Hypothesis Tests for One Sample
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Hypothesis Testing and Likelihood Ratio Tests
Hypottthesiiis tttestttiiing and llliiikellliiihood ratttiiio tttesttts Y We will adopt the following model for observed data. The distribution of Y = (Y1, ..., Yn) is parameter considered known except for some paramett er ç, which may be a vector ç = (ç1, ..., çk); ç“Ç, the paramettter space. The parameter space will usually be an open set. If Y is a continuous random variable, its probabiiillliiittty densiiittty functttiiion (pdf) will de denoted f(yy;ç) . If Y is y probability mass function y Y y discrete then f(yy;ç) represents the probabii ll ii tt y mass functt ii on (pmf); f(yy;ç) = Pç(YY=yy). A stttatttiiistttiiicalll hypottthesiiis is a statement about the value of ç. We are interested in testing the null hypothesis H0: ç“Ç0 versus the alternative hypothesis H1: ç“Ç1. Where Ç0 and Ç1 ¶ Ç. hypothesis test Naturally Ç0 § Ç1 = ∅, but we need not have Ç0 ∞ Ç1 = Ç. A hypott hesii s tt estt is a procedure critical region for deciding between H0 and H1 based on the sample data. It is equivalent to a crii tt ii call regii on: a critical region is a set C ¶ Rn y such that if y = (y1, ..., yn) “ C, H0 is rejected. Typically C is expressed in terms of the value of some tttesttt stttatttiiistttiiic, a function of the sample data. For µ example, we might have C = {(y , ..., y ): y – 0 ≥ 3.324}. The number 3.324 here is called a 1 n s/ n µ criiitttiiicalll valllue of the test statistic Y – 0 . S/ n If y“C but ç“Ç 0, we have committed a Type I error. -
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
TRANSCRIPT: 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 TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing. 2 TRANSCRIPT: Let’s being the lecture with a summary of the logic process that underlies hypothesis testing. 3 TRANSCRIPT: It is often impossible or otherwise not feasible to collect data on every individual within a population. Therefore, researchers rely on samples to help answer questions about populations. Hypothesis testing is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest. Hypothesis testing is one of the most commonly used inferential procedures. Hypothesis testing will combine many of the concepts we have already covered, including z‐scores, probability, and the distribution of sample means. To conduct a hypothesis test, we first state a hypothesis about a population, predict the characteristics of a sample of that population (that is, we predict that a sample will be representative of the population), obtain a sample, then collect data from that sample and analyze the data to see if it is consistent with our hypotheses. 4 TRANSCRIPT: The process of hypothesis testing begins by stating a hypothesis about the unknown population. Actually we state two opposing hypotheses. The first hypothesis we state –the most important one –is the null hypothesis. The null hypothesis states that the treatment has no effect. In general the null hypothesis states that there is no change, no difference, no effect, and otherwise no relationship between the independent and dependent variables. -
The Scientific Method: Hypothesis Testing and Experimental Design
Appendix I The Scientific Method The study of science is different from other disciplines in many ways. Perhaps the most important aspect of “hard” science is its adherence to the principle of the scientific method: the posing of questions and the use of rigorous methods to answer those questions. I. Our Friend, the Null Hypothesis As a science major, you are probably no stranger to curiosity. It is the beginning of all scientific discovery. As you walk through the campus arboretum, you might wonder, “Why are trees green?” As you observe your peers in social groups at the cafeteria, you might ask yourself, “What subtle kinds of body language are those people using to communicate?” As you read an article about a new drug which promises to be an effective treatment for male pattern baldness, you think, “But how do they know it will work?” Asking such questions is the first step towards hypothesis formation. A scientific investigator does not begin the study of a biological phenomenon in a vacuum. If an investigator observes something interesting, s/he first asks a question about it, and then uses inductive reasoning (from the specific to the general) to generate an hypothesis based upon a logical set of expectations. To test the hypothesis, the investigator systematically collects data, either with field observations or a series of carefully designed experiments. By analyzing the data, the investigator uses deductive reasoning (from the general to the specific) to state a second hypothesis (it may be the same as or different from the original) about the observations. -
