Extending Hypothesis Testing. p-values & confidence intervals

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Extending Hypothesis Testing. p-values & confidence intervals"

Transcription

1 Extending Hypothesis Testing p-values & confidence intervals

2 So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by using a statistic and an associated measure of the probability of observing our statistic, given the current state or null hypothesis.

3 Next: We will use the p-value to: make inferences about the population assign a level of confidence

4 Review the Steps Phase 1: State the Question 1. Evaluate and describe the data 2. Review assumptions 3. State the question-in the form of hypotheses Phase 2: Decide How to Answer the Question 4. Decide on a summary number-a statistic-that reflects the question 5. How could random variation affect that statistic? 6. State a decision rule, using the statistic, to answer the question

5 Detailed Steps (cont) Phase 3: Answer the Question 7. Calculate the statistic 8. Make a statistical decision 9. State the substantive conclusion Phase 4: Communicate the Answer to the Question 10. Document our understanding with text, tables, or figures

6 Clarify & Generalize the steps Step 2: Assumptions: Representative: Is the observed data representative of the population? Independence: Are the observations (responses of interest) independent? Size: Is the size of the sample large enough to make generalizations to the population at large?

7 Size Assumption So, how large is large enough? Rule-of-thumb: N large enough to expect to see five of each of the two outcomes Both of the following must be true: p 0 n > 5 (1 p 0 ) n > 5

8 In the CPR study p 0 = 0.06 and n = 278, so: p 0 n = = > 5 (1 p 0 ) n = (1 0.06) 278 = > 5

9 Common Mistakes Using the observed proportion rather than the hypothesized proportion Compare the observed number of events of interest to five Why Not? We always operate under the assumption that the null hypothesis is true so use the null proportion!

10 Step 2 is particularly important: If the data do not meet the assumptions, then the statistical tests applied to test the hypothesis will not be valid Only proceed to steps 3 10 if the assumptions are met

11 Step 4 For the CPR example, we used a specific statistic, the proportion p The statistic and decision rules can be more generally defined and applied to all situations for testing a proportion.

12 Review CPR Simulation H0: The population survival proportion is 0.06 or less if the observed proportion p (x = 23 survivors or less). HA: The population survival proportion is larger than 0.06 if the observed proportion p > (x = 24 or more survivors).

13 Recall for the CPR simulations, the results looked similar to a normal distribution

14 Applied vs. Theoretical The smooth curve is the theoretical distribution of a normal curve under the null hypothesis Centered on the population value (p0 = 0.06) with proportions farther away from this center being less likely to occur Use the theoretical distribution to determine if our observed proportion is different from our assumed proportion

15 General Test Statistic observed p - hypothesized p standard error of the hypothesized p z = pˆ p0 p0 0 n ( 1 p )

16 observed proportion assumed proportion z = standard error of p 0 p pˆ p0 ( 1 ) 0 p0 n

17 Why p 0? Calculate the test statistic under the assumption that the null hypothesis is true. We are not concerned about how the variability of the observed data will affect our hypothesis testing result We believe the null hypothesis and the variability in the observed data should be assumed to be the same as the variability under the null hypothesis.

18 Z-score Using the z-score allows us to use a decision rule based on the standard normal distribution, rather than the proportion, p. The standard normal distribution ~N(0,1) The cut-off for the decision rule does not change for different values of p, n, and p 0.

19 For an a = 0.05, the z value is 1.645, ( 5% of the N(0,1) values are greater than 1.645)

20 General Decision Rule H 0 : proportion p p 0. Choose this if z z critical and p-value α. H A : proportion p > p 0. Choose this if z > z critical and p-value < α.

21 Clarify Steps w/ CPR Example 1. Evaluate and describe the data We observed n = 278 CPR patients who received instructions by phone, of whom x = 29 survived to hospital discharge. The characteristic of interest is survival proportion, p = 29/278 = The intent is to compare the outcomes in this study to a = 0.06 survival rate presumed to be typical.

