 49 shipping orders was examined and found to have average weight of 11 pounds. Assume that the stdev. of the weights (σ) is 2.8 pounds. The null and alternative hypotheses in this problem are: H 0 : μ = 10 ; H a : μ > 10. The test statistic for this problem is the standardized version of X Z = X μ σ / n Decision:? STA286 week 10 4

5 Testing Process Hypothesis testing is a proof by contradiction. The testing process has four steps: Step 1: Assume H 0 is true. Step 2: Use statistical theory to make a statistic (function of the data) that includes H 0. 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 H 0 is true. Step 4: If the probability we calculated in step 3 is high it means that the sample is likely under H 0 and so we have no evidence against H 0. If the probability is low it means that the sample is unlikely under H 0. This in turn means one of two things; either H 0 is false or we are unlucky and H 0 is true. STA286 week 10 5

6 Example STA286 week 10 6

7 Graphical Representation Let S n 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 H 0. It is called the critical region. C is the set of all samples for which we fail to reject H 0. It is called the acceptance region. STA286 week 10 7

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 H 0 when in fact H 0 is true is called a type I error. The error made by failing to reject the null hypothesis H 0 when in fact H 0 is false is called a type II error. STA286 week 10 8

9 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... STA286 week 10 9

10 Power The probability that a fixed size α test will reject H 0 when H 0 is false is called the power of the test. Power is not about an error. We want high power. Example STA286 week 10 10

11 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: STA286 week 10 11

12 Simple versus Composite Hypothesis Recall, a simple hypothesis completely specifies the distribution. A composite does not. When testing a simple null hypothesis versus a composite alternative, the power of the test is a function of the parameter of interest. In addition, the power is also affected by the sample size. STA286 week 10 12

13 Example STA286 week 10 13

14 Test for Mean of Normal Population σ 2 is known Suppose X 1,, X n is a random sample from a N(μ, σ 2 ) distribution where σ 2 is known. We are interested in testing hypotheses about μ. The test statistics is the standardized version of the sample mean. We could test three sets of hypotheses X STA286 week 10 14

15 Example The Pfft Light Bulb Company claims that the mean life of its 2 watt bulbs is 1300 hours. Suspecting that the claim is too high, Nalph Rader gathered a random sample of 64 bulbs and tested each. He found the average life to be 1295 hours. Test the company's claim using α = Assume σ = 20 hours. STA286 week 10 15

16 Exercise A standard intelligence examination has been given for several years with an average score of 80 and a standard deviation of 7. If 25 students taught with special emphasis on reading skill, obtain a mean grade of 83 on the examination, is there reason to believe that the special emphasis changes the result on the test? Use α = STA286 week 10 16

17 Minitab Example Data on the Degree of Reading Power (DRP) scores for 44 students are recorded after they took a reading course. We wish to test whether the mean DRP of these students is greater than the mean DRP in the population which is known to be 32. The MINITAB output for the test is given below. Z-Test Test of mu = vs mu > The assumed sigma = 11.0 Variable N Mean StDev SE Mean Z P DRP Scor MINITAB Command Stat > Basic Statistics >1Sample Z and select Test mean STA286 week 10 17

18 Test for Mean of Normal Population σ 2 is unknown Suppose X 1,, X n is a random sample from a N(μ, σ 2 ) distribution where σ 2 is unknown, n is small and we are interested in testing hypotheses about μ. The test statistics is... STA286 week 10 18

19 Example In a metropolitan area, the concentration of cadmium (Cd) in leaf lettuce was measured in 6 representative gardens where sewage sludge was used as fertilizer. The following measurements (in mg/kg of dry weight) were obtained. Cd: Is there evidence that the mean concentration of Cd is higher than 12? STA286 week 10 19

