Stat 200 Testing Summary Page 1

 Martha Carson
 9 months ago
 Views:
Transcription
1 Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece Itervals 1001 α% largesample cofidece iterval for a populatio mea µ X z α σ, X + z α σ, where z α is the costat such that P Z z α = 1 α Notes a The values of z α for various cofidece coefficiets are readily obtaied from the table of ormal probabilities I particular, 1001 α% z α 90% % % 58 b Whe σ is ukow, ad is large say, bigger tha 30, the it is usual to replace σ i the cofidece iterval formula by its estimator s c The cofidece iterval result holds as log as X i have fiite variace, ot just whe the X i s are ormal If, however, X i is ormal, i = 1,,, the the cofidece iterval above is exact ie has exact cofidece coefficiet 1001 α% 1001 α% large sample cofidece iterval for a differece i meas µ X µ Y X Y z α σ X + σ Y, X Y + z α σ X + σ Y Whe the true variaces σ X ad σ Y are ukow ad ad are large, the s X ad s Y are used i place of σx ad σ Y, respectively, i the above formula 1001 α% large sample cofidece iterval for a proportio p ˆp z α ˆp1 ˆp, ˆp + z α ˆp1 ˆp 1001 α% large sample cofidece iterval for a differece i proportios p 1 p ˆp 1 ˆp z α ˆp 1 1 ˆp 1 + ˆp 1 ˆp, ˆp 1 ˆp + z α ˆp 1 1 ˆp 1 + ˆp 1 ˆp Large sample hypothesis tests
2 Stat 00 Testig Summary Page Note: oesided tests are give i paretheses ad square brackets [ ] These tests are valid for large say, > 30, ad populatio variaces ca be replaced by sample variaces wherever appropriate Large sample hypothesis test for a populatio mea µ We wish to test H 0 : µ = µ 0 agaist the alterative H A : µ µ 0 H A : µ > µ 0 [H A : µ < µ 0 ] The test statistic is Z = X µ 0 σ/ or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a differece i meas µ X µ Y We wish to test H 0 : µ X µ Y = D 0 agaist the alterative H A : µ X µ Y D 0 H A : µ X µ Y > D 0 [H A : µ X µ Y < D 0 ] The test statistic is Z = X Y D 0 σ X + σ Y or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a proportio p We wish to test H 0 : p = p 0 agaist the alterative H A : p p 0 H A : p > p 0 [H A : p < p 0 ] The test statistic is Z = ˆp p 0 p0 1 p 0 or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ] Large sample hypothesis test for a differece i proportios p 1 p We wish to test H 0 : p 1 = p agaist the alterative H A : p 1 p H A : p 1 > p [H A : p 1 < p ] Let ˆp = X + Y / + deote the overall proportio of successes for the two populatios The test statistic is ˆp 1 ˆp 0 Z = 1 ˆp1 ˆp + 1 or if the observed value of Z exceeds z α [if the observed value of Z is smaller tha z α ]
3 Stat 00 Testig Summary Page 3 3 Smallsample cofidece itervals 1001 α% small sample cofidece iterval for a populatio mea µ X t 1 α/ s, X + t 1 α/ s, where t 1 α/ is the critical poit of a t 1 distributio such that P T t 1 α/ = 1 α, ie 1001 α% of the area uder the t 1 desity lies betwee t 1 α/ ad t 1 α/ 1001 α% small sample cofidece iterval for a differece i meas µ X µ Y from idepedet populatios To compute this iterval we eed to make the additioal assumptio that the variaces of the two populatios are equal ie that σx = σ Y The, we estimate this commo variace usig the pooled variace estimator, The cofidece iterval is the s pooled = 1s X + 1s Y + X Y t 1+ α/s pooled , X Y + t 1+ α/s pooled Note that the degrees of freedom used for the t statistic is α% small sample cofidece iterval for a differece i meas µ X µ Y from paired data The data is X 1, Y 1,, X, Y Form the differeces D 1 = X 1 Y 1,, D = X Y, ad let D ad s D deote the sample mea ad sample stadard deviatio of the differeces D i, i = 1,, The the iterval is D t 1 α/ sd, D + t 1 α/ sd 4 Small sample hypothesis tests Note: Oesided tests are give i paretheses ad square brackets [ ] Small sample hyptohesis test for a mea µ We wish to test H 0 : µ = µ 0 agaist H A : µ µ 0 H A : µ > µ 0 [H A : µ < µ 0 ] The test statistic is T = X µ 0 s/ We reject H 0 i favor of H A at sigificace level α if the observed value of T either exceeds t 1 α/ or is smaller tha t 1 α/ if the observed value of T exceeds t 1 α [if the observed value of T is smaller tha t 1 α] Note that the α/ th critical poit is used for twosided tests while the α th critical poit is used for oesided tests
