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

Size: px
Start display at page:

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

Transcription

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

2 Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis H 0 contains a statement of equality such as, = or. Alternative hypothesis H a contains a statement of inequality such as <, or > If I am false, you are true Complementary Statements If I am false, you are true Accept or reject null, never prove null is true

3 Writing Hypotheses Write the claim about the population. Write its complement. Either hypothesis, can represent the claim. Example: A hospital claims its ambulance response time is less than 10 minutes. claim Example: A consumer magazine claims the proportion of cell phone calls made during evenings and weekends is at most 60%. claim H 0 always contains the = condition

4 Hypothesis Test Strategy 1) Assume the equality condition in the null hypothesis is true, regardless of whether the claim is represented by the null or alternative hypothesis. 2) Collect data from a random sample taken from the population and calculate the necessary sample statistics. 3a) If the sample statistic has a low probability of being drawn from a population in which the null hypothesis is true, you will reject H 0. (i.e. you will support the alternative hypothesis.) 3b) If the probability is not low enough, fail to reject H 0.

5 Errors and Level of Significance Decision Do not reject H 0 Reject H 0 Actual Truth of H 0 H 0 True Correct Decision Type I Error H 0 False Type II Error Correct Decision Type I Error: Null hypothesis is true but reject it. Level of significance, (e.g. 0.01, 0.05, 0.10) Maximum probability of committing a Type I Error. Type II Error: Null hypothesis is false but accept it. Probability of Type II Error 1-β (power of test) Innocent until proven guilty Beyond reasonable doubt

6 Types of Hypothesis Tests H a is more probable One-tail test Right-tail test H a is more probable Left-tail test H a is more probable Two-tail test

7 The P-value is the probability of obtaining a sample statistic with a value as or more extreme than the one determined by the sample data. P-value = indicated area Area in left tail P-values Area in right tail z For a left tail test If z is negative, twice the area in the left tail z z For a two-tail test z For a right tail test If z is positive, twice the area in the right tail

8 Finding P-values: 1-tail Test The test statistic for a right-tail test is z = Find the P-value. Area in right tail z = 1.56 The area to the right of z = 1.56 is = The P-value is

9 Finding P-values: 2-tail Test The test statistic for a two-tail test is z = Find the corresponding P-value. z = 2.63 The area to the left of z = 2.63 is The P-value is 2(0.0043) =

10 Test Decisions with P-values The decision about whether there is enough evidence to reject the null hypothesis can be made by comparing the P-value to the value of, the arbitrary level of significance of the test. If reject the null hypothesis. If fail to reject the null hypothesis.

11 Method I: Using P-values The P-value of a hypothesis test is Make your decision at the 0.05 level of significance. Compare the P-value to. Since > 0.05, fail to reject H 0. If P = , what is your decision if 1) Since, reject H 0. 2) Since > 0.01, fail to reject H 0.

12 Interpreting the Decision Claim Claim is H 0 Claim is H a Decision Reject H 0 Fail to reject H 0 There is enough evidence to reject the claim. There is not enough evidence to reject the claim. There is enough evidence to support the claim. There is not enough evidence to support the claim.

13 Steps in a Hypothesis Test 1. Write the null and alternative hypothesis. Write H 0 and H a as mathematical statements (H 0 always contains the = symbol). 2. State the level of significance. Maximum probability of rejecting the null hypothesis when it is true. (Making a Type I error.) 3. Identify the sampling distribution. Sampling distribution is the distribution for the test statistic assuming that H 0 is true and that the experiment is repeated an infinite number of times.

14 Steps in a Hypothesis Test 4. Find the test statistic and standardize it. Perform calculations to standardize your sample statistic. 5. Calculate the P-value for the test statistic. This is the probability of obtaining your test statistic or one that is more extreme from the sampling distribution.

15 6. Make your decision. If the P-value is less than α, reject H 0. If the P value is greater than α, fail to reject H Interpret your decision. If the claim is the null hypothesis, you will either reject the claim or determine there is not enough evidence to reject the claim. If the claim is the alternative hypothesis, you will either support the claim or determine there is not enough evidence to support the claim.

16 Section 7.2 Hypothesis Testing for the Mean (n 30)

17 The z-test for a Mean The z-test is a statistical test for a population mean. The z-test can be used: (1) if the population is normal and σ is known or (2) when the sample size, n, is at least 30. Test statistic is the sample mean test statistic is z. and the standardized When n ³ 30, use s in place of.

