STAT 135 Lab 7 Distributions derived from the normal distribution, and comparing independent samples.
|
|
- Paula Holmes
- 5 years ago
- Views:
Transcription
1 STAT 135 Lab 7 Distributions derived from the normal distribution, and comparing independent samples. Rebecca Barter March 16, 2015
2 The χ 2 distribution
3 The χ 2 distribution We have seen several instances of test statistics which follow a χ 2 distribution, for example, the generalized likelihood ratio, Λ 2 log(λ) χ 2 df and the Pearson χ 2 goodness-of-fit test statistic X 2 = n (O i E i ) 2 E i i=1 χ 2 n 1 dim(θ) It turns out that the χ 2 distribution is very related to the N(0, 1) distribution.
4 The χ 2 distribution The χ 2 distribution can be generated from the sum of squared normals: If Z N(0, 1), then Z 2 χ 2 1 and more generally, if Z i N(0, 1), then Z Z 2 n χ 2 n from which it is easy to conclude that if U χ 2 n and V χ 2 m, then U + V χ 2 m+n that is, to get the distribution of the sum of χ 2 random variables, just add the degrees of freedom.
5 The χ 2 distribution The χ 2 n distribution is also related to the gamma distribution: ( n U χ 2 n U Gamma 2, 1 ) 2 so we can write the density of the χ 2 n distribution as f (x) = 1 2 n/2 Γ( n 2 )x n 2 1 e x 2, x 0
6 The t distribution
7 The t distribution If Z N(0, 1) is independent of U χ 2 n, then Z U n t n The t n distribution has density f n (x) = ( ) (n+1)/2 Γ((n + 1)/2) 1 + t2 nπγ(n/2) n
8 The t distribution Note that if X i N(µ, σ 2 ), then X n µ σ/ n N(0, 1) However, if we don t know σ, we can estimate it by s 2 = 1 n 1 and our distribution becomes n (X i X ) 2 i=1 X n µ s/ n t n 1
9 The F -distribution
10 The F-distribution If U χ 2 m and V χ 2 n are independent, then The F m,n distribution has density f (w) = F = U/m V /n F m,n Γ((m + n)/2) ( m ) ( m/2 w m/ m ) (m+n)/2 Γ(m/2)Γ(n/2) n n w
11 Moment generating functions
12 Moment generating functions The moment generating function (MGF) for a random variable X is given by M(t) = E (e ) tx and the MGF of a random variable uniquely determines the corresponding distribution.
13 Moment generating functions The MGF of X is defined by M(t) = E (e ) tx A useful formula: The kth derivative of the MGF of X, evaluated at zero, is equal to the kth moment of X : E ( X k) = d k M dt k t=0
14 Moment generating functions Some nice properties of the moment generating function (show them yourself!): If X has MGF M X (t), and Y = a + bx, then Y has MGF M Y (t) = e at M X (bt) If X and Y are independent with MGF s M X and M Y and Z = X + Y, then M Z (t) = M X (t)m Y (t)
15 Exercise
16 Exercise: moment generating functions For each of the following distributions, calculate the moment generating function, and the first two moments. 1. X Poisson(λ) 2. Z = X + Y, where X Poisson(λ) and Y Poisson(µ) are independent 3. Z N(0, 1)
17 Comparing Independent Samples from two normal populations with common but unknown variance t-test
18 Comparing independent samples So far, we have been working towards making inferences about parameters from a single population. For example, we have looked at conducting hypothesis tests for the (unknown) population mean µ, such as H 0 : µ = 0 H 1 : µ > 0 Given a sample X 1,..., X n from our population of interest, we would look at the sample mean, X n, to see if we have enough evidence against H 0 in favor of H 1 (for example, by calculating a p-value)
19 Comparing independent samples What if we were instead interested in comparing parameters from two different populations? Suppose that: the first population has (unknown) mean µ 1, and the second population has (unknown) mean µ 2. Then we might test the hypothesis: against H 0 : µ 1 = µ 2 H 1 : µ 1 > µ 2 or H 1 : µ 1 < µ 2 or H 1 : µ 1 µ 2
20 Comparing independent samples from two normal populations with common but unknown variance Suppose that we have observed samples: X 1 = x 1,..., X n = x n from pop 1 with unknown variance σx 2 Y 1 = y 1,..., Y m = y m from pop 2 with unknown variance σy 2 Assume common variance: σx 2 = σ2 Y We might want to test against H 0 : µ 1 = µ 2 H 1 : µ 1 > µ 2 What test statistic might we look at to see if we have evidence against H 0 in favor of H 1?