20 MINITAB commands: Stat > Basic Statistics > 1-Sample t MINITAB outputs for the above problem: T-Test of the Mean Test of mu = vs mu > Variable N Mean StDev SE Mean T P Cd T Confidence Intervals Variable N Mean StDev SE Mean 95.0 % CI Cd (6.79, 29.21) STA286 week 10 20

21 Test for Mean of a Non-Normal Population Suppose X 1,, X n are iid from some distribution with E(X i )=μ and Var(X i )= σ 2. Further suppose that n is large and we are interested in testing hypotheses about μ. Since n is large the CLT applies to the sample mean and the test statistics is again the standardized version of the sample mean X, that is we use the z-test. If the variance of the population is unknown the result of the test is approximately correct. STA286 week 10 21

22 Example Binomial Distribution Suppose X 1,,X n are random sample from Bernoulli(θ) distribution. We are interested in testing hypotheses about θ STA286 week 10 22

23 Test on Pairs of Means Case I Suppose are iid 2 independent of that 1,..., X n N μ 1 x, σ x Y1,..., Yn2 2 are iid N μ σ. Further, suppose that n 1 and n 2 are large or that 2 2 and σ are known. We are interested in testing H 0 : μ x = μ y versus a one sided or a two sided alternative Then, X ( ) ( ) y, y σ x y STA286 week 10 23

24 Test on Pairs of Means Case II Suppose are iid 2 independent of that are 1,..., X n N μ, 1 x σ Y x 1,..., Yn2 2 iid N μ σ. 2 Further, suppose that σ 2 x and σ y are unknown but we assume they are equal to σ 2. We are interested in testing H 0 : μ x - μ y = δ versus a one sided or a two sided alternative Then, X ( ) ( ) y, y STA286 week 10 24

25 Test on Pairs of Means Case III Suppose are iid 2 independent of that are 1,..., X n N μ, 1 x σ Y x 1,..., Yn2 2 iid N μ σ. 2 Further, suppose that σ 2 x and σ y are unknown but we can not assume that they are equal to. We are interested in testing H 0 : μ x - μ y = δ versus a one sided or a two sided alternative Then, X ( ) ( ) y, y STA286 week 10 25

26 Example The strength of concrete depends, to some extent, on the method used for drying it. Two drying methods were tested on independently specimens yielding the following results We can assume that the strength of concrete using each of these methods follows a normal distribution with the same variance. Do the methods appear to produce concrete with different mean strength? Use α = STA286 week 10 26

27 Test for a Single Variance Suppose X 1,, X n is a random sample from a N(μ, σ 2 ) distribution. We are interested in testing H : σ 2 = σ 2 versus a one sided or a 0 0 two sided alternative Then STA286 week 10 27

28 Test on Pairs of Variances 2 Suppose 1,..., X n are iid N μ independent of that 1 x, σ Y x 1,..., Yn 2 2 are iid N μ σ. 2 2 We are interested in testing H σ = σ versus a one sided or a two sided alternative Then X ( ) ( ) y, y 0 : x y STA286 week 10 28

29 Example STA286 week 10 29

Hypothesis Testing Introduction

Hypothesis Testing Introduction Hypothesis Testing Introduction Hypothesis: A conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. A hypothesis can

More information

Hypothesis Testing. Lecture 10

Hypothesis Testing. Lecture 10 Lecture 10 Hypothesis Testing A hypothesis is a conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. There are two types

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

Hypothesis Testing. Steps for a hypothesis test:

Hypothesis Testing. Steps for a hypothesis test: Hypothesis Testing Steps for a hypothesis test: 1. State the claim H 0 and the alternative, H a 2. Choose a significance level or use the given one. 3. Draw the sampling distribution based on the assumption

More information

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. 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.

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

Do the following using Mintab (1) Make a normal probability plot for each of the two curing times.