18 The z-test for a Mean (P-value) Example: A cereal company claims the mean sodium content in one serving of its cereal is no more than 230 mg. You work for a national health service and are asked to test this claim. You find that a random sample of 52 servings has a mean sodium content of 232 mg and a standard deviation of 10 mg. At α= 0.05, do you have enough evidence to reject the company s claim? 1. Write the null and alternative hypothesis. 2. State the level of significance. = Determine the sampling distribution. Since the sample size is at least 30, the sampling distribution is normal (Central Limit Theorem, Chptr. 5).

19 4. Find the test statistic and standardize it. n = Calculate the P-value for the test statistic. Since this is a right-tail test, the P-value is the area found to the right of z = 1.44 in the normal distribution. From the table P = , P = Test statistic s = 10 (n>30) Area in right tail z = 1.44

20 6. Make your decision. Compare the P-value to. Since > 0.05, fail to reject H Interpret your decision. There is not enough evidence to reject the claim that the mean sodium content of one serving of its cereal is no more than 230 mg.

21 Method II: Rejection Regions Sampling distribution for Rejection Region z z 0 Critical Value z 0 Rejection region is the range of values for which the null hypothesis is not probable. Always in the direction of the alternative hypothesis. Its area is equal to. A critical value separates rejection region from the nonrejection region.

22 Critical Values Rejection region Rejection region z 0 Find z 0 for a left-tail test with =.01. z 0 = 2.33 Rejection region z 0 Find z 0 for a right-tail test with =.05. Rejection region z 0 = z 0 z 0 Find z 0 and z 0 for a two-tail test with α =.01 z 0 = and z 0 = 2.575

23 Using Critical Values to Make Test Decisions 1. Write the null and alternative hypothesis. Write H 0 and H a as mathematical statements. Remember H 0 always contains the = symbol. 2. State the level of significance. The maximum probability of rejecting the null hypothesis when it is actually true (Type I Error.) 3. Identify the sampling distribution. The distribution for the test statistic assuming that the equality condition in H 0 is true and that the experiment is repeated an infinite number of times.

24 4. Find the critical value. Rejection Region z 0 6. Find the test statistic. 5. Find the rejection region. Critical value separates rejection region of the sampling distribution from the non-rejection region. Area of the critical region is equal to the level of significance of the test. Perform the calculations to standardize your sample statistic.

25 7. Make your decision. If the test statistic falls in the critical region, reject H 0. Otherwise, fail to reject H Interpret your decision. If the claim is the null hypothesis, you will either reject the claim or determine there is not enough evidence to reject the claim. If the claim is the alternative hypothesis, you will either support the claim or determine there is not enough evidence to support the claim.

26 The z-test for a Mean Example: A cereal company claims the mean sodium content in one serving of its cereal is no more than 230 mg. You work for a national health service and are asked to test this claim. You find that a random sample of 52 servings has a mean sodium content of 232 mg and a standard deviation of 10 mg. At α = 0.05, do you have enough evidence to reject the company s claim? 1. Write the null and alternative hypothesis. 2. State the level of significance. = Determine the sampling distribution. Since the sample size is at least 30, the sampling distribution is normal.

27 Since H a contains the > symbol, this is a right-tail test. z 0 Rejection region 4. Find the critical value , α = 0.05, cum. area = Find the rejection region. 6. Find the test statistic and standardize it. n = 52 = 232 s = Make your decision. Test statistic does not fall in the rejection region, so fail to reject H 0 8. Interpret your decision. z = 1.44 There is not enough evidence to reject the company s claim that there is at most 230 mg of sodium in one serving of its cereal.

28 Using the P-value of a Test to Compare Areas Rejection area 0.05 z 0 z Test statistic = 0.05 z 0 = z = 1.23 P = For a P-value decision, compare areas. If reject H 0. If fail to reject H 0. For a critical value decision, decide if z is in the rejection region If z is in the rejection region, reject H 0. If z is not in the rejection region, fail to reject H 0. *Decision same, critical values & z-scores, P-values & areas*

29 Section 7.3 Hypothesis Testing for the Mean (n < 30)

30 The t Sampling Distribution Find the critical value t 0 for a left-tailed test given α = 0.01 and n = 18. Area in left tail d.f. = 18 1 = 17 t 0 = t 0 Find the critical values t 0 and t 0 for a two-tailed test given = 0.05 and n = 11. d.f. = 11 1 = 10 t 0 = and t 0 = t 0 t 0 *If > 30 d.f. then ~normal*

31 Testing Small Sample Example: A university says the mean number of classroom hours per week for full-time faculty is A random sample of classroom hours for full-time faculty for one week is listed below. You work for a student organization and are asked to test this claim. At α= 0.01, do you have enough evidence to reject the university s claim? Write the null and alternative hypothesis 2. State the level of significance = Determine the sampling distribution Since the sample size is 8, the sampling distribution is a t-distribution with 8 1 = 7 d.f.