22 2. Review assumptions There are three assumptions: Representativeness: From the design of the study, it is clear that subjects are representative of cardiacarrest victims in cities with a quick-response emergency system. Independence: The response of one cardiac-arrest victim does not depend on the response of others. The subjects are independent. Sufficient size: Since, n = = > 5, and (1 ) n = (1 0.06) 278 = > 5, this assumption is valid.

23 3. State the question in the form of hypotheses The intent is to show that phone-cpr is superior to doing nothing. Thus, the alternative hypothesis is that there are higher than 6% survival rates: H 0 : p 0.06 H A : p > 0.06.

24 4. Decide on a summary number a statistic that reflects the question We ll use the z-score: z = pˆ p0 ( 1 p ) p0 0 n

25 5. How could random variation affect that statistic? If the null hypothesis is true, then z is zero. Since the assumptions are met, z is normally distributed. Large values of z reflect higher survival proportions and thus favor the alternative hypothesis.

26 6. State a decision rule, using the statistic, to answer the question General Choose to believe (at α = 0.05): H 0 : Choose this if p-value α H A : Choose this if p-value < α For CPR Example, for an α = 0.05: H 0 : p 0.06 Choose this if p-value 0.05 H A : p > Choose this if p-value < 0.05

27 7. Calculate the statistic z pˆ p = = p0 0 n 278 ( 1 p ) 0.06( ) = = Recall that a z-value to the right of 3 is unlikely. In fact, the associated p-value is p = (we ll talk about calculating p-values later).

28 8. Make a statistical decision Reject the null hypothesis since p-value < The observed value of the summary statistic is larger than what is expected by chance alone.

29 9. State the substantive conclusion We conclude that the survival proportion is larger than 0.06.

30 10. Document our understanding with text, tables, or figures Does dispatcher-instructed bystander-administered CPR improve the chances of survival? Without this intervention it is presumed that the survival probability will be unchanged (at 6%). From this study, which used n = 278 patients, we observed p = (x = 29 survived until hospital discharge). The observed rate was compared to the hypothesized rate using the z test statistic. We reject the hypothesis p 0.06 in favor of the alternative hypothesis that the survival probability is larger than 6% (z = 3.09, p-value = ).

31 Universal Decision Rule H 0 : null-hypothesis. Choose this if p-value α (usually 0.05). H A : alternative-hypothesis. Choose this if p-value < α (usually 0.05).

32 How do we determine p-values? p-values can be determined from standard normal tables, such as Table A.1 in the Statistical Sleuth.,715 Tedious and you need to be careful what the table gives as the proportion it could be the opposite of what you are looking for! Use a calculator

33

34 Calculation note: Software might return a p-value as 0 or not possible Determine the number of decimal places the calculator reports (when it will return a 0 value) Then report p < or p <

35 Confidence Intervals Often, researchers want to use a less rigid approach to hypothesis testing by estimating the parameter and placing upper and lower bounds (or limits) on the estimate. The interval is called a confidence interval.

36 The confidence interval approach allows us to make statements about a population parameter without referring to hypotheses Also gives a range of values that reflects our degree of certainty.

37 Definitions Inference: An inference is a conclusion that patterns observed in the data are present in the broader population. Statistical Inference: A statistical inference is an inference justified by a probability model (distribution) linking the data to the broader population. Parameter: A parameter is an unknown numerical value describing a feature of a distribution.

38 More Definitions Statistic: A statistic is any value that can be calculated from the observed data. Estimate: An estimate is a statistic used as a guess (or estimate) of a parameter.

39 General Definition estimate ± (reliability coefficient) (standard error) Estimating a parameter with an interval involves three components: The point estimate. The standard error of the estimate. This describes how much variability we expect. A reliability coefficient. This describes our degree of certainty.

40 Estimate Calculate the observed proportion: p = x / n In the CPR case p =

41 Standard Error The standard error we use here is different from that used in hypothesis testing. Recall that earlier we were in the mind-set of hypothesis testing. Here we are not doing hypothesis testing here. We re just estimating a confidence interval based upon the observed data

42 Standard Error of p-hat SE p ˆ = pˆ ( 1 pˆ) n Note that the standard error of the estimate gets smaller as n gets larger. We expect less variability in an estimate if we use more data to make the estimate.