21 Comparing independent samples from two normal populations with common but unknown variance If H 0 was true, we might expect X n Ȳm 0 if H 1 was true, we might expect X n Ȳm > 0. Note that our p-value (the probability, assuming H 0 true, of seeing something at least as extreme as what we have observed) would be given by P H0 ( X Ȳ > x ȳ) To calculate this probability, we need to identify the distribution of X Ȳ.
22 Comparing independent samples from two normal populations with common but unknown variance It turns out that t = ( X n Ȳm) s 1 n + 1 m t n+m 2 where the denominator is an estimate of the standard deviation of X Ȳ (since the true variance is unknown), and s 2 = (n 1)s2 X + (m 1)s2 Y n + m 2 is the pooled variance of X 1,..., X n and Y 1,..., Y m
23 Comparing independent samples from two normal populations with common but unknown variance To test H 0 : µ 1 = µ 2 against H 1 : µ 1 > µ 2 Our p-value can be calculated by P H0 ( X Ȳ > x ȳ) = P ( X Ȳ ) ( x ȳ) > 1 s n m s n + 1 m = P t n+m 2 > ( x ȳ) 1 s n + 1 m This test is called a two-sample t-test
24 Exercise
25 Exercise: comparing independent samples from two normal populations with common but unknown variance Suppose I am interested in comparing the average delivery time for two pizza companies. To test this hypothesis, I ordered 7 pizzas from pizza company A and recorded the delivery times: (20.4, 24.2, 15.4, 21.4, 20.2, 18.5, 21.5) and 5 pizzas from pizza company B: (20.2, 16.9, 18.5, 17.3, 20.5) I know that the delivery times follow normal distributions, but I don t know the mean or variance. Do I have enough evidence to conclude that the average delivery times for each company are different?
26 A nonparametric test for comparing Independent Samples from two arbitrary populations Mann-Whitney test
27 Comparing Independent Samples from two arbitrary populations Suppose that we are interested in comparing two populations, but we don t have any information about the distribution of our two populations. We observe data X 1 = x 1,..., X n = x n IID with unknown cts distribution F Y 1 = y 1,..., Y m = y m IID with unknown cts distribution G Suppose that we want to test the null hypothesis H 0 : F = G We could test H 0 using a nonparametric test (a test that makes no distributional assumptions on the data) based on ranks.
28 Comparing independent samples from two arbitrary populations Let R i be the rank of X i (when the ranks are taken over all of the X i s and Y i s combined). What is the expected value of n i=1 R i under H 0? First note that since under H 0 all ranks are equally likely, P(R 1 = k) = 1 n + m Thus E(R 1 ) = 1P(R 1 = 1) + 2P(R 1 = 2) (n + m)p(r 1 = n + m) = 1 ( (n + m)) n + m = 1 ( ) (n + m)(n + m + 1) n + m 2 = n + m + 1 2
29 Comparing independent samples from two arbitrary populations We just showed that E(R 1 ) = n + m So that if H 0 : F = G is true, we have ( n ) n(n + m + 1) E R i = 2 i=1 Thus a value of n i=1 R i that is significantly larger or smaller than n(n+m+1) 2 would provide evidence against H 0 : F = G.