32 Since H a contains the symbol, this is a two-tail test. 4. Find the critical values. t 0 t Find the test statistic and standardize it n = 8 = s = Make your decision. t = 1.08 does not fall in the rejection region, so fail to reject H 0 at = Interpret your decision. 5. Find the rejection region. There is not enough evidence to reject the university s claim that faculty spend a mean of 11 classroom hours.

33 Section 7.4 Hypothesis Testing for Proportions

34 Test for Proportions p is the population proportion of successes. The test statistic is. (the proportion of sample successes) If and the sampling distribution for is normal. The standardized test statistic is:

35 Test for Proportions Example: A communications industry spokesperson claims that over 40% of Americans either own a cellular phone or have a family member who does. In a random survey of 1,036 Americans, 456 said they or a family member owned a cellular phone. Test the spokesperson s claim at α = What can you conclude? 1. Write the null and alternative hypothesis. 2. State the level of significance. = 0.05

36 3. Determine the sampling distribution. 1036(.40) > 5 and 1036(.60) > 5. The sampling distribution is normal. Rejection region 4. Find the critical value. 5. Find the rejection region Find the test statistic and standardize it. n = 1036 x = Make your decision. What is p-value? z = 2.63 falls in the rejection region, so reject H 0 8. Interpret your decision. There is enough evidence to support the claim that over 40% of Americans own a cell phone or have a family member who does.

37 Section 7.5 Hypothesis Testing for Variance and Standard Deviation

38 Critical Values for s 2 is the test statistic for the population variance. Its sampling distribution is a χ 2 distribution with n 1 d.f. Find a χ 2 0 critical value for a left-tail test when n = 17 and = χ 2 0 = Find critical values χ 2 0 for a two-tailed test when n = 12, = χ 2 L = and χ 2 R = The standardized test statistic is

39 Test for Example: A state school administrator says that the standard deviation of test scores for 8th grade students who took a life-science assessment test is less than 30. You work for the administrator and are asked to test this claim. You find that a random sample of 10 tests has a standard deviation of At α = 0.01, do you have enough evidence to support the administrator s claim? Assume test scores are normally distributed. 1. Write the null and alternative hypothesis. 2. State the level of significance. = Determine the sampling distribution. The sampling distribution is χ 2 with 10 1 = 9 d.f. 1- or 2-tailed test?

40 4. Find the critical value Find the test statistic. n = 10 s = Make your decision. 5. Find the rejection region. χ 2 = does not fall in the rejection region, so fail to reject H 0 8. Interpret your decision. There is not enough evidence to support the administrator s claim that the standard deviation is less than 30.

Hypothesis Testing --- One Mean

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

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

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

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

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion

Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion Learning Objectives Upon successful completion of Chapter 8, you will be able to: Understand terms. State the null and alternative

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

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

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

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

Hypothesis Testing: Two Means, Paired Data, Two Proportions

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

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

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

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

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

Chapter 7 TEST OF HYPOTHESIS

Chapter 7 TEST OF HYPOTHESIS Chapter 7 TEST OF HYPOTHESIS In a certain perspective, we can view hypothesis testing just like a jury in a court trial. In a jury trial, the null hypothesis is similar to the jury making a decision of

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

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

Hypothesis Testing. Reminder of Inferential Statistics. Hypothesis Testing: Introduction

Hypothesis Testing. Reminder of Inferential Statistics. Hypothesis Testing: Introduction Hypothesis Testing PSY 360 Introduction to Statistics for the Behavioral Sciences Reminder of Inferential Statistics All inferential statistics have the following in common: Use of some descriptive statistic

More information

Hypothesis Testing. Hypothesis Testing

Hypothesis Testing. Hypothesis Testing Hypothesis Testing Daniel A. Menascé Department of Computer Science George Mason University 1 Hypothesis Testing Purpose: make inferences about a population parameter by analyzing differences between observed

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

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

Study Guide for the Final Exam

Study Guide for the Final Exam Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

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

In the past, the increase in the price of gasoline could be attributed to major national or global

In the past, the increase in the price of gasoline could be attributed to major national or global Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

More information

Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1) Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

More information

Lesson 9 Hypothesis Testing

Lesson 9 Hypothesis Testing Lesson 9 Hypothesis Testing Outline Logic for Hypothesis Testing Critical Value Alpha (α) -level.05 -level.01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis

More information

Name: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%.

Name: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%. Chapter 7-8 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. Please indicate which program

More information

Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!

Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice! Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!) Part A - Multiple Choice Indicate the best choice

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

Stats Review Chapters 9-10

Stats Review Chapters 9-10 Stats Review Chapters 9-10 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

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

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

More information

Chapter 7: One-Sample Inference

Chapter 7: One-Sample Inference Chapter 7: One-Sample Inference Now that you have all this information about descriptive statistics and probabilities, it is time to start inferential statistics. There are two branches of inferential

More information

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

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7. THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

More information

Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test.

Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. HYPOTHESIS TESTING Learning Objectives Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. Know how to perform a hypothesis test

More information

Final Exam Practice Problem Answers

Final Exam Practice Problem Answers Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

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

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

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

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

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

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

Testing Hypotheses About Proportions

Testing Hypotheses About Proportions 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

More information

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

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

Statistics Review PSY379

Statistics Review PSY379 Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

More information

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on

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

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Ch. 10 Chi SquareTests and the F-Distribution 10.1 Goodness of Fit 1 Find Expected Frequencies Provide an appropriate response. 1) The frequency distribution shows the ages for a sample of 100 employees.

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

Introduction. Statistics Toolbox

Introduction. Statistics Toolbox Introduction A hypothesis test is a procedure for determining if an assertion about a characteristic of a population is reasonable. For example, suppose that someone says that the average price of a gallon

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

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

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

Pearson's Correlation Tests

Pearson's Correlation Tests Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation

More information

How To Test For Significance On A Data Set

How To Test For Significance On A Data Set 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

Comparing Means in Two Populations

Comparing Means in Two Populations Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

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

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

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

Module 2 Probability and Statistics

Module 2 Probability and Statistics Module 2 Probability and Statistics BASIC CONCEPTS Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The standard deviation of a standard normal distribution

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

z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

More information

Using Excel for inferential statistics

Using Excel for inferential statistics FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

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

DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript

DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript JENNIFER ANN MORROW: Welcome to "Introduction to Hypothesis Testing." My name is Dr. Jennifer Ann Morrow. In today's demonstration,

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

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

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

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

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

Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics

Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Jan Krhovják Basic Idea Behind the Statistical Tests Generated random sequences properties as sample drawn from uniform/rectangular

More information

Section 12 Part 2. Chi-square test

Section 12 Part 2. Chi-square test Section 12 Part 2 Chi-square test McNemar s Test Section 12 Part 2 Overview Section 12, Part 1 covered two inference methods for categorical data from 2 groups Confidence Intervals for the difference of

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

Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

More information

Chapter 4: Statistical Hypothesis Testing

Chapter 4: Statistical Hypothesis Testing Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics - Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin

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

Opgaven Onderzoeksmethoden, Onderdeel Statistiek

Opgaven Onderzoeksmethoden, Onderdeel Statistiek Opgaven Onderzoeksmethoden, Onderdeel Statistiek 1. What is the measurement scale of the following variables? a Shoe size b Religion c Car brand d Score in a tennis game e Number of work hours per week

More information

ELEMENTARY STATISTICS

ELEMENTARY STATISTICS ELEMENTARY STATISTICS Study Guide Dr. Shinemin Lin Table of Contents 1. Introduction to Statistics. Descriptive Statistics 3. Probabilities and Standard Normal Distribution 4. Estimates and Sample Sizes

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

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. Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A researcher for an airline interviews all of the passengers on five randomly

More information

Normal distributions in SPSS

Normal distributions in SPSS Normal distributions in SPSS Bro. David E. Brown, BYU Idaho Department of Mathematics February 2, 2012 1 Calculating probabilities and percents from measurements: The CDF.NORMAL command 1. Go to the Variable

More information

Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

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. 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

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

More information

Chi-square test Fisher s Exact test

Chi-square test Fisher s Exact test Lesson 1 Chi-square test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions

More information

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application

More information

Difference of Means and ANOVA Problems

Difference of Means and ANOVA Problems Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly

More information

Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015 Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

6.4 Normal Distribution

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

More information

Tests for One Proportion

Tests for One Proportion Chapter 100 Tests for One Proportion Introduction The One-Sample Proportion Test is used to assess whether a population proportion (P1) is significantly different from a hypothesized value (P0). This is

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

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

Test Positive True Positive False Positive. Test Negative False Negative True Negative. Figure 5-1: 2 x 2 Contingency Table

Test Positive True Positive False Positive. Test Negative False Negative True Negative. Figure 5-1: 2 x 2 Contingency Table ANALYSIS OF DISCRT VARIABLS / 5 CHAPTR FIV ANALYSIS OF DISCRT VARIABLS Discrete variables are those which can only assume certain fixed values. xamples include outcome variables with results such as live

More information

Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

More information

1.5 Oneway Analysis of Variance

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

More information