4 Stat 00 Testig Summary Page 4 Small sample hypothesis test for a differece i meas µ X µ Y from idepedet populatios We wish to test H 0 : µ X µ Y = D 0 agaist H A : µ X µ Y D 0 H A : µ X µ Y > D 0 [H A : µ X µ Y < D 0 ] The test statistic is T = X Y D 0 s pooled 1 + 1, where s pooled is the pooled variace estimator s pooled = 1s X + 1s Y / + We reject H 0 i favor of H A at sigificace level α if the observed value of T e ither exceeds t 1 + α/ or is smaller tha t 1 + α/ if the observed value of T exceeds t 1 + α [if the observed value of T is smaller tha t 1 + α] freedom for this test is + ot 1 Note the degrees of Small sample hypothesis test for a differece i meas µ X µ Y from paired data We are iterested i testig H 0 : µ X µ Y = 0 agaist H A : µ X µ Y 0 H A : µ X µ Y > 0 [H A : µ X µ Y < 0] The test is based o the differeces D i = X i Y i, i = 1,,, so is equivalet to testig H 0 : µ D = 0 agaist H A : µ D 0 H A : µ D > 0 [H A : µ D < 0], where µ D is the mea of the populatio of differeces This is just a oesample test for the mea µ D, so the test statistic is T = D 0 s D / We reject H 0 i favor of H A at sigificace level α if the observed value of T e ither exceeds t 1 α/ or is smaller tha t 1 α/ if the observed value of T exceeds t 1 α [if the observed value of T is smaller tha t 1 α] 5 Tests ad cofidece itervals for the variace 1001 α% cofidece iterval for the populatio variace σ 1s χ 1 α/, 1s χ, 11 α/ where χ 1 α/ is the critical poit of the χ 1 distributio such that if X χ 1 the P X χ 1α/ = 1 α/ Hypothesis test for the variace σ We wish to test the hypothesis H 0 : σ = σ0 agaist H A : σ σ 0 H A : σ > σ 0 [H A : σ < σ 0] The test statistic is χ = 1s σ0 We reject H 0 i favor of H A if the observed χ value either exceeds χ 1 α/ or is smaller tha χ 1 1 α/ if the observed value of χ exceeds χ 1 α [if the observed value of χ is smaller
5 Stat 00 Testig Summary Page 5 tha χ 1 1 α] Note that the α/ th critical poit of the χ 1 distributio is used for the twotailed test, while the α th critical poit is used for the oesided test Hypothesis test for the equality of variaces Recall that i order to carry out a test of the hypothesis that µ X = µ Y from idepedet populatios, it was ecessary to assume σx = σ Y We ca formally test this assumptio as follows: we wish to test H 0 : σx = σ Y H A : σx > σ Y [H A : σx > σ Y ] The test statistic is agaist H A : σ X σ Y F = s X s Y We reject H 0 i favor of H A at sigificace level α if the observed F value exceeds F 1 1, 1α/, the upper α/ th critical poit of a F 1 1, 1 distributio, is falls below F 1 1, 11 α/ if the observed value of F exceeds F 1 1, 1α [if the observed value of F is smaller tha F 1 1, 11 α Tables for both the F ad χ distributios for various degrees of freedom are give at the back of the book Note that i order to coduct some of the tests above we eed use the special idetity for F distributios: F,m 1 α = 1 F m, α
Statistics 20: Final Exam Solutions Summer Session 2007
1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationStatistics. Chapter 10 TwoSample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 101
Statistics Chapter 0 TwoSample Tests Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0 Learig Objectives I this chapter, you lear How to use hypothesis testig for comparig the differece
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationChapter 4 Tests of Hypothesis
Dr. Moa Elwakeel [ 5 TAT] Chapter 4 Tests of Hypothesis 4. statistical hypothesis more. A statistical hypothesis is a statemet cocerig oe populatio or 4.. The Null ad The Alterative Hypothesis: The structure
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationLecture 9: Independent Groups & Repeated Measures ttest
Brittay s ote 4/6/207 Lecture 9: Idepedet s & Repeated Measures ttest Review: Sigle Sample ztest Populatio (otreatmet) Sample (treatmet) Need to kow mea ad stadard deviatio Problem with this? Sigle
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationChapter 22: What is a Test of Significance?