30 Example: Comparing independent samples from two arbitrary populations Suppose, for example, that we had X = (6.2, 3.7, 4.7, 1.3) Then our ranks are Y = (1.2, 0.8, 1.4, 2.5, 1.1) rank(x ) = (9, 7, 8, 4) rank(y ) = (3, 1, 5, 6, 2) and we have that the sum of the ranks of the X i s and Y i s are n sum of rank(x ) := R i = 28 i=1 m sum of rank(y ) = R i = 17 i=1
31 Comparing independent samples from two arbitrary populations In our example, we saw that the sums of the ranks of the X i s was larger than the sums of the ranks of the Y i s. But how can we tell if this difference is enough to conclude that H 0 : F = G is false (that the two samples came from different distributions)?
32 Comparing independent samples from two arbitrary populations Suppose that the rank of X i is R i the rank of Y i is R i Note that under H 0, if n = m, we would expect that Recall that n R i = i=1 ( n ) E R i = i=1 m i=1 R i n(n + m + 1) 2 similarly we should have that the expected sum of the R i s is ( m ) E R i m(n + m + 1) = 2 i=1
33 Comparing independent samples from two arbitrary populations Thus if H 0 were true, we would expect that n R i + i=1 m R i n 1 (n + m + 1) i=1 where n 1 is the sample size of the smaller population: n 1 = min(n, m). Based on this idea, we can use the Mann-Whitney test to test H 0.
34 Comparing independent samples from two arbitrary populations The Mann-Whitney test can be conducted as follows 1. Concatenate the X i and Y j into a single vector Z 2. Let n 1 be the sample size of the smaller sample 3. Compute R = sum of the ranks of the smaller sample in Z 4. Compute R = n 1 (m + n + 1) R 5. Compute R = min(r, R ) 6. Compare the value of R to critical values in a table: if the value is less than or equal to the tabulated value, reject the null that F = G
35
36
37 Example: Comparing independent samples from two arbitrary populations Back to our example, where X = (6.2, 3.7, 4.7, 1.3) rank(x ) = (9, 7, 8, 4) Y = (1.2, 0.8, 1.4, 2.5, 1.1) rank(y ) = (3, 1, 5, 6, 2) Here n 1 = 4. The smaller sample is X and so the sum of the ranks of X are: R = sum of rank(x ) := n R i = 28 i=1 Next, we compute R = n 1 (m + n + 1) R = 4 ( ) 28 = 12
38 Example: Comparing independent samples from two arbitrary populations so that R = sum of rank(x ) := n R i = 28 i=1 R = n 1 (m + n + 1) R = 4 ( ) 28 = 12 R = min(r, R ) = 12 The critical value in the table for a two-sided test, with α = 0.05 (n 1 = 4, n 2 = 5) is 11. Thus, our value is not less than or equal to the critical value, so we fail to reject H 0.
39 Exercise
40 Exercise: Rice Chapter 11, Exercise 21 A study was done to compare the performances of engine bearings made of different compounds. Ten bearings of each type were tested. The times until failure (in units of millions of cycles) for each type are given below. Type I Type II Use normal theory to test the hypothesis that there is no difference between the two types of bearings. 2. Test the same hypothesis using a nonparametric method. 3. Which of the methods that of part (a) or that of part (b) do you think is better in this case?
STAT 135 Lab 8 Hypothesis Testing Review, Mann-Whitney Test by Normal Approximation, and Wilcoxon Signed Rank Test.