Do the following using Mintab (1) Make a normal probability plot for each of the two curing times. SMAM 314 Computer Assignment 4 1. An experiment was performed to determine the effect of curing time on the comprehensive strength of concrete blocks. Two independent random samples of 14 blocks were prepared

More information

1 SAMPLE SIGN TEST. Non-Parametric Univariate Tests: 1 Sample Sign Test 1. A non-parametric equivalent of the 1 SAMPLE T-TEST.

1 SAMPLE SIGN TEST. Non-Parametric Univariate Tests: 1 Sample Sign Test 1. A non-parametric equivalent of the 1 SAMPLE T-TEST. Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

More information

A) 0.1554 B) 0.0557 C) 0.0750 D) 0.0777

A) 0.1554 B) 0.0557 C) 0.0750 D) 0.0777 Math 210 - Exam 4 - Sample Exam 1) What is the p-value for testing H1: µ < 90 if the test statistic is t=-1.592 and n=8? A) 0.1554 B) 0.0557 C) 0.0750 D) 0.0777 2) The owner of a football team claims that

More information

Hypothesis Testing or How to Decide to Decide Edpsy 580

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

More information

AP Statistics Hypothesis Testing Chapter 9. Intro to Significance Tests

AP Statistics Hypothesis Testing Chapter 9. Intro to Significance Tests Intro to Significance Tests Name Hr For the following pairs, indicate whether they are legitimate hypotheses and why. 1. 2. 3. 4. For each situation, state the null and alternate hypothesis. (Define your

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

One-Sample t-test. Example 1: Mortgage Process Time. Problem. Data set. Data collection. Tools

One-Sample t-test. Example 1: Mortgage Process Time. Problem. Data set. Data collection. Tools One-Sample t-test Example 1: Mortgage Process Time Problem A faster loan processing time produces higher productivity and greater customer satisfaction. A financial services institution wants to establish

More information

Hypothesis testing for µ:

Hypothesis testing for µ: University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative

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

MAT X Hypothesis Testing - Part I

MAT X Hypothesis Testing - Part I MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

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

Power & Effect Size power Effect Size

Power & Effect Size power Effect Size Power & Effect Size Until recently, researchers were primarily concerned with controlling Type I errors (i.e. finding a difference when one does not truly exist). Although it is important to make sure

More information

Hypothesis Testing. Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University

Hypothesis Testing. Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University Hypothesis Testing Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University 1 AMU / Bon-Tech, LLC, Journi-Tech Corporation Copyright 2015 Learning Objectives Upon successful

More information

Note you select the cumulative probability to get the probability from 0 to 200. Cumulative Distribution Function. Binomial with n = 704 and p = 0.

Note you select the cumulative probability to get the probability from 0 to 200. Cumulative Distribution Function. Binomial with n = 704 and p = 0. GSB420 LAB3 Instructions and Assignment: Probability Distribution and Hypothesis Tests using Minitab 1. Binomial Distribution Example Ichiro Suzuki s hit. In 2004, Ichiro Suzuki of the Seattle Mariners

More information

Paired 2 Sample t-test

Paired 2 Sample t-test Variations of the t-test: Paired 2 Sample 1 Paired 2 Sample t-test Suppose we are interested in the effect of different sampling strategies on the quality of data we recover from archaeological field surveys.

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

Statistics 641 - EXAM II - 1999 through 2003

Statistics 641 - EXAM II - 1999 through 2003 Statistics 641 - EXAM II - 1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the

More information

CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING

CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized

More information

Sample Problems for Hypothesis Test

Sample Problems for Hypothesis Test Sample Problems for Hypothesis Test 1. The Bureau of Labor Statistics reported that the average yearly income of dentists in the year 2012 was $110,000. A sample of 81 dentists, which was taken in 2013,

More information

Case Study Call Centre Hypothesis Testing

Case Study Call Centre Hypothesis Testing is often thought of as an advanced Six Sigma tool but it is a very useful technique with many applications and in many cases it can be quite simple to use. Hypothesis tests are used to make comparisons

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

Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in

Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical

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

HYPOTHESIS TESTING: POWER OF THE TEST

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,

More information

Chapter 7. Section Introduction to Hypothesis Testing

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

More information

Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test.

Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test. Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations.

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

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

Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

More information

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

Regression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between

More information

CHAPTER 9 HYPOTHESIS TESTING

CHAPTER 9 HYPOTHESIS TESTING CHAPTER 9 HYPOTHESIS TESTING The TI-83 Plus and TI-84 Plus fully support hypothesis testing. Use the key, then highlight TESTS. The options used in Chapter 9 are given on the two screens. TESTING A SINGLE

More information

2 Sample t-test (unequal sample sizes and unequal variances)

2 Sample t-test (unequal sample sizes and unequal variances) Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing

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

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

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

More information

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

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

More information

Data Science and Statistics in Research: unlocking the power of your data

Data Science and Statistics in Research: unlocking the power of your data Data Science and Statistics in Research: unlocking the power of your data Session 2.4: Hypothesis testing II 1/ 25 OUTLINE Independent t-test Paired t-test Chi-squared test 2/ 25 Independent t-test 3/

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

Lecture Notes Module 1

Lecture Notes Module 1 Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

More information

University of Chicago Graduate School of Business. Business 41000: Business Statistics

University of Chicago Graduate School of Business. Business 41000: Business Statistics Name: University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas. 2. Throughout

More information

Chapter 1 Hypothesis Testing

Chapter 1 Hypothesis Testing Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,

More information

Chapter 11-12 1 Review

Chapter 11-12 1 Review Chapter 11-12 Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data

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

Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Two-Sample T-Tests Assuming Equal Variance (Enter Means) Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

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

CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS

CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS CHAPTER 11 CHI-SQUARE AND F DISTRIBUTIONS CHI-SQUARE TESTS OF INDEPENDENCE (SECTION 11.1 OF UNDERSTANDABLE STATISTICS) In chi-square tests of independence we use the hypotheses. H0: The variables are independent

More information

Hypothesis Testing. April 21, 2009

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

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

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

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

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

Chapter 9, Part A Hypothesis Tests. Learning objectives

Chapter 9, Part A Hypothesis Tests. Learning objectives Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population

More information

fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson

fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson Contents What Are These Demos About? How to Use These Demos If This Is Your First Time Using Fathom Tutorial: An Extended Example

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

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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.

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

Chapter 9: Hypothesis Testing Sections

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

More information

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

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

More information

AMS 5 HYPOTHESIS TESTING

AMS 5 HYPOTHESIS TESTING AMS 5 HYPOTHESIS TESTING Hypothesis Testing Was it due to chance, or something else? Decide between two hypotheses that are mutually exclusive on the basis of evidence from observations. Test of Significance

More information

Hypothesis Testing - II

Hypothesis Testing - II -3σ -2σ +σ +2σ +3σ Hypothesis Testing - II Lecture 9 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in Probability & Statistics in Engineering Today in P&S -3σ -2σ +σ +2σ +3σ Review: Hypothesis

More information

8 6 X 2 Test for a Variance or Standard Deviation

8 6 X 2 Test for a Variance or Standard Deviation Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96 1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

More information

AP STATISTICS (Warm-Up Exercises)

AP STATISTICS (Warm-Up Exercises) AP STATISTICS (Warm-Up Exercises) 1. Describe the distribution of ages in a city: 2. Graph a box plot on your calculator for the following test scores: {90, 80, 96, 54, 80, 95, 100, 75, 87, 62, 65, 85,

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

Review #2. Statistics

Review #2. Statistics Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of

More information

Sample Size Determination

Sample Size Determination Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

More information

p ˆ (sample mean and sample

p ˆ (sample mean and sample Chapter 6: Confidence Intervals and Hypothesis Testing When analyzing data, we can t just accept the sample mean or sample proportion as the official mean or proportion. When we estimate the statistics