Hypothesis Testing – Examples and Case Studies
Hypothesis Testing – Chapter 23 Examples and Case Studies Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a test statistic. 3. Use the test statistic to determine the p-value. 4. The result is statistically significant if the p-value is less than or equal to the level of significance. Often media only presents results of step 4. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 2 23.2 Testing Hypotheses About Proportions and Means If the null and alternative hypotheses are expressed in terms of a population proportion, mean, or difference between two means and if the sample sizes are large … … the test statistic is simply the corresponding standardized score computed assuming the null hypothesis is true; and the p-value is found from a table of percentiles for standardized scores. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 3 Example 2: Weight Loss for Diet vs Exercise Did dieters lose more fat than the exercisers? Diet Only: sample mean = 5.9 kg sample standard deviation = 4.1 kg sample size = n = 42 standard error = SEM1 = 4.1/ √42 = 0.633 Exercise Only: sample mean = 4.1 kg sample standard deviation = 3.7 kg sample size = n = 47 standard error = SEM2 = 3.7/ √47 = 0.540 measure of variability = [(0.633)2 + (0.540)2] = 0.83 Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 4 Example 2: Weight Loss for Diet vs Exercise Step 1. -
Tests of Hypotheses Using Statistics
Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables . -
Understanding Statistical Hypothesis Testing: the Logic of Statistical Inference
Review Understanding Statistical Hypothesis Testing: The Logic of Statistical Inference Frank Emmert-Streib 1,2,* and Matthias Dehmer 3,4,5 1 Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University, 33100 Tampere, Finland 2 Institute of Biosciences and Medical Technology, Tampere University, 33520 Tampere, Finland 3 Institute for Intelligent Production, Faculty for Management, University of Applied Sciences Upper Austria, Steyr Campus, 4040 Steyr, Austria 4 Department of Mechatronics and Biomedical Computer Science, University for Health Sciences, Medical Informatics and Technology (UMIT), 6060 Hall, Tyrol, Austria 5 College of Computer and Control Engineering, Nankai University, Tianjin 300000, China * Correspondence: [email protected]; Tel.: +358-50-301-5353 Received: 27 July 2019; Accepted: 9 August 2019; Published: 12 August 2019 Abstract: Statistical hypothesis testing is among the most misunderstood quantitative analysis methods from data science. Despite its seeming simplicity, it has complex interdependencies between its procedural components. In this paper, we discuss the underlying logic behind statistical hypothesis testing, the formal meaning of its components and their connections. Our presentation is applicable to all statistical hypothesis tests as generic backbone and, hence, useful across all application domains in data science and artificial intelligence. Keywords: hypothesis testing; machine learning; statistics; data science; statistical inference 1. Introduction We are living in an era that is characterized by the availability of big data. In order to emphasize the importance of this, data have been called the ‘oil of the 21st Century’ [1]. However, for dealing with the challenges posed by such data, advanced analysis methods are needed. -
Statistical Significance and Statistical Error in Antitrust Analysis
STATISTICAL SIGNIFICANCE AND STATISTICAL ERROR IN ANTITRUST ANALYSIS PHILLIP JOHNSON EDWARD LEAMER JEFFREY LEITZINGER* Proof of antitrust impact and estimation of damages are central elements in antitrust cases. Generally, more is needed for these purposes than simple ob- servational evidence regarding changes in price levels over time. This is be- cause changes in economic conditions unrelated to the behavior at issue also may play a role in observed outcomes. For example, prices of consumer elec- tronics have been falling for several decades because of technological pro- gress. Against that backdrop, a successful price-fixing conspiracy may not lead to observable price increases but only slow their rate of decline. There- fore, proof of impact and estimation of damages often amounts to sorting out the effects on market outcomes of illegal behavior from the effects of other market supply and demand factors. Regression analysis