STAT 135 Lab 8 Hypothesis Testing Review, Mann-Whitney Test by Normal Approximation, and Wilcoxon Signed Rank Test. Rebecca Barter March 30, 2015 Mann-Whitney Test Mann-Whitney Test Recall that the Mann-Whitney
More informationSTAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis
STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis Rebecca Barter April 6, 2015 Multiple Testing Multiple Testing Recall that when we were doing two sample t-tests, we were testing the equality
More informationSTAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots. March 8, 2015
STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots March 8, 2015 The duality between CI and hypothesis testing The duality between CI and hypothesis
More informationLecture The Sample Mean and the Sample Variance Under Assumption of Normality
Math 408 - Mathematical Statistics Lecture 13-14. The Sample Mean and the Sample Variance Under Assumption of Normality February 20, 2013 Konstantin Zuev (USC) Math 408, Lecture 13-14 February 20, 2013
More informationSTAT 135 Lab 3 Asymptotic MLE and the Method of Moments
STAT 135 Lab 3 Asymptotic MLE and the Method of Moments Rebecca Barter February 9, 2015 Maximum likelihood estimation (a reminder) Maximum likelihood estimation Suppose that we have a sample, X 1, X 2,...,
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 informationStatistical Inference: Estimation and Confidence Intervals Hypothesis Testing
Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses 9.2 Testing Simple Hypotheses 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.6 Comparing the Means of Two
More informationLecture 15. Hypothesis testing in the linear model
14. Lecture 15. Hypothesis testing in the linear model Lecture 15. Hypothesis testing in the linear model 1 (1 1) Preliminary lemma 15. Hypothesis testing in the linear model 15.1. Preliminary lemma Lemma
More informationCentral Limit Theorem ( 5.3)
Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately
More informationReview: 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
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationComparison of Two Population Means
Comparison of Two Population Means Esra Akdeniz March 15, 2015 Independent versus Dependent (paired) Samples We have independent samples if we perform an experiment in two unrelated populations. We have
More informationPSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests
PSY 307 Statistics for the Behavioral Sciences Chapter 20 Tests for Ranked Data, Choosing Statistical Tests What To Do with Non-normal Distributions Tranformations (pg 382): The shape of the distribution
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationChapter 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
More informationGROUPED 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
More informationEC2001 Econometrics 1 Dr. Jose Olmo Room D309
EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population:
More informationNon-parametric Inference and Resampling
Non-parametric Inference and Resampling Exercises by David Wozabal (Last update. Juni 010) 1 Basic Facts about Rank and Order Statistics 1.1 10 students were asked about the amount of time they spend surfing
More informationBusiness Statistics. Lecture 10: Course Review
Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,
More informationHypothesis Testing hypothesis testing approach
Hypothesis Testing In this case, we d be trying to form an inference about that neighborhood: Do people there shop more often those people who are members of the larger population To ascertain this, we
More informationSTAT 135 Lab 11 Tests for Categorical Data (Fisher s Exact test, χ 2 tests for Homogeneity and Independence) and Linear Regression
STAT 135 Lab 11 Tests for Categorical Data (Fisher s Exact test, χ 2 tests for Homogeneity and Independence) and Linear Regression Rebecca Barter April 20, 2015 Fisher s Exact Test Fisher s Exact Test
More informationINTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
More information1 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
More informationSampling Distributions
Sampling Distributions In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of
More informationLecture 10: Generalized likelihood ratio test
Stat 200: Introduction to Statistical Inference Autumn 2018/19 Lecture 10: Generalized likelihood ratio test Lecturer: Art B. Owen October 25 Disclaimer: These notes have not been subjected to the usual
More informationSTATISTICS 4, S4 (4769) A2
(4769) A2 Objectives To provide students with the opportunity to explore ideas in more advanced statistics to a greater depth. Assessment Examination (72 marks) 1 hour 30 minutes There are four options
More informationPHP2510: 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
More informationDistribution-Free Procedures (Devore Chapter Fifteen)
Distribution-Free Procedures (Devore Chapter Fifteen) MATH-5-01: Probability and Statistics II Spring 018 Contents 1 Nonparametric Hypothesis Tests 1 1.1 The Wilcoxon Rank Sum Test........... 1 1. Normal
More informationS D / n t n 1 The paediatrician observes 3 =
