Section 5.4: Hypothesis testing for μ
|
|
- Allyson Maxwell
- 5 years ago
- Views:
Transcription
1 Section 5.4: Hypothesis testing for μ Possible claims or hypotheses: Ball bearings have μ = 1 cm Medicine decreases blood pressure For testing hypotheses, we set up a null (H 0 ) and alternative (H a ) hypothesis. Maybe H 0 μ = 1 cm H a μ 1 1 cm ball bearing diameters Maybe H 0 μ 120 mm H a μ > 120 Blood Pressures when on drug. Maybe H 0 μ 120 mm H a μ < 120 Blood Pressures when on drug. Null hypothesis Hypothesis accepted unless sufficient evidence to the contrary Law Case H 0 Innocent H a Guilty o In a court of law, the defendant is innocent until "proven" guilty. o Burden of proof is to disprove innocent = H 0. For statistical hypothesis tests o The null hypothesis, H 0, is innocent until proven guilty. For example suppose regulations limit factory effluent to 1 ppm Copper H 0 μ 1 ppm H a μ > 1 ppm Burden of proof: to show noncompliance H 0 μ > 1 ppm H a μ 1 ppm Burden of proof: to show compliance Often, in research, the burden of proof is on showing that your research hypothesis has merit, The research hypothesis is most often H a, and H 0 states that research hypothesis doesn t hold. The burden of proof is on you to show that the research hypothesis has merit. Example: New Blood Pressure medicine: μ = change in BP Research hypothesis: Drug reduces BP H 0 μ 0 Drug doesn t help H a μ < 0 Drug helps Example: Suppose we are checking a sand 3 minute timer.
2 Suppose we will accept the manufacturer's claim unless we have sufficient evidence to the contrary. o H 0 μ = 180 seconds o H a μ 180 To test the claim, we measure the time elapsed n=50 times o y = sample average of n = 50 trials. o See if y is close enough to 180. o How far from 180 should y be before we reject μ = 180? o Possible rule: Reject H 0 μ = 180 if y > σ y or y < σ y With this rule o P(reject H 0 μ = 180 μ = 180) = 0.05 o P(reject H 0 H 0 true) = 0.05 o The notation α is used for P(type I error) = α o In this case, α = For α = 0.01 using Z table probabilities we reject H 0 if y 180 > 2.58σ y o Say σ = 5 sec n = 50 α = 0.05 o σ y = σ n = 5 50 = = 1 2 = o 1.96σ y = 1.96(0.707) = 1.4
3 In general suppose we are testing a two-sided alternative where the H a values are on two sides of the H 0 value. H 0 μ = μ 0 (e. g. μ 0 = 180 H 0 μ = 180) H a μ μ 0 For an α level test, reject H 0 if y μ 0 > z α 2 σ y y μ 0 > z σ α 2 y Z > z α 2 In this chapter, z α 2 means percentile 100(1 α 2) z can be found at the bottom of a t-table (section 5.7) z = 1.96 o Suppose y = 181.4, we know that σ = 5, and we test at the α=0.05 level. o σ y = σ = 5 = 1.4 n 50 o z = = 2.53 > 1.96 Reject H o If α = 0.01 Reject H 0 if z > o Since 2.53 < 2.58, we do not reject H 0 at the α = 0.01 level o For α = 0.01 compared to α = 0.05, we are less willing to tolerate rejecting H 0 when H 0 is true (claiming the timer doesn t have advertised mean rejecting H 0 is harder).