43 CPR Example For n = 278 and = 0.104, the associated standard error is: SE p ˆ pˆ ˆ = = n ( 1 p) 0.104( ) 278 =

44 Reliability coefficient The reliability coefficient reflects how sure we want to be: 95% sure 90% sure 99% sure Based on the standard normal for those proportions

45

46 Reliability Coefficients Commonly Used For 90% confidence, use z = For 95% confidence, use z = For 99% confidence, use z =

47 Confidence Interval pˆ z SE p ˆ ± ( 1 α 2) ( ) ± ( 0.068, 0.140)

48 Using a sentence: In the first case study there were 29 survivors (out of n = 278 studied) yielding a 95% confidence interval on the population survival proportion of [0.068, 0.140]. That is, We re 95% confident that the survival proportion is between and

49 Is there a 100% CI? Yes, it is [0,1] But this is a silly answer and doesn t make a conclusive statement about the population estimate. This is the same for all proportions!

50 Using the 10 Steps The Changes to the 10 Steps are minimal: 3. State the question (CI) 4. Decide on a summary statistic that reflects the question (CI formula). 5. How could random variation affect that statistic? (If the assumptions are met, then this interval will cover the population proportion 95% of the time )

51 6. Determine the reliability coefficient and standard error to be used in the CI 7. Calculate the interval 8. Compare the interval to comparison value (If there is a comparison value, does the interval include it?)

52 9. State the substantive conclusion: Something like: We estimate the population proportion of to be [lower, upper] with 95% confidence perhaps which does not include the hypothesized value of. 10. Document our understanding with text

53 Summary We have looked at several methods to assess and describe data and underlying populations. We can use simulations, z-scores, p-values, or confidence intervals about an estimate to make conclusions about observed data and broader populations. Next, we ll look at sample size and precision of estimates and the design of a study to estimate population proportions.

Steps to Answering the Questions with Data

Steps to Answering the Questions with Data Hypothesis Testing Steps to Answering the Questions with Data How does science advance knowledge? How do we answer questions about the world using observations? Generally, science forms a question and

More information

Chapter 8. Hypothesis Testing

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

More information

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

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

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

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

AP Statistics 2002 Scoring Guidelines

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

More information

Water Quality Problem. Hypothesis Testing of Means. Water Quality Example. Water Quality Example. Water quality example. Water Quality Example

Water Quality Problem. Hypothesis Testing of Means. Water Quality Example. Water Quality Example. Water quality example. Water Quality Example Water Quality Problem Hypothesis Testing of Means Dr. Tom Ilvento FREC 408 Suppose I am concerned about the quality of drinking water for people who use wells in a particular geographic area I will test

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 19: Confidence Intervals for Percentages Tessa L. Childers-Day UC Berkeley 28 July 2014 By the end of this lecture... You will be able to: Estimate

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

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

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

Homework 6 Solutions

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

More information

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

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

More information

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

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

Point and Interval Estimates

Point and Interval Estimates Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

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

More information

Homework 5 Solutions

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

More information

Chapter 7 Part 2. Hypothesis testing Power

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

More information

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

Announcements. Unit 4: Inference for numerical variables Lecture 1: Bootstrap, paired, and two sample. Rent in Durham.

Announcements. Unit 4: Inference for numerical variables Lecture 1: Bootstrap, paired, and two sample. Rent in Durham. Announcements Announcements Unit 4: Inference for numerical variables Lecture 1: Bootstrap, paired, and two sample Statistics 101 Mine Çetinkaya-Rundel February 26, 2013 Extra credit due Thursday at the

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

Hypothesis testing S2

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

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 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

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

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check

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

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

Basic concepts and introduction to statistical inference

Basic concepts and introduction to statistical inference Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence

More information

Chapter Five: Paired Samples Methods 1/38

Chapter Five: Paired Samples Methods 1/38 Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery

More information

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

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

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

6. Statistical Inference: Significance Tests

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

More information

How to Conduct a Hypothesis Test

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

More information

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

AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

More information

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ 1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