Chapter 22: What is a Test of Sigificace? Thought Questio Assume that the statemet If it s Saturday, the it s the weeked is true. followig statemets will also be true? Which of the If it s the weeked,
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationQuestion 1: Exercise 8.2
Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.
More informationy ij = µ + α i + ɛ ij,
STAT 4 ANOVA Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i
More informationConfidence Level We want to estimate the true mean of a random variable X economically and with confidence.
Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio
More informationAssessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions
Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the
More informationTable 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab
Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationCH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions
CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe, ad two, sided tolerace itervals based o either a distributiofree (oparametric) method or a method based o a ormality assumptio (parametric). A twosided
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationFirst Year Quantitative Comp Exam Spring, Part I  203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationAsymptotic distribution of the firststage Fstatistic under weak IVs
November 6 Eco 59A WEAK INSTRUMENTS III Testig for Weak Istrumets From the results discussed i Weak Istrumets II we kow that at least i the case of a sigle edogeous regressor there are weakidetificatiorobust
More informationRegression. Correlation vs. regression. The parameters of linear regression. Regression assumes... Random sample. Y = α + β X.
Regressio Correlatio vs. regressio Predicts Y from X Liear regressio assumes that the relatioship betwee X ad Y ca be described by a lie Regressio assumes... Radom sample Y is ormally distributed with
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationStat 3411 Spring 2011 Assignment 6 Answers
Stat 3411 Sprig 2011 Aigmet 6 Awer (A) Awer are give i 10 3 cycle (a) 149.1 to 187.5 Sice 150 i i the 90% 2ided cofidece iterval, we do ot reject H 0 : µ 150 v i favor of the 2ided alterative H a : µ
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More informationBHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13
BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the
More informationSampling, Sampling Distribution and Normality
4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
More informationCTL.SC0x Supply Chain Analytics
CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos
More informationAsymptotic Results for the Linear Regression Model
Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationHypothesis tests and confidence intervals
Hypothesis tests ad cofidece itervals The 95% cofidece iterval for µ is the set of values, µ 0, such that the ull hypothesis H 0 : µ = µ 0 would ot be rejected by a twosided test with α = 5%. The 95%
More informationf(x)dx = 1 and f(x) 0 for all x.
OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More information2.2. Central limit theorem.
36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral LidebergFeller CLT, it is most stadard
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationHomework for 4/9 Due 4/16
Name: ID: Homework for 4/9 Due 4/16 1. [ 136] It is covetioal wisdom i military squadros that pilots ted to father more girls tha boys. Syder 1961 gathered data for military fighter pilots. The sex of
More informationTesting Statistical Hypotheses with Fuzzy Data
Iteratioal Joural of Statistics ad Systems ISS 973675 Volume 6, umber 4 (), pp. 44449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH email: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationEstimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches
Iteratioal Joural of Mathematical Aalysis Vol. 8, 2014, o. 48, 23752383 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ijma.2014.49287 Estimatig Cofidece Iterval of Mea Usig Classical, Bayesia,
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA SA 0B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology eachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More information1 Hypothesis test of a mean vector
THE UNIVERSITY OF CHICAGO Booth School of Busiess Busiess 41912, Sprig Quarter 2010, Mr Ruey S Tsay Lecture: Iferece about sample mea Key cocepts: 1 Hotellig s T 2 test 2 Likelihood ratio test 3 Various
More informationGrant MacEwan University STAT 151 Formula Sheet Final Exam Dr. Karen Buro
Grat MacEwa Uiverity STAT 151 Formula Sheet Fial Exam Dr. Kare Buro Decriptive Statitic Sample Variace: = i=1 (x i x) 1 = Σ i=1x i (Σ i=1 x i) 1 Sample Stadard Deviatio: = Sample Variace = Media: Order
More informationSTAT331. Example of Martingale CLT with Cox s Model
STAT33 Example of Martigale CLT with Cox s Model I this uit we illustrate the Martigale Cetral Limit Theorem by applyig it to the partial likelihood score fuctio from Cox s model. For simplicity of presetatio
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationTables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.
Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER Orgaizig ad Summarizig Data Relative frequecy frequecy um of all frequecie Cla midpoit: The um of coecutive lower
More informationTRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN
HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries  November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi
More informationA RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider ksample ad chage poit problems for idepedet data i a
More informationCONFIDENCE INTERVALS
CONFIDENCE INTERVALS CONFIDENCE INTERVALS Documets prepared for use i course B01.1305, New York Uiversity, Ster School of Busiess The otio of statistical iferece page 3 This sectio describes the tasks
More informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 705041010,
More informationEcon 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.
Eco 371 Exam #1 Multiple Choice (5 poits each): For each of the followig, select the sigle most appropriate optio to complete the statemet 1) The probability of a outcome a) is the umber of times that
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationCE3502 Environmental Monitoring, Measurements, and Data Analysis (EMMA) Spring 2008 Final Review
CE35 Evirometal Moitorig, Meauremet, ad Data Aalyi (EMMA) Sprig 8 Fial Review I. Topic:. Decriptive tatitic: a. Mea, Stadard Deviatio, COV b. Bia (accuracy), preciio, Radom v. ytematic error c. Populatio
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationChapter VII Measures of Correlation
Chapter VII Measures of Correlatio A researcher may be iterested i fidig out whether two variables are sigificatly related or ot. For istace, he may be iterested i kowig whether metal ability is sigificatly
More informationV. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany
PROBABILITY AND STATISTICS Vol. III  Correlatio Aalysis  V. Nollau CORRELATION ANALYSIS V. Nollau Istitute of Mathematical Stochastics, Techical Uiversity of Dresde, Germay Keywords: Radom vector, multivariate
More informationAdvanced Engineering Mathematics Exercises on Module 4: Probability and Statistics
Advaced Egieerig Mathematics Eercises o Module 4: Probability ad Statistics. A survey of people i give regio showed that 5% drak regularly. The probability of death due to liver disease, give that a perso
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. BetaBinomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 12631277 HIKARI Ltd, www.mhikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
More informationImportant Concepts not on the AP Statistics Formula Sheet
Part I: IQR = Q 3 Q 1 Test for a outlier: 1.5(IQR) above Q 3 or below Q 1 The calculator will ru the test for you as log as you choose the boplot with the oulier o it i STATPLOT Importat Cocepts ot o the
More informationLecture 11 October 27
STATS 300A: Theory of Statistics Fall 205 Lecture October 27 Lecturer: Lester Mackey Scribe: Viswajith Veugopal, Vivek Bagaria, Steve Yadlowsky Warig: These otes may cotai factual ad/or typographic errors..
More informationBIOSTATISTICAL METHODS FOR TRANSLATIONAL & CLINICAL RESEARCH
BIOSAISICAL MEHODS FOR RANSLAIONAL & CLINICAL RESEARCH Direct Bioassays: REGRESSION APPLICAIONS COMPONENS OF A BIOASSAY he subject is usually a aimal, a huma tissue, or a bacteria culture, he aget is usually
More informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationStatistical Inference
Solved Exercises ad Problems of Statistical Iferece David Casado Complutese Uiversity of Madrid Faculty of Ecoomic ad Busiess Scieces Departmet of Statistics ad Operatioal Research II David Casado de Lucas
More information18. Twosample problems for population means (σ unknown)
8. Twosamle roblems for oulatio meas (σ ukow) The Practice of Statistics i the Life Scieces Third Editio 04 W.H. Freema ad Comay Objectives (PSLS Chater 8) Comarig two meas (σ ukow) Twosamle situatios
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More information