4 o An equivalent approach would be to see if μ = 180 is in the (two-sided) 95% confidence interval. o Here the 95% confidence interval is ± 1.96(0.707) ± to o Reject H 0 since μ = 180 is not included. o The confidence interval gives more info than just Reject H 0. o We know how different from 180 the true mean, μ, might be. Opinion: You are better off using confidence intervals rather than hypothesis tests. 3 cases H 0 μ = μ 0 H a μ μ 0 2-sided H 0 μ μ 0 H a μ < μ 0 1-sided H 0 μ μ 0 H a μ > μ 0 1-sided H a μ μ 0 A 2-sideed alternative Reject H 0 if y is far from μ 0 in either direction Reject H 0 if z > z α 2 y μ 0 > z SE α 2 y
5 H a μ < μ 0 A 1-sided alternative Reject H 0 if y is far below μ 0 Reject H 0 if z < z α y μ 0 < z α SE y H a μ > μ 0 A 1-sided alternative Reject H 0 if y is far above μ 0 Reject H 0 if z < z α y μ 0 > z α SE y
6 o α = P(type I error) = P(reject H 0 H 0 is true) (Actually worst case) o H 0 μ μ 0 H a μ < μ 0 o α = max μ μ 0 P(Reject H 0 ) o The H 0 case most likely mistaken for H a is μ = μ 0. o α = P(Reject H 0 μ = μ 0 ) o In many books, H 0 is written with = sign. o H a μ = 0 H a μ < 0 Think again of a law case: H 0 Innoccent Decision Acquit Convict Truth = Correct Type I error (α) Innocent Truth= Guilty Type II error (β) Correct o α = P(type I error) = P(reject H 0 H 0 is true) In this example, α = P(convict innoc) o β = P(type II error) = P(accept H 0 H 0 is false) In this example, β = P(acquit guilty) o 1 β = Power = P(reject H 0 H 0 is false) o β = P(accept H 0 H 0 is false) o β depends on how false H 0 is. Example: Light bulbs with manufacturers' claim of mean lifetime of 1000 hours. o Suppose we accept the manufacturer's claim unless we have sufficient evidence to the contrary o H 0 μ = 1000 hours H a μ 1000 hours o α = P(Reject H 0 μ = 1000 μ = 1000) o β = P(Don t reject H 0 μ = 1000 μ 1000) o β is different for different values of μ 1000
7 Calculating Power Back to 3 minute timer. H 0 μ = 180 H a μ 180 α = 0.05 o Reject H 0 if: o y > SE Y or y < SE Y If μ = 181.5, then Power = P(y > SE Y μ = 181.5) + P(y < SE Y μ = 181.5) = P z > SE Y = P z > P z < SE Y SE Y SE Y Suppose n = 100 σ = 5 SE Y = σ = 5 = 0.5 n 100 Power = P(z > ) + P(z < ) = P(z > 1.04) + P(z < 4.96) P(z > 1.04) = For 2-sided test following the step above Power P z > z α 2 μ 0 μ α SE Y For μ = with 2-sided alternative and α = 0.05 Power P z > = P(z > ) = P(z > 3.04) = The farther the truth is from H 0 μ = μ 0, the greater the power. o For testing µ=180 the power of rejecting H 0 is greater when µ=182.5 than when µ=181.5
8 Section 5.5: Choosing n Based on Power Deciding n based on power considerations requires α to be specified o α = P(type I error) = P(Reject H 0 H 0 is true ) β to be specified for some particular alternative value of µ o β = P(type II error) = P(reject H 0 some speci ied H a = μ α value) o Example: Sand Timer H 0 = 180 H a 180 Suppose we decide to o Set α = 0.05 o Require power of 0.80 to reject H 0 if μ = 181 Solve for n: ( ) σ = n ( )σ ( ) n = 196 = n
9 In general going through these steps for a 2-sided alternative n z α 2+z β 2 σ 2 = μ 2 α μ 0 (difference to detect) The solution is only approximate because we ignored a small probability on one side of the normal curve. This is close enough as long as is big enough. In the same way for a one-sided test: n = z α+z β 2 σ 2 2 Here we are not ignoring one side of the normal curve so this is an equlity. The required sample size increases as o σ 2 increases o decreases o α, β error rates decrease o z α, z β increase
10 Section 5.6: P-values Rather than just saying Reject H 0, we want to say how convincingly H 0 is rejected. Suppose for example H 0 μ = 180 H a μ 180 y = n = 100 σ = 5 SE y = σ = 5 = 0.5 n 100 z = y 180 = = 1.2 = 2.4 SE y The observed mean is 2.4 SE's away from H 0 μ = 180 If H 0 μ = 180 is true, we expect z to be around 0. Evidence against H 0 is values of z far from 0 in either direction. The p-value answers o What is the prob y of a z this far or farther from expected if H 0 is true? o What is the prob y of this much or more evidence against H 0 if H 0 is true? o What is the prob y of this much evidence against defendant if defendant is innocent?
