Stat 200 -Testing Summary Page 1

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Stat 200 -Testing Summary Page 1"

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% large-sample 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: oe-sided 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 Small-sample 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: Oe-sided 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 two-sided tests while the α th critical poit is used for oe-sided 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 oe-sample 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 two-tailed test, while the α th critical poit is used for the oe-sided 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

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 information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (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 XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, 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 o-users, x - y. Such studies are sometimes viewed

More information

Statistics. Chapter 10 Two-Sample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 10-1

Statistics. Chapter 10 Two-Sample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 10-1 Statistics Chapter 0 Two-Sample 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 information

STATISTICAL INFERENCE

STATISTICAL 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 information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 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 large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Chapter 4 Tests of Hypothesis

Chapter 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 information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 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:

(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 information

Sampling Distributions, Z-Tests, Power

Sampling Distributions, Z-Tests, Power Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

Joint 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 information

The 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.

The 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 information

Lecture 9: Independent Groups & Repeated Measures t-test

Lecture 9: Independent Groups & Repeated Measures t-test Brittay s ote 4/6/207 Lecture 9: Idepedet s & Repeated Measures t-test Review: Sigle Sample z-test Populatio (o-treatmet) Sample (treatmet) Need to kow mea ad stadard deviatio Problem with this? Sigle

More information

Topic 6 Sampling, hypothesis testing, and the central limit theorem

Topic 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 information

Parameter, Statistic and Random Samples

Parameter, 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 information

Statistical inference: example 1. Inferential Statistics

Statistical 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 information

Chapter 22: What is a Test of Significance?

Chapter 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 information

Chapter 1 (Definitions)

Chapter 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 information

Question 1: Exercise 8.2

Question 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 information

y ij = µ + α i + ɛ ij,

y 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 information

Confidence Level We want to estimate the true mean of a random variable X economically and with confidence.

Confidence 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 information

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions

Assessment 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 information

Table 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

Table 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 information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 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 information

CH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions

CH19 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 information

NCSS Statistical Software. Tolerance Intervals

NCSS Statistical Software. Tolerance Intervals Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

More information

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: 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 information

The standard deviation of the mean

The 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 information

Binomial Distribution

Binomial 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 information

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. 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 information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First 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 I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

More information

Chapter 6 Principles of Data Reduction

Chapter 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 information

UCLA STAT 110B Applied Statistics for Engineering and the Sciences

UCLA 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 information

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality A goodess-of-fit 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 information

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

More information

Confidence Intervals for the Population Proportion p

Confidence 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 information

Section 14. Simple linear regression.

Section 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 information

Asymptotic distribution of the first-stage F-statistic under weak IVs

Asymptotic distribution of the first-stage F-statistic 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 weak-idetificatio-robust

More information

Regression. Correlation vs. regression. The parameters of linear regression. Regression assumes... Random sample. Y = α + β X.

Regression. 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 information

Testing Statistical Hypotheses for Compare. Means with Vague Data

Testing Statistical Hypotheses for Compare. Means with Vague Data Iteratioal Mathematical Forum 5 o. 3 65-6 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 information

STA 4032 Final Exam Formula Sheet

STA 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 information

Stat 3411 Spring 2011 Assignment 6 Answers

Stat 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% 2-ided cofidece iterval, we do ot reject H 0 : µ 150 v i favor of the 2-ided alterative H a : µ

More information

STAT 203 Chapter 18 Sampling Distribution Models

STAT 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 information

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

BHW #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 information

Sampling, Sampling Distribution and Normality

Sampling, 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 information

Simple Linear Regression

Simple 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 information

CTL.SC0x Supply Chain Analytics

CTL.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 information

Asymptotic Results for the Linear Regression Model

Asymptotic 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 information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians 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 information

Hypothesis tests and confidence intervals

Hypothesis 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 two-sided test with α = 5%. The 95%

More information

f(x)dx = 1 and f(x) 0 for all x.

f(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 information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 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 information

2.2. Central limit theorem.

2.2. 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 Lideberg-Feller CLT, it is most stadard

More information

MASSACHUSETTS 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

MASSACHUSETTS 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 information

Homework for 4/9 Due 4/16

Homework for 4/9 Due 4/16 Name: ID: Homework for 4/9 Due 4/16 1. [ 13-6] 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 information

Testing Statistical Hypotheses with Fuzzy Data

Testing Statistical Hypotheses with Fuzzy Data Iteratioal Joural of Statistics ad Systems ISS 973-675 Volume 6, umber 4 (), pp. 44-449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui

More information

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

More information

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

It 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 information

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. 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 information

Probability and statistics: basic terms

Probability 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 information

Statistical Theory MT 2009 Problems 1: Solution sketches

Statistical 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 information

Lecture 4. Random variable and distribution of probability

Lecture 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 e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za

More information

Estimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches

Estimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches Iteratioal Joural of Mathematical Aalysis Vol. 8, 2014, o. 48, 2375-2383 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49287 Estimatig Cofidece Iterval of Mea Usig Classical, Bayesia,

More information

Lecture 1 Probability and Statistics

Lecture 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 information

UCLA STAT 110B Applied Statistics for Engineering and the Sciences

UCLA 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 information

Element sampling: Part 2

Element 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 information

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.

IE 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 information

1 Hypothesis test of a mean vector

1 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 information

Grant MacEwan University STAT 151 Formula Sheet Final Exam Dr. Karen Buro

Grant 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 information

STAT331. Example of Martingale CLT with Cox s Model

STAT331. 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 information

A LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!

A 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 information

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS R775 Philips Res. Repts 26,414-423, 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 information

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.

Tables 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 information

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN

TRACEABILITY 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 information

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a

More information

CONFIDENCE INTERVALS

CONFIDENCE 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 information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The 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 information

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

Some 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 70504-1010,

More information

Econ 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.

Econ 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 information

Review 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

Review 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 information

CE3502 Environmental Monitoring, Measurements, and Data Analysis (EMMA) Spring 2008 Final Review

CE3502 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 information

Section 11.8: Power Series

Section 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 information

Chapter VII Measures of Correlation

Chapter 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 information

V. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany

V. 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 information

Advanced Engineering Mathematics Exercises on Module 4: Probability and Statistics

Advanced 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 information

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters

More information

Regression with an Evaporating Logarithmic Trend

Regression 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 information

Important Concepts not on the AP Statistics Formula Sheet

Important 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 information

Lecture 11 October 27

Lecture 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 information

BIOSTATISTICAL METHODS FOR TRANSLATIONAL & CLINICAL RESEARCH

BIOSTATISTICAL 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 information

Probability and Statistics

Probability 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 information

Introduction to Probability and Statistics Twelfth Edition

Introduction 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 information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 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 information

Statistical Inference

Statistical 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 information

18. Two-sample problems for population means (σ unknown)

18. Two-sample problems for population means (σ unknown) 8. Two-samle 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) Two-samle situatios

More information

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 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 z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

MASSACHUSETTS 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 information

DISTRIBUTION LAW Okunev I.V.

DISTRIBUTION 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