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

Hypothesis Testing: p-value

Hypothesis Testing: p-value STAT 101 Dr. Kari Lock Morgan Paul the Octopus Hypothesis Testing: SECTION 4.2 andomization distribution http://www.youtube.com/watch?v=3esgpumj9e Hypotheses In 2008, Paul the Octopus predicted 8 World

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

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

AP Statistics 2009 Scoring Guidelines Form B

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

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

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

Chapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited Chapter 8 Professor April 20, 2014 In Chapter 8, we continue our study of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample

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

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

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

More information

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

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

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

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

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

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

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

More information

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

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

Statistical Foundations:

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

More information

Unit 29 Chi-Square Goodness-of-Fit Test

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

More information

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

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

More information

The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker

The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker HYPOTHESIS TESTING PHILOSOPHY 1 The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker Question: So I'm hypothesis testing. What's the hypothesis I'm testing? Answer: When you're

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

E205 Final: Version B

E205 Final: Version B Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

More information

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

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

More information

Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA. Analysis Of Variance Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

More information

BIOSTATISTICS QUIZ ANSWERS

BIOSTATISTICS QUIZ ANSWERS BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never

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

Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

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

The Paired t-test and Hypothesis Testing. John McGready Johns Hopkins University

The Paired t-test and Hypothesis Testing. John McGready Johns Hopkins University This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

Linear Regression with One Regressor

Linear Regression with One Regressor Linear Regression with One Regressor Michael Ash Lecture 10 Analogy to the Mean True parameter µ Y β 0 and β 1 Meaning Central tendency Intercept and slope E(Y ) E(Y X ) = β 0 + β 1 X Data Y i (X i, Y

More information

Example Hypotheses. Chapter 8-2: Basics of Hypothesis Testing. A newspaper headline makes the claim: Most workers get their jobs through networking

Example Hypotheses. Chapter 8-2: Basics of Hypothesis Testing. A newspaper headline makes the claim: Most workers get their jobs through networking Chapter 8-2: Basics of Hypothesis Testing Two main activities in statistical inference are using sample data to: 1. estimate a population parameter forming confidence intervals 2. test a hypothesis or

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

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

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

More information

Hypothesis tests, confidence intervals, and bootstrapping

Hypothesis tests, confidence intervals, and bootstrapping Hypothesis tests, confidence intervals, and bootstrapping Business Statistics 41000 Fall 2015 1 Topics 1. Hypothesis tests Testing a mean: H0 : µ = µ 0 Testing a proportion: H0 : p = p 0 Testing a difference

More information

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE 11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which

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

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

Hypothesis Testing. Hypothesis Testing CS 700

Hypothesis Testing. Hypothesis Testing CS 700 Hypothesis Testing CS 700 1 Hypothesis Testing! Purpose: make inferences about a population parameter by analyzing differences between observed sample statistics and the results one expects to obtain if

More information

Testing: is my coin fair?

Testing: is my coin fair? Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

More information

CHAPTER 15: Tests of Significance: The Basics

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

More information

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4)

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Assume that the change in daily closing prices for stocks on the New York Stock Exchange is a random

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

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

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

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

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

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

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

More information

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

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

More information

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

Unit 21 Student s t Distribution in Hypotheses Testing

Unit 21 Student s t Distribution in Hypotheses Testing Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between

More information

Multiple random variables

Multiple random variables Multiple random variables Multiple random variables We essentially always consider multiple random variables at once. The key concepts: Joint, conditional and marginal distributions, and independence of

More information

2011 # AP Exam Solutions 2011 # # # #1 5/16/2011

2011 # AP Exam Solutions 2011 # # # #1 5/16/2011 2011 AP Exam Solutions 1. A professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength. (a) Speed is measured by the time required

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

2 Precision-based sample size calculations

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

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

Sample Size Planning, Calculation, and Justification

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

More information

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

Analysis of Variance ANOVA

Analysis of Variance ANOVA Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

More information

Inference, Sampling, and Confidence. Inference. Inference

Inference, Sampling, and Confidence. Inference. Inference Inference, Sampling, and Confidence Inference defined Sampling Statistics and parameters Sampling distribution Confidence and standard error Estimation Precision and accuracy Estimating sample size 1 Inference

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

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