11 p-value = P(z > 2.4) + P(z < 2.4) = 2(0.0082) = Small p-values are bad for H 0. o If the data result is far from what s expected for H 0, the p-value is small and H 0 Reject H 0 if p-value < α. α tells us how unusual the data have to be in order to give up on H 0. For a p-value = o Since p 0.05, we reject H 0 if α = 0.05 o Since p > 0.01, Do not reject H 0 if α = 0.01 Suppose for Blood Pressure changes H 0 μ 0 H a μ < 0 y = 1.8 n = 10 σ = 3 SE y = 3 10 = z = = p-value = P(z < 1.9) = Since p 0.05, reject H 0 if α = 0.05 Since p > 0.01, do not reject H 0 if α = 0.01
12 In general H 0 μ = μ 0 p-value = P( z > z data ) H a μ μ 0 H 0 μ μ 0 p-value = P(z > z data ) H a μ > μ 0 H 0 μ μ 0 p-value = P(z < z data ) H a μ < μ 0
Section 9.1 (Part 2) (pp ) Type I and Type II Errors
Section 9.1 (Part 2) (pp. 547-551) Type I and Type II Errors Because we are basing our conclusion in a significance test on sample data, there is always a chance that our conclusions will be in error.
More informationUnit 19 Formulating Hypotheses and Making Decisions
Unit 19 Formulating Hypotheses and Making Decisions Objectives: To formulate a null hypothesis and an alternative hypothesis, and to choose a significance level To identify the Type I error and the Type
More informationTests about a population mean
October 2 nd, 2017 Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 1: Descriptive statistics Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter 8: Confidence
More informationStatistical Process Control (contd... )
Statistical Process Control (contd... ) ME522: Quality Engineering Vivek Kumar Mehta November 11, 2016 Note: This lecture is prepared with the help of material available online at https://onlinecourses.science.psu.edu/
More informationECO220Y Hypothesis Testing: Type I and Type II Errors and Power Readings: Chapter 12,
ECO220Y Hypothesis Testing: Type I and Type II Errors and Power Readings: Chapter 12, 12.7-12.9 Winter 2012 Lecture 15 (Winter 2011) Estimation Lecture 15 1 / 25 Linking Two Approaches to Hypothesis Testing
More informationHypothesis tests
6.1 6.4 Hypothesis tests Prof. Tesler Math 186 February 26, 2014 Prof. Tesler 6.1 6.4 Hypothesis tests Math 186 / February 26, 2014 1 / 41 6.1 6.2 Intro to hypothesis tests and decision rules Hypothesis
More informationHypothesis Testing. ECE 3530 Spring Antonio Paiva
Hypothesis Testing ECE 3530 Spring 2010 Antonio Paiva What is hypothesis testing? A statistical hypothesis is an assertion or conjecture concerning one or more populations. To prove that a hypothesis is
More informationIn any hypothesis testing problem, there are two contradictory hypotheses under consideration.