More information

Estimation of σ 2, the variance of ɛ

Estimation of σ 2, the variance of ɛ Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated

More information

Chapter 9: Hypothesis Tests of a Single Population

Chapter 9: Hypothesis Tests of a Single Population Chapter 9: Hypothesis Tests of a Single Population Department of Mathematics Izmir University of Economics Week 12 2014-2015 Introduction In this chapter we will focus on Example developing hypothesis

More information

Elements of Hypothesis Testing (Summary from lecture notes)

Elements of Hypothesis Testing (Summary from lecture notes) Statistics-20090 MINITAB - Lab 1 Large Sample Tests of Hypothesis About a Population Mean We use hypothesis tests to make an inference about some population parameter of interest, for example the mean

More information

General Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test

General Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics

More information

Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test)

Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test) Hypothesis Test for Mean Using Given Data (Standard Deviation Known-z-test) A hypothesis test is conducted when trying to find out if a claim is true or not. And if the claim is true, is it significant.

More information

Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections Chapter 9: Hypothesis Testing Sections 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 Means of Two Normal

More information

Introductory Statistics Hypothesis Testing Review( Critical Value Approach)

Introductory Statistics Hypothesis Testing Review( Critical Value Approach) MULTIPLE CHOICE. Introductory Statistics Hypothesis Testing Review( Critical Value Approach) Determine the critical value(s) for a one-mean z-test. 1) A left-tailed test with α = 0.02. A) ±1.96 B) ±2.054

More information

Tests of Hypotheses Using Statistics

Tests of Hypotheses Using Statistics Tests of Hypotheses Using Statistics Adam Massey and Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract We present the various methods of hypothesis testing that one

More information

12.8 ONE-SIDED AND TWO-SIDED HYPOTHESIS TESTING

12.8 ONE-SIDED AND TWO-SIDED HYPOTHESIS TESTING 12.8 ONE-SIDED AND TWO-SIDED HYPOTHESIS TESTING In Section 12.2 we considered the normal-distribution-based z test of the population mean null hypothesis H : 98. F 0 We based our decision to reject H0

More information

Construct a scatterplot for the given data. 2) x Answer:

Construct a scatterplot for the given data. 2) x Answer: Review for Test 5 STA 2023 spr 2014 Name Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents

More information

3. Nonparametric methods

3. Nonparametric methods 3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

More information

STAT 350 Practice Final Exam Solution (Spring 2015)

STAT 350 Practice Final Exam Solution (Spring 2015) PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

More information

Hypothesis Testing. Concept of Hypothesis Testing

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

More information

Two Related Samples t Test

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

More information

Lecture Outline. Hypothesis Testing. Simple vs. Composite Testing. Stat 111. Hypothesis Testing Framework

Lecture Outline. Hypothesis Testing. Simple vs. Composite Testing. Stat 111. Hypothesis Testing Framework Stat 111 Lecture Outline Lecture 14: Intro to Hypothesis Testing Sections 9.1-9.3 in DeGroot 1 Hypothesis Testing Consider a statistical problem involving a parameter θ whose value is unknown but must

More information

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

More information

Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

More information

The Basics of a Hypothesis Test

The Basics of a Hypothesis Test Overview The Basics of a Test Dr Tom Ilvento Department of Food and Resource Economics Alternative way to make inferences from a sample to the Population is via a Test A hypothesis test is based upon A

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

Chapter III. Testing Hypotheses

Chapter III. Testing Hypotheses Chapter III Testing Hypotheses R (Introduction) A statistical hypothesis is an assumption about a population parameter This assumption may or may not be true The best way to determine whether a statistical

More information

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4) Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

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

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

Solutions to Worksheet on Hypothesis Tests

Solutions to Worksheet on Hypothesis Tests s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality

More information

Measuring the Power of a Test

Measuring the Power of a Test Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection

More information

Sampling and Hypothesis Testing

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

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