Non-parametric tests Paired t-test A paediatrician measured the blood cholesterol of her patients and was worried to note that some had levels over 00mg/100ml To investigate whether dietary regulation
More informationMATH4427 Notebook 5 Fall Semester 2017/2018
MATH4427 Notebook 5 Fall Semester 2017/2018 prepared by Professor Jenny Baglivo c Copyright 2009-2018 by Jenny A. Baglivo. All Rights Reserved. 5 MATH4427 Notebook 5 3 5.1 Two Sample Analysis: Difference
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter Seven Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Seven Notes Spring 2011 1 / 42 Outline
More informationChapter 3. Comparing two populations
Chapter 3. Comparing two populations Contents Hypothesis for the difference between two population means: matched pairs Hypothesis for the difference between two population means: independent samples Two
More informationTwo Sample Problems. Two sample problems
Two Sample Problems Two sample problems The goal of inference is to compare the responses in two groups. Each group is a sample from a different population. The responses in each group are independent
More informationSTAT 135 Lab 2 Confidence Intervals, MLE and the Delta Method
STAT 135 Lab 2 Confidence Intervals, MLE and the Delta Method Rebecca Barter February 2, 2015 Confidence Intervals Confidence intervals What is a confidence interval? A confidence interval is calculated
More informationStatistics: revision
NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 29 / 30 April 2004 Department of Experimental Psychology University of Cambridge Handouts: Answers
More informationSampling Distributions
In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of random sample. For example,
More informationStatistical Hypothesis Testing
Statistical Hypothesis Testing Dr. Phillip YAM 2012/2013 Spring Semester Reference: Chapter 7 of Tests of Statistical Hypotheses by Hogg and Tanis. Section 7.1 Tests about Proportions A statistical hypothesis
More informationAMS7: 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
More informationSession 3 The proportional odds model and the Mann-Whitney test
Session 3 The proportional odds model and the Mann-Whitney test 3.1 A unified approach to inference 3.2 Analysis via dichotomisation 3.3 Proportional odds 3.4 Relationship with the Mann-Whitney test Session
More informationComparison of Two Samples
2 Comparison of Two Samples 2.1 Introduction Problems of comparing two samples arise frequently in medicine, sociology, agriculture, engineering, and marketing. The data may have been generated by observation
More information7 Random samples and sampling distributions
7 Random samples and sampling distributions 7.1 Introduction - random samples We will use the term experiment in a very general way to refer to some process, procedure or natural phenomena that produces
More informationChapter 7: Special Distributions
This chater first resents some imortant distributions, and then develos the largesamle distribution theory which is crucial in estimation and statistical inference Discrete distributions The Bernoulli
More informationTopic 22 Analysis of Variance
Topic 22 Analysis of Variance Comparing Multiple Populations 1 / 14 Outline Overview One Way Analysis of Variance Sample Means Sums of Squares The F Statistic Confidence Intervals 2 / 14 Overview Two-sample
More informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationSection 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
More informationSEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics
SEVERAL μs AND MEDIANS: MORE ISSUES Business Statistics CONTENTS Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study POST-HOC ANALYSIS
More informationStatistics Handbook. All statistical tables were computed by the author.
Statistics Handbook Contents Page Wilcoxon rank-sum test (Mann-Whitney equivalent) Wilcoxon matched-pairs test 3 Normal Distribution 4 Z-test Related samples t-test 5 Unrelated samples t-test 6 Variance
More informationSTA442/2101: Assignment 5
STA442/2101: Assignment 5 Craig Burkett Quiz on: Oct 23 rd, 2015 The questions are practice for the quiz next week, and are not to be handed in. I would like you to bring in all of the code you used to
More informationLab #12: Exam 3 Review Key
Psychological Statistics Practice Lab#1 Dr. M. Plonsky Page 1 of 7 Lab #1: Exam 3 Review Key 1) a. Probability - Refers to the likelihood that an event will occur. Ranges from 0 to 1. b. Sampling Distribution
More informationLecture 14: ANOVA and the F-test
Lecture 14: ANOVA and the F-test S. Massa, Department of Statistics, University of Oxford 3 February 2016 Example Consider a study of 983 individuals and examine the relationship between duration of breastfeeding
More informationCh 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
More informationChapter 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
More informationNon-parametric (Distribution-free) approaches p188 CN
Week 1: Introduction to some nonparametric and computer intensive (re-sampling) approaches: the sign test, Wilcoxon tests and multi-sample extensions, Spearman s rank correlation; the Bootstrap. (ch14
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.0 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.
More informationHypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More information4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures
Non-parametric Test Stephen Opiyo Overview Distinguish Parametric and Nonparametric Test Procedures Explain commonly used Nonparametric Test Procedures Perform Hypothesis Tests Using Nonparametric Procedures
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future
More informationUnit 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)
More informationOne-Way Analysis of Variance. With regression, we related two quantitative, typically continuous variables.
One-Way Analysis of Variance With regression, we related two quantitative, typically continuous variables. Often we wish to relate a quantitative response variable with a qualitative (or simply discrete)
More informationCherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009 there were participants
18.650 Statistics for Applications Chapter 5: Parametric hypothesis testing 1/37 Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009
More informationi=1 X i/n i=1 (X i X) 2 /(n 1). Find the constant c so that the statistic c(x X n+1 )/S has a t-distribution. If n = 8, determine k such that
Math 47 Homework Assignment 4 Problem 411 Let X 1, X,, X n, X n+1 be a random sample of size n + 1, n > 1, from a distribution that is N(µ, σ ) Let X = n i=1 X i/n and S = n i=1 (X i X) /(n 1) Find the
More informationEconomics 520. Lecture Note 19: Hypothesis Testing via the Neyman-Pearson Lemma CB 8.1,
Economics 520 Lecture Note 9: Hypothesis Testing via the Neyman-Pearson Lemma CB 8., 8.3.-8.3.3 Uniformly Most Powerful Tests and the Neyman-Pearson Lemma Let s return to the hypothesis testing problem
More information[Chapter 6. Functions of Random Variables]
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
More informationChapter 12. Analysis of variance
Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout
More informationCHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)
FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7
MA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7 1 Random Vectors Let a 0 and y be n 1 vectors, and let A be an n n matrix. Here, a 0 and A are non-random, whereas y is
More informationEpidemiology Principles of Biostatistics Chapter 10 - Inferences about two populations. John Koval
Epidemiology 9509 Principles of Biostatistics Chapter 10 - Inferences about John Koval Department of Epidemiology and Biostatistics University of Western Ontario What is being covered 1. differences in
More informationUnit 9: Inferences for Proportions and Count Data
Unit 9: Inferences for Proportions and Count Data Statistics 571: Statistical Methods Ramón V. León 1/15/008 Unit 9 - Stat 571 - Ramón V. León 1 Large Sample Confidence Interval for Proportion ( pˆ p)
More informationPaired comparisons. We assume that
To compare to methods, A and B, one can collect a sample of n pairs of observations. Pair i provides two measurements, Y Ai and Y Bi, one for each method: If we want to compare a reaction of patients to
More informationChapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.
Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:
More informationMath 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
More informationNonparametric tests. Timothy Hanson. Department of Statistics, University of South Carolina. Stat 704: Data Analysis I
1 / 16 Nonparametric tests Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I Nonparametric one and two-sample tests 2 / 16 If data do not come from a normal
More informationChapter 15: Nonparametric Statistics Section 15.1: An Overview of Nonparametric Statistics
Section 15.1: An Overview of Nonparametric Statistics Understand Difference between Parametric and Nonparametric Statistical Procedures Parametric statistical procedures inferential procedures that rely
More informationOne sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:
One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get
More informationLecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2
Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y
More informationLecture 21: October 19
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 21: October 19 21.1 Likelihood Ratio Test (LRT) To test composite versus composite hypotheses the general method is to use
More informationCorrelation 1. December 4, HMS, 2017, v1.1
Correlation 1 December 4, 2017 1 HMS, 2017, v1.1 Chapter References Diez: Chapter 7 Navidi, Chapter 7 I don t expect you to learn the proofs what will follow. Chapter References 2 Correlation The sample
More informationProbability- the good parts version. I. Random variables and their distributions; continuous random variables.