8.1 Hypotheses and Test Procedures: A hypothesis One example of a hypothesis is p =.5, if we are testing if a new formula for a soda is preferred to the old formula (p=.5 assumes that they are preferred
More informationChapter 5: HYPOTHESIS TESTING
MATH411: Applied Statistics Dr. YU, Chi Wai Chapter 5: HYPOTHESIS TESTING 1 WHAT IS HYPOTHESIS TESTING? As its name indicates, it is about a test of hypothesis. To be more precise, we would first translate
More informationHypothesis for Means and Proportions
November 14, 2012 Hypothesis Tests - Basic Ideas Often we are interested not in estimating an unknown parameter but in testing some claim or hypothesis concerning a population. For example we may wish
More informationMathematical statistics
October 20 th, 2018 Lecture 17: Tests of Hypotheses Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
More informationECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12
ECO220Y Review and Introduction to Hypothesis Testing Readings: Chapter 12 Winter 2012 Lecture 13 (Winter 2011) Estimation Lecture 13 1 / 33 Review of Main Concepts Sampling Distribution of Sample Mean
More informationHypotheses and Errors
Hypotheses and Errors Jonathan Bagley School of Mathematics, University of Manchester Jonathan Bagley, September 23, 2005 Hypotheses & Errors - p. 1/22 Overview Today we ll develop the standard framework
More informationChapter Three. Hypothesis Testing
3.1 Introduction The final phase of analyzing data is to make a decision concerning a set of choices or options. Should I invest in stocks or bonds? Should a new product be marketed? Are my products being
More informationAnnouncements. Unit 3: Foundations for inference Lecture 3: Decision errors, significance levels, sample size, and power.
Announcements Announcements Unit 3: Foundations for inference Lecture 3:, significance levels, sample size, and power Statistics 101 Mine Çetinkaya-Rundel October 1, 2013 Project proposal due 5pm on Friday,
More informationPreliminary Statistics. Lecture 5: Hypothesis Testing
Preliminary Statistics Lecture 5: Hypothesis Testing Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Elements/Terminology of Hypothesis Testing Types of Errors Procedure of Testing Significance
More informationMathematical statistics
November 1 st, 2018 Lecture 18: Tests about a population mean Overview 9.1 Hypotheses and test procedures test procedures errors in hypothesis testing significance level 9.2 Tests about a population mean
More informationFirst we look at some terms to be used in this section.
8 Hypothesis Testing 8.1 Introduction MATH1015 Biostatistics Week 8 In Chapter 7, we ve studied the estimation of parameters, point or interval estimates. The construction of CI relies on the sampling
More informationHypotheses Test Procedures. Is the claim wrong?
Hypotheses Test Procedures MATH 2300 Sections 9.1 and 9.2 Is the claim wrong? An oil company representative claims that the average price for gasoline in Lubbock is $2.30 per gallon. You think the average
More information280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE Tests of Statistical Hypotheses
280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE 9-1.2 Tests of Statistical Hypotheses To illustrate the general concepts, consider the propellant burning rate problem introduced earlier. The null
More informationIntroduction to Statistics
MTH4106 Introduction to Statistics Notes 6 Spring 2013 Testing Hypotheses about a Proportion Example Pete s Pizza Palace offers a choice of three toppings. Pete has noticed that rather few customers ask
More informationHypothesis Testing The basic ingredients of a hypothesis test are
Hypothesis Testing The basic ingredients of a hypothesis test are 1 the null hypothesis, denoted as H o 2 the alternative hypothesis, denoted as H a 3 the test statistic 4 the data 5 the conclusion. The
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests
Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1 Chapter Topics Hypothesis Testing Methodology Z Test
More informationMathematical Statistics
Mathematical Statistics MAS 713 Chapter 8 Previous lecture: 1 Bayesian Inference 2 Decision theory 3 Bayesian Vs. Frequentist 4 Loss functions 5 Conjugate priors Any questions? Mathematical Statistics
More informationLecture Testing Hypotheses: The Neyman-Pearson Paradigm
Math 408 - Mathematical Statistics Lecture 29-30. Testing Hypotheses: The Neyman-Pearson Paradigm April 12-15, 2013 Konstantin Zuev (USC) Math 408, Lecture 29-30 April 12-15, 2013 1 / 12 Agenda Example:
More informationChapter 7: Hypothesis Testing
Chapter 7: Hypothesis Testing *Mathematical statistics with applications; Elsevier Academic Press, 2009 The elements of a statistical hypothesis 1. The null hypothesis, denoted by H 0, is usually the nullification
More informationFor use only in [the name of your school] 2014 S4 Note. S4 Notes (Edexcel)
