BINF702 SPRING 2015 Chapter 7 Hypothesis Testing: One-Sample Inference
|
|
|
- Flora Hood
- 5 years ago
- Views:
Transcription
1 BINF702 SPRING 2015 Chapter 7 Hypothesis Testing: One-Sample Inference BINF702 SPRING 2014 Chapter 7 Hypothesis Testing 1
2 Section 7.9 One-Sample c 2 Test for the Variance of a Normal Distribution Eq (One-Sample c 2 Test for the Variance of a Normal Distribution (Two- Sided Alternative) We compute the test statistic X 2 = (n-1)s 2 /s 2 0 The rejection region is given by X c or X n1, / 2 n1,1 / 2 The acceptance region is given by c X c c n1, / 2 n1,1 / 2 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 2
3 Section 7.9 One-Sample c 2 Test for the Variance of a Normal Distribution Eq (p-value for a One- Sample c 2 Test for the Variance of a Normal Distribution (Two-Sided Alternative) Test Statistic X 2 n1 s 2 s 2 0 Case I s s p-value given by 2* Pr( Y X Y ~ c ) n1 Case II s s p-value given by 2* Pr( Y X Y ~ c ) n1 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 3
4 Section 7.9 One-Sample c 2 Test for the Variance of a Normal Distribution Ex What is s 2 and the p-value? BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 4
5 7.10 One-Sample Test for a Binomial Distribution Equation 7.42 (One-Sample Test for a Binomial Proportion Normal-Theory Method (Two-Sided Alternative) Let the test statistic be given by z pˆ p p q / n The rejection region is given by The acceptence region is given by z z or z The test should only be used if np q / 2 1 / 2 z z z / 2 1 / 2 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 5
6 7.10 One-Sample Test for a Binomial Distribution Equation 7.43 Computation of the p-value for the One- Sample Binomial Test Normal-Theory Method (Two-Sided Alternative) Let the test statistic be given by z If pˆ p q p 0 / n pˆ p p-value=2 z If pˆ p 0 p-value 2 1 z BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 6
7 Approximate Binomial Testing in R prop.test(x, n, p = NULL, alternative = c("two.sided", "less","greater"), conf.level = 0.95, correct = TRUE) Problem 7.10 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 7
8 7.10 One-Sample Test for a Binomial Distribution Eq (Computation of the p-value for the One- Sample Binomial Test-Exact Method(Two-Sided Alternative) x n k nk If pˆ p0, p 2P X x min 2 p0 1 p0,1 k 0 k n n k nk If pˆ p0, p 2P X x min 2 p0 1 p0,1 kxk BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 8
9 7.10 One-Sample Test for a Binomial Distribution (Exact Method) binom.test(x, n, p = 0.5, alternative = c("two.sided", "less", "greater"), conf.level = 0.95) Ex BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 9
10 7.10 One-Sample Test for a Binomial Distribution (Exact Method) We note that this is different from the value in the text or the value using the equation in the text. How do you compute this using R? BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 10
11 7.10 One-Sample Test for a Binomial Distribution Eq (Power for the One-Sample Binomial Test (Two-Sided Alternative)) The power of the one-sample binomial test for the hypothesis H 0 : p = p 0 vs. H 1 : p!=p 0 for the specific alternative p = p 1 is given by pq pq z p p n where np q / pq 0 0 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 11
12 7.10 One-Sample Test for a Binomial Distribution Equation 7.46 (Sample-Size Estimation for the One-Sample Test (Two-Sided Alternative)) Suppose we wish to test H 0 : p = p 0 versus H 1 : p!= p 0. The sample size needed to conduct a two-sided test with significance level and power 1 b versus the specified alternative hypothesis p = p 1 is n p q z z p p / 2 1b pq pq BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 12
13 7.10 One-Sample Test for a Binomial Distribution Can you write a function to implement Equation 7.46? BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 13
14 7.10 One-Sample Test for a Binomial Distribution Our function in action on Ex BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 14
15 Section 7.11 One-Sample Inference for the Poisson Distribution Eq (One Sample Inference for the Poisson Distribution (Small-Sample Test Critical Value Method) Let X be a Poisson random variable with expected value = m. To test the hypothesis H 0 : m = m 0 versus H 1 : m!= m 0 using a two-sided test with significance level, 1. Obtain the two-sided 100% x (1-) confidence interval for m based on the observed value x of X. Denote the confidence interval (c 1, c 2 ) 2. If m 0 < c 1 or m 0 > c 2, then reject H 0, if c 1 <= m 0 <= c 2, then accept H 0. BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 15
16 Section 7.11 One-Sample Inference for the Poisson Distribution Eq (One-Sample Inference for the Poisson Distribution (Small-Sample Test p-value Method) Let m = expected value of a Poisson distribution. To test the hypothesis H 0 : m = m 0 versus H 1 : m!= m 0, 1. Compute x = observed number of deaths in the study population 2. Under H 0, the random variable X will follow a Poisson distribution with parameter m 0. Thus the two-sided p-value is given by min 2 x k 0 e min 21 0 k m0 k! m x k 0 e,1 if x 0 k m0 k! m m 0,1 if x m 0 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 16