Probability- the good arts version I. Random variables and their distributions; continuous random variables. A random variable (r.v) X is continuous if its distribution is given by a robability density
More informationWe know from STAT.1030 that the relevant test statistic for equality of proportions is:
2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which
More informationAsymptotic Statistics-VI. Changliang Zou
Asymptotic Statistics-VI Changliang Zou Kolmogorov-Smirnov distance Example (Kolmogorov-Smirnov confidence intervals) We know given α (0, 1), there is a well-defined d = d α,n such that, for any continuous
More informationSTAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test
STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test Rebecca Barter April 13, 2015 Let s now imagine a dataset for which our response variable, Y, may be influenced by two factors,
More information7.3 The Chi-square, F and t-distributions
7.3 The Chi-square, F and t-distributions Ulrich Hoensch Monday, March 25, 2013 The Chi-square Distribution Recall that a random variable X has a gamma probability distribution (X Gamma(r, λ)) with parameters
More informationOne sample problem. sample mean: ȳ = . sample variance: s 2 = sample standard deviation: s = s 2. y i n. i=1. i=1 (y i ȳ) 2 n 1
One sample problem Population mean E(y) = µ, variance: Var(y) = σ 2 = E(y µ) 2, standard deviation: σ = σ 2. Normal distribution: y N(µ, σ 2 ). Standard normal distribution: z N(0, 1). If y N(µ, σ 2 ),
More informationStat 529 (Winter 2011) Experimental Design for the Two-Sample Problem. Motivation: Designing a new silver coins experiment
Stat 529 (Winter 2011) Experimental Design for the Two-Sample Problem Reading: 2.4 2.6. Motivation: Designing a new silver coins experiment Sample size calculations Margin of error for the pooled two sample
More informationNonparametric hypothesis tests and permutation tests
Nonparametric hypothesis tests and permutation tests 1.7 & 2.3. Probability Generating Functions 3.8.3. Wilcoxon Signed Rank Test 3.8.2. Mann-Whitney Test Prof. Tesler Math 283 Fall 2018 Prof. Tesler Wilcoxon
More informationRank-Based Methods. Lukas Meier
Rank-Based Methods Lukas Meier 20.01.2014 Introduction Up to now we basically always used a parametric family, like the normal distribution N (µ, σ 2 ) for modeling random data. Based on observed data
More informationThe Components of a Statistical Hypothesis Testing Problem
Statistical Inference: Recall from chapter 5 that statistical inference is the use of a subset of a population (the sample) to draw conclusions about the entire population. In chapter 5 we studied one
More informationSTA 2101/442 Assignment 3 1
STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance
More information7.2 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
More informationregression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist
regression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist sales $ (y - dependent variable) advertising $ (x - independent variable)
More information1 Hypothesis testing for a single mean
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 informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationSTAT 512 sp 2018 Summary Sheet
STAT 5 sp 08 Summary Sheet Karl B. Gregory Spring 08. Transformations of a random variable Let X be a rv with support X and let g be a function mapping X to Y with inverse mapping g (A = {x X : g(x A}
More information5 Introduction to the Theory of Order Statistics and Rank Statistics
5 Introduction to the Theory of Order Statistics and Rank Statistics This section will contain a summary of important definitions and theorems that will be useful for understanding the theory of order
More informationNormal (Gaussian) distribution The normal distribution is often relevant because of the Central Limit Theorem (CLT):
Lecture Three Normal theory null distributions Normal (Gaussian) distribution The normal distribution is often relevant because of the Central Limit Theorem (CLT): A random variable which is a sum of many
More informationKeppel, G. & Wickens, T.D. Design and Analysis Chapter 2: Sources of Variability and Sums of Squares
Keppel, G. & Wickens, T.D. Design and Analysis Chapter 2: Sources of Variability and Sums of Squares K&W introduce the notion of a simple experiment with two conditions. Note that the raw data (p. 16)
More informationMultiple Regression Analysis
Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 +... β k x k + u 2. Inference 0 Assumptions of the Classical Linear Model (CLM)! So far, we know: 1. The mean and variance of the OLS estimators
More information