s (Edexcel) Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 1 Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 2 Copyright www.pgmaths.co.uk - For AS,
More informationSTAT Chapter 8: Hypothesis Tests
STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More information20 Hypothesis Testing, Part I
20 Hypothesis Testing, Part I Bob has told Alice that the average hourly rate for a lawyer in Virginia is $200 with a standard deviation of $50, but Alice wants to test this claim. If Bob is right, she
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More informationPreliminary Statistics Lecture 5: Hypothesis Testing (Outline)
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 5: Hypothesis Testing (Outline) Gujarati D. Basic Econometrics, Appendix A.8 Barrow M. Statistics
More informationME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV
Theory of Engineering Experimentation Chapter IV. Decision Making for a Single Sample Chapter IV 1 4 1 Statistical Inference The field of statistical inference consists of those methods used to make decisions
More informationThe Purpose of Hypothesis Testing
Section 8 1A:! An Introduction to Hypothesis Testing The Purpose of Hypothesis Testing See s Candy states that a box of it s candy weighs 16 oz. They do not mean that every single box weights exactly 16
More informationHypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals
Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals Lecture 9 Justin Kern April 9, 2018 Measuring Effect Size: Cohen s d Simply finding whether a
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu.30 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. .30
More informationCHAPTER 8. Test Procedures is a rule, based on sample data, for deciding whether to reject H 0 and contains:
CHAPTER 8 Test of Hypotheses Based on a Single Sample Hypothesis testing is the method that decide which of two contradictory claims about the parameter is correct. Here the parameters of interest are
More informationLECTURE 5. Introduction to Econometrics. Hypothesis testing
LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will
More information6.4 Type I and Type II Errors
6.4 Type I and Type II Errors Ulrich Hoensch Friday, March 22, 2013 Null and Alternative Hypothesis Neyman-Pearson Approach to Statistical Inference: A statistical test (also known as a hypothesis test)
More informationMath 101: Elementary Statistics Tests of Hypothesis
Tests of Hypothesis Department of Mathematics and Computer Science University of the Philippines Baguio November 15, 2018 Basic Concepts of Statistical Hypothesis Testing A statistical hypothesis is an
More informationWhat is a Hypothesis?
What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill in this city is μ = $42 population proportion Example:
More information23. MORE HYPOTHESIS TESTING
23. MORE HYPOTHESIS TESTING The Logic Behind Hypothesis Testing For simplicity, consider testing H 0 : µ = µ 0 against the two-sided alternative H A : µ µ 0. Even if H 0 is true (so that the expectation
More informationStatistical Inference. Hypothesis Testing
Statistical Inference Hypothesis Testing Previously, we introduced the point and interval estimation of an unknown parameter(s), say µ and σ 2. However, in practice, the problem confronting the scientist
More informationOn Assumptions. On Assumptions
On Assumptions An overview Normality Independence Detection Stem-and-leaf plot Study design Normal scores plot Correction Transformation More complex models Nonparametric procedure e.g. time series Robustness
More informationIntroduction to Statistics
MTH4106 Introduction to Statistics Notes 15 Spring 2013 Testing hypotheses about the mean Earlier, we saw how to test hypotheses about a proportion, using properties of the Binomial distribution It is
More informationWith our knowledge of interval estimation, we can consider hypothesis tests
Chapter 10 Hypothesis Testing 10.1 Testing Hypotheses With our knowledge of interval estimation, we can consider hypothesis tests An Example of an Hypothesis Test: Statisticians at Employment and Immigration
More informationBias Variance Trade-off
Bias Variance Trade-off The mean squared error of an estimator MSE(ˆθ) = E([ˆθ θ] 2 ) Can be re-expressed MSE(ˆθ) = Var(ˆθ) + (B(ˆθ) 2 ) MSE = VAR + BIAS 2 Proof MSE(ˆθ) = E((ˆθ θ) 2 ) = E(([ˆθ E(ˆθ)]
More informationHypothesis Testing. ) the hypothesis that suggests no change from previous experience
Hypothesis Testing Definitions Hypothesis a claim about something Null hypothesis ( H 0 ) the hypothesis that suggests no change from previous experience Alternative hypothesis ( H 1 ) the hypothesis that
More informationStatistical Inference: Uses, Abuses, and Misconceptions
Statistical Inference: Uses, Abuses, and Misconceptions Michael W. Trosset Indiana Statistical Consulting Center Department of Statistics ISCC is part of IU s Department of Statistics, chaired by Stanley