is a statistical technique widely employed by econo- mists to identify the role played by one factor among those that simultane- ously determine market outcomes. In this way, regression analysis is well suited to proof of impact and estimation of damages in antitrust cases. For that reason, regression models have become commonplace in antitrust litigation.1 In our experience, one aspect of regression results that often attracts spe- cific attention in that environment is the statistical significance of the esti- mates. As is discussed below, some courts, participants in antitrust litigation, and commentators maintain that stringent levels of statistical significance should be a threshold requirement for the results of regression analysis to be used as evidence regarding impact and damages. -
Statistical Proof of Discrimination: Beyond "Damned Lies"
Washington Law Review Volume 68 Number 3 7-1-1993 Statistical Proof of Discrimination: Beyond "Damned Lies" Kingsley R. Browne Follow this and additional works at: https://digitalcommons.law.uw.edu/wlr Part of the Labor and Employment Law Commons Recommended Citation Kingsley R. Browne, Statistical Proof of Discrimination: Beyond "Damned Lies", 68 Wash. L. Rev. 477 (1993). Available at: https://digitalcommons.law.uw.edu/wlr/vol68/iss3/2 This Article is brought to you for free and open access by the Law Reviews and Journals at UW Law Digital Commons. It has been accepted for inclusion in Washington Law Review by an authorized editor of UW Law Digital Commons. For more information, please contact [email protected]. Copyright © 1993 by Washington Law Review Association STATISTICAL PROOF OF DISCRIMINATION: BEYOND "DAMNED LIES" Kingsley R. Browne* Abstract Evidence that an employer's work force contains fewer minorities or women than would be expected if selection were random with respect to race and sex has been taken as powerful-and often sufficient-evidence of systematic intentional discrimina- tion. In relying on this kind of statistical evidence, courts have made two fundamental errors. The first error is assuming that statistical analysis can reveal the probability that observed work-force disparities were produced by chance. This error leads courts to exclude chance as a cause when such a conclusion is unwarranted. The second error is assuming that, except for random deviations, the work force of a nondiscriminating employer would mirror the racial and sexual composition of the relevant labor force. This assumption has led courts inappropriately to shift the burden of proof to employers in pattern-or-practice cases once a statistical disparity is shown. -
Exploratory Versus Confirmative Testing in Clinical Trials
xperim & E en Gaus et al., Clin Exp Pharmacol 2015, 5:4 l ta a l ic P in h DOI: 10.4172/2161-1459.1000182 l a C r m f Journal of o a l c a o n l o r g u y o J ISSN: 2161-1459 Clinical and Experimental Pharmacology Research Article Open Access Interpretation of Statistical Significance - Exploratory Versus Confirmative Testing in Clinical Trials, Epidemiological Studies, Meta-Analyses and Toxicological Screening (Using Ginkgo biloba as an Example) Wilhelm Gaus, Benjamin Mayer and Rainer Muche Institute for Epidemiology and Medical Biometry, University of Ulm, Germany *Corresponding author: Wilhelm Gaus, University of Ulm, Institute for Epidemiology and Medical Biometry, Schwabstrasse 13, 89075 Ulm, Germany, Tel : +49 731 500-26891; Fax +40 731 500-26902; E-mail: [email protected] Received date: June 18 2015; Accepted date: July 23 2015; Published date: July 27 2015 Copyright: © 2015 Gaus W, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract The terms “significant” and “p-value” are important for biomedical researchers and readers of biomedical papers including pharmacologists. No other statistical result is misinterpreted as often as p-values. In this paper the issue of exploratory versus confirmative testing is discussed in general. A significant p-value sometimes leads to a precise hypothesis (exploratory testing), sometimes it is interpret as “statistical proof” (confirmative testing). A p-value may only be interpreted as confirmative, if (1) the hypothesis and the level of significance were established a priori and (2) an adjustment for multiple testing was carried out if more than one test was performed. -
Chapter 8 Key Ideas Hypothesis (Null and Alternative)