17 Section 7.11 One-Sample Inference for the Poisson Distribution Ex Can you solve this using R as a calculator? BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 17
18 Section 7.11 One-Sample Inference for the Poisson Distribution Def The standardized mortality ratio (SMR) is define by 100% x O/E = 100% x the observed number of deaths in the study population divided by the expected number of deaths in the study population under the assumption that the mortality rates for the study population are the same as those for the general population. For nonfatal conditions the standardized mortality ratio is sometimes known as the standardized morbidity ratio. BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 18
19 Section 7.11 One-Sample Inference for the Poisson Distribution Eq (One-Sample Inference for the Poisson Distribution (Large-Sample Test) Let m = expected value of a Poisson random variable. To test the hypothesis H 0 : m = m 0 versus H 1 : m!= m 0 (1) Compute x = observed number of events in the study population (2) Compute the test statistic X 2 2 x m SMR m0 100 (3) For a two-sided test at level, H is rejected if X and m 1 ~ c under H 2 2 c1,1 2 2 H0 is accepted if X c1,1 0 BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 19
20 Section 7.11 One-Sample Inference for the Poisson Distribution Eq (continued) (4) The exact p-value is given by Pr( c X ) 1 / (5) This test should only be used if m 10 (6) In addition, an approximate 100% x (1- ) CI for m is given by x z x BINF702 SPRING 2013 Chapter 7 Hypothesis Testing 20
BINF 702 SPRING Chapter 8 Hypothesis Testing: Two-Sample Inference. BINF702 SPRING 2014 Chapter 8 Hypothesis Testing: Two- Sample Inference 1
BINF 702 SPRING 2014 Chapter 8 Hypothesis Testing: Two-Sample Inference Two- Sample Inference 1 A Poster Child for two-sample hypothesis testing Ex 8.1 Obstetrics In the birthweight data in Example 7.2,
MATH 240. Chapter 8 Outlines of Hypothesis Tests
MATH 4 Chapter 8 Outlines of Hypothesis Tests Test for Population Proportion p Specify the null and alternative hypotheses, ie, choose one of the three, where p is some specified number: () H : p H : p
CH.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
1 Statistical inference for a population mean
1 Statistical inference for a population mean 1. Inference for a large sample, known variance Suppose X 1,..., X n represents a large random sample of data from a population with unknown mean µ and known
CHAPTER 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
SPRING 2007 EXAM C SOLUTIONS
SPRING 007 EXAM C SOLUTIONS Question #1 The data are already shifted (have had the policy limit and the deductible of 50 applied). The two 350 payments are censored. Thus the likelihood function is L =
Visual 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
Design of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments
Design of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments The hypothesis testing framework The two-sample t-test Checking assumptions, validity Comparing more that
MATH 728 Homework 3. Oleksandr Pavlenko
MATH 78 Homewor 3 Olesandr Pavleno 4.5.8 Let us say the life of a tire in miles, say X, is normally distributed with mean θ and standard deviation 5000. Past experience indicates that θ = 30000. The manufacturer
GROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION
FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89
Tests for Population Proportion(s)
Tests for Population Proportion(s) Esra Akdeniz April 6th, 2016 Motivation We are interested in estimating the prevalence rate of breast cancer among 50- to 54-year-old women whose mothers have had breast
Introduction 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
STAT 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
Formulas and Tables. for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. ˆp E p ˆp E Proportion
Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 3: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx)
BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings
BIO5312 Biostatistics Lecture 6: Statistical hypothesis testings Yujin Chung October 4th, 2016 Fall 2016 Yujin Chung Lec6: Statistical hypothesis testings Fall 2016 1/30 Previous Two types of statistical
Chapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
CHAPTER EIGHT TESTS OF HYPOTHESES
11/18/213 CAPTER EIGT TESTS OF YPOTESES (8.1) Definition: A statistical hypothesis is a statement concerning one population or more. 1 11/18/213 8.1.1 The Null and The Alternative ypotheses: The structure
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd Basic Statistics Sample size?
ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Basic Statistics Sample size? Sample size determination: text section 2-4-2 Page 41 section 3-7 Page 107 Website::http://www.stat.uiowa.edu/~rlenth/Power/
TUTORIAL 8 SOLUTIONS #
TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level
Evaluating Hypotheses
Evaluating Hypotheses IEEE Expert, October 1996 1 Evaluating Hypotheses Sample error, true error Confidence intervals for observed hypothesis error Estimators Binomial distribution, Normal distribution,
Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
, 0 x < 2. a. Find the probability that the text is checked out for more than half an hour but less than an hour. = (1/2)2
Math 205 Spring 206 Dr. Lily Yen Midterm 2 Show all your work Name: 8 Problem : The library at Capilano University has a copy of Math 205 text on two-hour reserve. Let X denote the amount of time the text
Math 141. Lecture 10: Confidence Intervals. Albyn Jones 1. jones/courses/ Library 304. Albyn Jones Math 141
Math 141 Lecture 10: Confidence Intervals Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Inference Suppose X Binomial(n, p). Inference about p includes the topics: Inference
STAT Chapter 9: Two-Sample Problems. Paired Differences (Section 9.3)
STAT 515 -- Chapter 9: Two-Sample Problems Paired Differences (Section 9.3) Examples of Paired Differences studies: Similar subjects are paired off and one of two treatments is given to each subject in
PubH 5450 Biostatistics I Prof. Carlin. Lecture 13
PubH 5450 Biostatistics I Prof. Carlin Lecture 13 Outline Outline Sample Size Counts, Rates and Proportions Part I Sample Size Type I Error and Power Type I error rate: probability of rejecting the null
p An interval estimator of a parameter θ is of the form θl < θ < θu at a
Chapter 7 of Devore CONFIDENCE INTERVAL ESTIMATORS An interval estimator of a parameter θ is of the form θl < θ < θu at a confidence pr (or a confidence coefficient) of 1 α. When θl =, < θ < θu is called
THE ROYAL STATISTICAL SOCIETY 2015 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 3
THE ROYAL STATISTICAL SOCIETY 015 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 3 The Society is providing these solutions to assist candidates preparing for the examinations in 017. The solutions are
Person-Time Data. Incidence. Cumulative Incidence: Example. Cumulative Incidence. Person-Time Data. Person-Time Data
Person-Time Data CF Jeff Lin, MD., PhD. Incidence 1. Cumulative incidence (incidence proportion) 2. Incidence density (incidence rate) December 14, 2005 c Jeff Lin, MD., PhD. c Jeff Lin, MD., PhD. Person-Time
Introduction to Statistical Analysis
Introduction to Statistical Analysis Changyu Shen Richard A. and Susan F. Smith Center for Outcomes Research in Cardiology Beth Israel Deaconess Medical Center Harvard Medical School Objectives Descriptive
Practice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
Unobservable Parameter. Observed Random Sample. Calculate Posterior. Choosing Prior. Conjugate prior. population proportion, p prior:
Pi Priors Unobservable Parameter population proportion, p prior: π ( p) Conjugate prior π ( p) ~ Beta( a, b) same PDF family exponential family only Posterior π ( p y) ~ Beta( a + y, b + n y) Observed
Sections 7.1 and 7.2. This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics
Sections 7.1 and 7.2 This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Estimate the value of a population parameter Test some claim (or
(8 One- and Two-Sample Test Of Hypothesis)
324 Stat Lecture Notes (8 One- and Two-Sample Test Of ypothesis) ( Book*: Chapter 1,pg319) Probability& Statistics for Engineers & Scientists By Walpole, Myers, Myers, Ye Definition: A statistical hypothesis
STA Module 10 Comparing Two Proportions
STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare
Medical statistics part I, autumn 2010: One sample test of hypothesis
Medical statistics part I, autumn 2010: One sample test of hypothesis Eirik Skogvoll Consultant/ Professor Faculty of Medicine Dept. of Anaesthesiology and Emergency Medicine 1 What is a hypothesis test?