More information8.1-4 Test of Hypotheses Based on a Single Sample
8.1-4 Test of Hypotheses Based on a Single Sample Example 1 (Example 8.6, p. 312) A manufacturer of sprinkler systems used for fire protection in office buildings claims that the true average system-activation
More informationTopic 17 Simple Hypotheses
Topic 17 Simple Hypotheses Terminology and the Neyman-Pearson Lemma 1 / 11 Outline Overview Terminology The Neyman-Pearson Lemma 2 / 11 Overview Statistical hypothesis testing is designed to address the
More informationPhysicsAndMathsTutor.com
1. A manager in a sweet factory believes that the machines are working incorrectly and the proportion p of underweight bags of sweets is more than 5%. He decides to test this by randomly selecting a sample
More informationSTAT 135 Lab 5 Bootstrapping and Hypothesis Testing
STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,
More informationexample: An observation X comes from a normal distribution with
Hypothesis test A statistical hypothesis is a statement about the population parameter(s) or distribution. null hypothesis H 0 : prior belief statement. alternative hypothesis H a : a statement that contradicts
More informationFurther Remarks about Hypothesis Tests
Further Remarks about Hypothesis Tests Engineering Statistics Section 8.5 Josh Engwer TTU 18 April 2016 Josh Engwer (TTU) Further Remarks about Hypothesis Tests 18 April 2016 1 / 14 PART I PART I: STATISTICAL
More informationBurden of Proof: Economic Analysis
Burden of Proof: Economic Analysis Burden of proof is often placed on the party who has readier access to knowledge about the fact in question. The design of burden of proof can be seen as a device for
More informationBasic Concepts of Inference
Basic Concepts of Inference Corresponds to Chapter 6 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT) with some slides by Jacqueline Telford (Johns Hopkins University) and Roy Welsch (MIT).
More informationSTAT 515 fa 2016 Lec Statistical inference - hypothesis testing
STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................
More informationPSY 305. Module 3. Page Title. Introduction to Hypothesis Testing Z-tests. Five steps in hypothesis testing
Page Title PSY 305 Module 3 Introduction to Hypothesis Testing Z-tests Five steps in hypothesis testing State the research and null hypothesis Determine characteristics of comparison distribution Five
More informationComparing Means from Two-Sample
Comparing Means from Two-Sample Kwonsang Lee University of Pennsylvania kwonlee@wharton.upenn.edu April 3, 2015 Kwonsang Lee STAT111 April 3, 2015 1 / 22 Inference from One-Sample We have two options to
More informationCONTINUOUS RANDOM VARIABLES
the Further Mathematics network www.fmnetwork.org.uk V 07 REVISION SHEET STATISTICS (AQA) CONTINUOUS RANDOM VARIABLES The main ideas are: Properties of Continuous Random Variables Mean, Median and Mode
More informationPage 312, Exercise 50
Millersville University Name Answer Key Department of Mathematics MATH 130, Elements of Statistics I, Homework 4 November 5, 2009 Page 312, Exercise 50 Simulation According to the U.S. National Center
More informationBusiness Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing
Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology
More informationPower of a test. Hypothesis testing
Hypothesis testing February 11, 2014 Debdeep Pati Power of a test 1. Assuming standard deviation is known. Calculate power based on one-sample z test. A new drug is proposed for people with high intraocular
More informationMEI STRUCTURED MATHEMATICS STATISTICS 2, S2. Practice Paper S2-B
MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS STATISTICS, S Practice Paper S-B Additional materials: Answer booklet/paper Graph paper MEI Examination formulae and tables (MF) TIME
More information8: Hypothesis Testing
Some definitions 8: Hypothesis Testing. Simple, compound, null and alternative hypotheses In test theory one distinguishes between simple hypotheses and compound hypotheses. A simple hypothesis Examples:
More informationPrecept 4: Hypothesis Testing
Precept 4: Hypothesis Testing Soc 500: Applied Social Statistics Ian Lundberg Princeton University October 6, 2016 Learning Objectives 1 Introduce vectorized R code 2 Review homework and talk about RMarkdown
More informationHypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =
Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,
More informationLECTURE 12 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING
LECTURE 1 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING INTERVAL ESTIMATION Point estimation of : The inference is a guess of a single value as the value of. No accuracy associated with it. Interval estimation
More informationBasic Statistics. 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation).