Chapter 8 Key Ideas Hypothesis (Null and Alternative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Significance Level, Power Section 8-1: Overview Confidence Intervals (Chapter 7) are great for estimating the value of a parameter, but in some situations estimation is not the goal. For example, suppose that researchers somehow know that the average number of fast food meals eaten per week by families in the 1990s was 3.5. Furthermore, they are interested in seeing if families today eat more fast food than in the 1990s. In this case, researchers don’t need to estimate how much fast food families eat today. All they need to know is whether that average is larger than 3.5. For the purposes of this example, let µ denote the average number of fast food meals eaten per week by families today. The researchers’ question could then be formulated as follows: Is the average number of meals eaten today the same as in the 1990s ( µ = 3.5) or is it more ( µ > 3.5)? There are two different possibilities, and the correct answer is impossible to know, since µ is unknown. However, the researchers could observe the sample mean and determine which possibility is the most likely . This process is called hypothesis testing . Definition Hypothesis – A claim or statement about a property of the population (e.g. “ µ = 3.5” from above) Hypothesis Test – A standard procedure for testing a claim about a property of the population. To guide this discussion of hypothesis testing, remember the “Rare Event Rule”: If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct. -
Inference Procedures Based on Order Statistics
INFERENCE PROCEDURES BASED ON ORDER STATISTICS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jesse C. Frey, M.S. * * * * * The Ohio State University 2005 Dissertation Committee: Approved by Prof. H. N. Nagaraja, Adviser Prof. Steven N. MacEachern Adviser ¨ ¨ Prof. Omer Ozturk¨ Graduate Program in Prof. Douglas A. Wolfe Statistics ABSTRACT In this dissertation, we develop several new inference procedures that are based on order statistics. Each procedure is motivated by a particular statistical problem. The first problem we consider is that of computing the probability that a fully- specified collection of independent random variables has a particular ordering. We derive an equal conditionals condition under which such probabilities can be computed exactly, and we also derive extrapolation algorithms that allow approximation and computation of such probabilities in more general settings. Romberg integration is one idea that is used. The second problem we address is that of producing optimal distribution-free confidence bands for a cumulative distribution function. We treat this problem both in the case of simple random sampling and in the more general case in which the sample consists of independent order statistics from the distribution of interest. The latter case includes ranked-set sampling. We propose a family of optimality criteria motivated by the idea that good confidence bands are narrow, and we develop theory that makes the identification and computation of optimal bands possible. The Brunn-Minkowski Inequality from the theory of convex bodies plays a key role in this work. -
Week 10: Causality with Measured Confounding
Week 10: Causality with Measured Confounding Brandon Stewart1 Princeton November 28 and 30, 2016 1These slides are heavily influenced by Matt Blackwell, Jens Hainmueller, Erin Hartman, Kosuke Imai and Gary King. Stewart (Princeton) Week 10: Measured Confounding November 28 and 30, 2016 1 / 176 Where We've Been and Where We're Going... Last Week I regression diagnostics This Week I Monday: F experimental Ideal F identification with measured confounding I Wednesday: F regression estimation Next Week I identification with unmeasured confounding I instrumental variables Long Run I causality with measured confounding ! unmeasured confounding ! repeated data Questions? Stewart (Princeton) Week 10: Measured Confounding November 28 and 30, 2016 2 / 176 1 The Experimental Ideal 2 Assumption of No Unmeasured Confounding 3 Fun With Censorship 4 Regression Estimators 5 Agnostic Regression 6 Regression and Causality 7 Regression Under Heterogeneous Effects 8 Fun with Visualization, Replication and the NYT 9 Appendix Subclassification Identification under Random Assignment Estimation Under Random Assignment Blocking Stewart (Princeton) Week 10: Measured Confounding November 28 and 30, 2016 3 / 176 1 The Experimental Ideal 2 Assumption of No Unmeasured Confounding 3 Fun With Censorship 4 Regression Estimators 5 Agnostic Regression 6 Regression and Causality 7 Regression Under Heterogeneous Effects 8 Fun with Visualization, Replication and the NYT 9 Appendix Subclassification Identification under Random Assignment Estimation Under Random Assignment Blocking Stewart