Stat 231 Exam 2 Fall 2013
Stat 231 Exam 2 Fall 2013 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1 1. Some IE 361 students worked with a manufacturer on quantifying the capability
A proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1!
Proportions A proportion is the fraction of individuals having a particular attribute. It is also the probability that an individual randomly sampled from the population will have that attribute Can range
GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs
STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...
How do we compare the relative performance among competing models?
How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model
9-6. Testing the difference between proportions /20
9-6 Testing the difference between proportions 1 Homework Discussion Question p514 Ex 9-6 p514 2, 3, 4, 7, 9, 11 (use both the critical value and p-value for all problems. 2 Objective Perform hypothesis
Solution: First note that the power function of the test is given as follows,
Problem 4.5.8: Assume the life of a tire given by X is distributed N(θ, 5000 ) Past experience indicates that θ = 30000. The manufacturere claims the tires made by a new process have mean θ > 30000. Is
Solutions to First Midterm Exam, Stat 371, Spring those values: = Or, we can use Rule 6: = 0.63.
Solutions to First Midterm Exam, Stat 371, Spring 2010 There are two, three or four versions of each question. The questions on your exam comprise a mix of versions. As a result, when you examine the solutions
Hypothesis Testing Chap 10p460
Hypothesis Testing Chap 1p46 Elements of a statistical test p462 - Null hypothesis - Alternative hypothesis - Test Statistic - Rejection region Rejection Region p462 The rejection region (RR) specifies
Basic 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).
Chapter 10: Inferences based on two samples
November 16 th, 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
Statistical Inference
Statistical Inference Classical and Bayesian Methods Class 7 AMS-UCSC Tue 31, 2012 Winter 2012. Session 1 (Class 7) AMS-132/206 Tue 31, 2012 1 / 13 Topics Topics We will talk about... 1 Hypothesis testing
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept
Math 152. Rumbos Fall Solutions to Exam #2
Math 152. Rumbos Fall 2009 1 Solutions to Exam #2 1. Define the following terms: (a) Significance level of a hypothesis test. Answer: The significance level, α, of a hypothesis test is the largest probability
Section 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples
Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means
Dr. Maddah ENMG 617 EM Statistics 10/12/12. Nonparametric Statistics (Chapter 16, Hines)
Dr. Maddah ENMG 617 EM Statistics 10/12/12 Nonparametric Statistics (Chapter 16, Hines) Introduction Most of the hypothesis testing presented so far assumes normally distributed data. These approaches
Chapter 7 Comparison of two independent samples
Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N
Lecture Slides. Elementary Statistics. Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3
Review. December 4 th, Review
December 4 th, 2017 Att. Final exam: Course evaluation Friday, 12/14/2018, 10:30am 12:30pm Gore Hall 115 Overview Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 6: Statistics and Sampling Distributions Chapter
Generalized linear models for binary data. A better graphical exploratory data analysis. The simple linear logistic regression model
Stat 3302 (Spring 2017) Peter F. Craigmile Simple linear logistic regression (part 1) [Dobson and Barnett, 2008, Sections 7.1 7.3] Generalized linear models for binary data Beetles dose-response example
LECTURE 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
Inferences for Proportions and Count Data
Inferences for Proportions and Count Data Corresponds to Chapter 9 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT), with some slides by Ramón V. León (University of Tennessee) 1 Inference
Mark Scheme (Results) June 2008
Mark (Results) June 8 GCE GCE Mathematics (6684/) Edexcel Limited. Registered in England and Wales No. 44967 June 8 6684 Statistics S Mark Question (a) (b) E(X) = Var(X) = ( ) x or attempt to use dx µ
Chapter 8 of Devore , H 1 :
Chapter 8 of Devore TESTING A STATISTICAL HYPOTHESIS Maghsoodloo A statistical hypothesis is an assumption about the frequency function(s) (i.e., PDF or pdf) of one or more random variables. Stated in
Hypothesis 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
Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 9 Inferences from Two Samples 9-1 Overview 9-2 Inferences About Two Proportions 9-3
7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between
7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation
Smoking Habits. Moderate Smokers Heavy Smokers Total. Hypertension No Hypertension Total
Math 3070. Treibergs Final Exam Name: December 7, 00. In an experiment to see how hypertension is related to smoking habits, the following data was taken on individuals. Test the hypothesis that the proportions
AMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
BIOSTATS Intermediate Biostatistics Spring 2017 Exam 2 (Units 3, 4 & 5) Practice Problems SOLUTIONS
BIOSTATS 640 - Intermediate Biostatistics Spring 2017 Exam 2 (Units 3, 4 & 5) Practice Problems SOLUTIONS Practice Question 1 Both the Binomial and Poisson distributions have been used to model the quantal
Single Sample Means. SOCY601 Alan Neustadtl
Single Sample Means SOCY601 Alan Neustadtl The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size
Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola 2001 by Addison Wesley Longman Publishing Company, Inc.
Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola 2001 by Addison Wesley Longman Publishing Company, Inc. Ch. 2: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s 2
Psychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
Formulas and Tables. for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. ˆp E p ˆp E Proportion.
Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Ch. 2: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx) 2 s Å n(n 2 1) Mean (frequency
Bios 6649: Clinical Trials - Statistical Design and Monitoring
Bios 6649: Clinical Trials - Statistical Design and Monitoring Spring Semester 2015 John M. Kittelson Department of Biostatistics & nformatics Colorado School of Public Health University of Colorado Denver
Multivariate Statistical Analysis
Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 9 for Applied Multivariate Analysis Outline Addressing ourliers 1 Addressing ourliers 2 Outliers in Multivariate samples (1) For
Chapter 9. Inferences from Two Samples. Objective. Notation. Section 9.2. Definition. Notation. q = 1 p. Inferences About Two Proportions
Chapter 9 Inferences from Two Samples 9. Inferences About Two Proportions 9.3 Inferences About Two s (Independent) 9.4 Inferences About Two s (Matched Pairs) 9.5 Comparing Variation in Two Samples Objective
Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text)
Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text) 1. A quick and easy indicator of dispersion is a. Arithmetic mean b. Variance c. Standard deviation
Permutation Tests. Noa Haas Statistics M.Sc. Seminar, Spring 2017 Bootstrap and Resampling Methods
Permutation Tests Noa Haas Statistics M.Sc. Seminar, Spring 2017 Bootstrap and Resampling Methods The Two-Sample Problem We observe two independent random samples: F z = z 1, z 2,, z n independently of
Lecture 15: Inference Based on Two Samples
Lecture 15: Inference Based on Two Samples MSU-STT 351-Sum17B (P. Vellaisamy: STT 351-Sum17B) Probability & Statistics for Engineers 1 / 26 9.1 Z-tests and CI s for (µ 1 µ 2 ) The assumptions: (i) X =
T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
Reference: Chapter 7 of Devore (8e)
Reference: Chapter 7 of Devore (8e) CONFIDENCE INTERVAL ESTIMATORS Maghsoodloo An interval estimator of a population parameter is of the form L < < u at a confidence Pr (or a confidence coefficient) of
This paper is not to be removed from the Examination Halls
~~ST104B ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104B ZB BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
BIOS 6222: Biostatistics II. Outline. Course Presentation. Course Presentation. Review of Basic Concepts. Why Nonparametrics.
BIOS 6222: Biostatistics II Instructors: Qingzhao Yu Don Mercante Cruz Velasco 1 Outline Course Presentation Review of Basic Concepts Why Nonparametrics The sign test 2 Course Presentation Contents Justification
STAT 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....................
Nonparametric Tests. Mathematics 47: Lecture 25. Dan Sloughter. Furman University. April 20, 2006
Nonparametric Tests Mathematics 47: Lecture 25 Dan Sloughter Furman University April 20, 2006 Dan Sloughter (Furman University) Nonparametric Tests April 20, 2006 1 / 14 The sign test Suppose X 1, X 2,...,
Discrete distribution. Fitting probability models to frequency data. Hypotheses for! 2 test. ! 2 Goodness-of-fit test
Discrete distribution Fitting probability models to frequency data A probability distribution describing a discrete numerical random variable For example,! Number of heads from 10 flips of a coin! Number
Chapter 5 Confidence Intervals
Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is
Probability theory and inference statistics! Dr. Paola Grosso! SNE research group!! (preferred!)!!