Basic Statistics There are three types of error: 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation). 2. Systematic error - always too high or too low
More informationStatistics 251: Statistical Methods
Statistics 251: Statistical Methods 1-sample Hypothesis Tests Module 9 2018 Introduction We have learned about estimating parameters by point estimation and interval estimation (specifically confidence
More informationCH.9 Tests of Hypotheses for a Single Sample
CH.9 Tests of Hypotheses for a Single Sample Hypotheses testing Tests on the mean of a normal distributionvariance known Tests on the mean of a normal distributionvariance unknown Tests on the variance
More informationCHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:
CHAPTER 9, 10 Hypothesis Testing Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. To
More informationMaking Inferences About Parameters
Making Inferences About Parameters Parametric statistical inference may take the form of: 1. Estimation: on the basis of sample data we estimate the value of some parameter of the population from which
More informationInterpretation of results through confidence intervals
Interpretation of results through confidence intervals Hypothesis tests Confidence intervals Hypothesis Test Reject H 0 : μ = μ 0 Confidence Intervals μ 0 is not in confidence interval μ 0 P(observed statistic
More informationRelating Graph to Matlab
There are two related course documents on the web Probability and Statistics Review -should be read by people without statistics background and it is helpful as a review for those with prior statistics
More informationCHAPTER 9: HYPOTHESIS TESTING
CHAPTER 9: HYPOTHESIS TESTING THE SECOND LAST EXAMPLE CLEARLY ILLUSTRATES THAT THERE IS ONE IMPORTANT ISSUE WE NEED TO EXPLORE: IS THERE (IN OUR TWO SAMPLES) SUFFICIENT STATISTICAL EVIDENCE TO CONCLUDE
More informationSection 10.1 (Part 2 of 2) Significance Tests: Power of a Test
1 Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test Learning Objectives After this section, you should be able to DESCRIBE the relationship between the significance level of a test, P(Type
More informationThe problem of base rates
Psychology 205: Research Methods in Psychology William Revelle Department of Psychology Northwestern University Evanston, Illinois USA October, 2015 1 / 14 Outline Inferential statistics 2 / 14 Hypothesis
More informationIntroductory Econometrics
Session 4 - Testing hypotheses Roland Sciences Po July 2011 Motivation After estimation, delivering information involves testing hypotheses Did this drug had any effect on the survival rate? Is this drug
More informationSamples and Populations Confidence Intervals Hypotheses One-sided vs. two-sided Statistical Significance Error Types. Statistiek I.
Statistiek I Sampling John Nerbonne CLCG, Rijksuniversiteit Groningen http://www.let.rug.nl/nerbonne/teach/statistiek-i/ John Nerbonne 1/41 Overview 1 Samples and Populations 2 Confidence Intervals 3 Hypotheses
More informationAP Statistics Ch 12 Inference for Proportions
Ch 12.1 Inference for a Population Proportion Conditions for Inference The statistic that estimates the parameter p (population proportion) is the sample proportion p ˆ. p ˆ = Count of successes in the
More informationDealing with the assumption of independence between samples - introducing the paired design.