Probability theory and inference statistics Dr. Paola Grosso SNE research group p.grosso@uva.nl paola.grosso@os3.nl (preferred) Roadmap Lecture 1: Monday Sep. 22nd Collecting data Presenting data Descriptive
A3. Statistical Inference
Appendi / A3. Statistical Inference / Mean, One Sample-1 A3. Statistical Inference Population Mean μ of a Random Variable with known standard deviation σ, and random sample of size n 1 Before selecting
Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8.
Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8. Coin A is flipped until a head appears, then coin B is flipped until
PHP2510: Principles of Biostatistics & Data Analysis. Lecture X: Hypothesis testing. PHP 2510 Lec 10: Hypothesis testing 1
PHP2510: Principles of Biostatistics & Data Analysis Lecture X: Hypothesis testing PHP 2510 Lec 10: Hypothesis testing 1 In previous lectures we have encountered problems of estimating an unknown population
CS 543 Page 1 John E. Boon, Jr.
CS 543 Machine Learning Spring 2010 Lecture 05 Evaluating Hypotheses I. Overview A. Given observed accuracy of a hypothesis over a limited sample of data, how well does this estimate its accuracy over
The t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary
Patrick Breheny October 13 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction Introduction What s wrong with z-tests? So far we ve (thoroughly!) discussed how to carry out hypothesis
Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences
Formulas and Tables by Mario F. Triola
Copyright 010 Pearson Education, Inc. Ch. 3: Descriptive Statistics x f # x x f Mean 1x - x s - 1 n 1 x - 1 x s 1n - 1 s B variance s Ch. 4: Probability Mean (frequency table) Standard deviation P1A or
Statistical Inference
Statistical Inference Classical and Bayesian Methods Class 6 AMS-UCSC Thu 26, 2012 Winter 2012. Session 1 (Class 6) AMS-132/206 Thu 26, 2012 1 / 15 Topics Topics We will talk about... 1 Hypothesis testing
Frequency table: Var2 (Spreadsheet1) Count Cumulative Percent Cumulative From To. Percent <x<=
A frequency distribution is a kind of probability distribution. It gives the frequency or relative frequency at which given values have been observed among the data collected. For example, for age, Frequency
STAT 512 MidTerm I (2/21/2013) Spring 2013 INSTRUCTIONS
STAT 512 MidTerm I (2/21/2013) Spring 2013 Name: Key INSTRUCTIONS 1. This exam is open book/open notes. All papers (but no electronic devices except for calculators) are allowed. 2. There are 5 pages in
Lecture #16 Thursday, October 13, 2016 Textbook: Sections 9.3, 9.4, 10.1, 10.2
STATISTICS 200 Lecture #16 Thursday, October 13, 2016 Textbook: Sections 9.3, 9.4, 10.1, 10.2 Objectives: Define standard error, relate it to both standard deviation and sampling distribution ideas. Describe
VTU Edusat Programme 16
VTU Edusat Programme 16 Subject : Engineering Mathematics Sub Code: 10MAT41 UNIT 8: Sampling Theory Dr. K.S.Basavarajappa Professor & Head Department of Mathematics Bapuji Institute of Engineering and
Unit 9: Inferences for Proportions and Count Data
Unit 9: Inferences for Proportions and Count Data Statistics 571: Statistical Methods Ramón V. León 12/15/2008 Unit 9 - Stat 571 - Ramón V. León 1 Large Sample Confidence Interval for Proportion ( pˆ p)
MIT Spring 2015
Assessing Goodness Of Fit MIT 8.443 Dr. Kempthorne Spring 205 Outline 2 Poisson Distribution Counts of events that occur at constant rate Counts in disjoint intervals/regions are independent If intervals/regions
CSE 312 Final Review: Section AA
CSE 312 TAs December 8, 2011 General Information General Information Comprehensive Midterm General Information Comprehensive Midterm Heavily weighted toward material after the midterm Pre-Midterm Material
Ch 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component