Dealing with the assumption of independence between samples - introducing the paired design. a) Suppose you deliberately collect one sample and measure something. Then you collect another sample in such
More informationHypothesis testing for µ:
University of California, Los Angeles Department of Statistics Statistics 10 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
More informationFinansiell Statistik, GN, 15 hp, VT2008 Lecture 10-11: Statistical Inference: Hypothesis Testing
Finansiell Statistik, GN, 15 hp, VT008 Lecture 10-11: Statistical Inference: Hypothesis Testing Gebrenegus Ghilagaber, PhD, Associate Professor April 1, 008 1 1 Statistical Inferences: Introduction Recall:
More informationStatistical Tests. Matthieu de Lapparent
Statistical Tests Matthieu de Lapparent matthieu.delapparent@epfl.ch Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique Fédérale de Lausanne
More informationClassroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats
Classroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats Materials Needed: Bags of popcorn, watch with second hand or microwave with digital timer. Instructions: Follow the instructions on the
More informationF79SM STATISTICAL METHODS
F79SM STATISTICAL METHODS SUMMARY NOTES 9 Hypothesis testing 9.1 Introduction As before we have a random sample x of size n of a population r.v. X with pdf/pf f(x;θ). The distribution we assign to X is
More informationStudy Ch. 9.3, #47 53 (45 51), 55 61, (55 59)
GOALS: 1. Understand that 2 approaches of hypothesis testing exist: classical or critical value, and p value. We will use the p value approach. 2. Understand the critical value for the classical approach
More informationDr. Allen Back. Nov. 21, 2016
Dr. Allen Back Nov. 21, 2016 , Type I/II Errors, α, and β Given H 0 : p = p 0, there are two ways an HT can report an inaccurate result: , Type I/II Errors, α, and β Given H 0 : p = p 0, there are two
More informationThe due date for this assignment is past. Your work can be viewed below, but no changes can be made.
WebAssign Mirka Martinez Math 3680 Homework 7 Devore Fall 2013 (Homework) Applied Statistics, Math 3680-Fall 2013, section 2, Fall 2013 Instructor: John Quintanilla Current Score : 136.5 / 130 Due : Friday,
More informationParameter Estimation, Sampling Distributions & Hypothesis Testing
Parameter Estimation, Sampling Distributions & Hypothesis Testing Parameter Estimation & Hypothesis Testing In doing research, we are usually interested in some feature of a population distribution (which
More informationCENTRAL LIMIT THEOREM (CLT)
CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic
More informationChapter 3 Multiple Regression Complete Example
Department of Quantitative Methods & Information Systems ECON 504 Chapter 3 Multiple Regression Complete Example Spring 2013 Dr. Mohammad Zainal Review Goals After completing this lecture, you should be
More informationElementary Statistics Triola, Elementary Statistics 11/e Unit 17 The Basics of Hypotheses Testing
(Section 8-2) Hypotheses testing is not all that different from confidence intervals, so let s do a quick review of the theory behind the latter. If it s our goal to estimate the mean of a population,
More informationChapter 7: Hypothesis Testing - Solutions
Chapter 7: Hypothesis Testing - Solutions 7.1 Introduction to Hypothesis Testing The problem with applying the techniques learned in Chapter 5 is that typically, the population mean (µ) and standard deviation
More informationVisual interpretation with normal approximation
Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation
More informationCHAPTER 93 SIGNIFICANCE TESTING
CHAPTER 93 SIGNIFICANCE TESTING EXERCISE 342 Page 981 1. Random samples of 1000 rings are drawn from the output of a machine periodically for inspection purposes. A defect rate of 5% is acceptable to the
More informationTopic 17: Simple Hypotheses
Topic 17: November, 2011 1 Overview and Terminology Statistical hypothesis testing is designed to address the question: Do the data provide sufficient evidence to conclude that we must depart from our
More informationSample Size and Power I: Binary Outcomes. James Ware, PhD Harvard School of Public Health Boston, MA
Sample Size and Power I: Binary Outcomes James Ware, PhD Harvard School of Public Health Boston, MA Sample Size and Power Principles: Sample size calculations are an essential part of study design Consider
More information