# UCLA STAT 110B Applied Statistics for Engineering and the Sciences

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 How to Implemet Wilcoxo s Rak est Rak the combied sample from smallest to largest. Let represet the sum of the raks of the first sample (A s). he, defied below, is the sum of the raks that the A s would have had if the observatios were raked from large to small. ( + + ) he Wilcoxo Rak Sum est H 0 :: the two populatio distributios are the same H a a :: the two populatios are i i some way differet he test statistic is the smaller of ad. Reject H 0 if the test statistic is less tha the critical value foud i able (a). able (a) is idexed by lettig populatio be the oe associated with the smaller sample size, ad populatio as the oe associated with, the larger sample size. Slide Slide 8 he wig stroke frequecies of two species If If several of measuremets bees were are are recorded tied, for a sample of each gets from the species the average of of the ad the raks 6 from species. they would have gotte, if if they Ca were you coclude that the distributios of wig ot tied! (See x 80) strokes differ for these two species? Use α.05. Species 35 (0) 5 (9) 90 (8) 88 () H Species 0 : the are the H 80 a : a : the the two species are are i i some way differet (3.5) 69 (). he sample with the smaller sample 80 size is called sample. (3.5) 85 (6). We rak the 0 observatios from 8 () smallest to largest, show i 8 (5) paretheses i the table. Slide 9 he Bee Problem Ca you coclude that the distributios of wig strokes differ for these two species? α.05. Species Reject Species H 0.. Data Calculate provides sufficiet evidece (0) 80 (3.5) idicatig a differece i i 5 the (9) 69 () ( + + ) distributios of of wig stroke 90 frequecies. (8) 80 (3.5) ( ) () (6) () (5). he test statistic is 0.. he critical value of from able (b) for a twotailed test with α/.05 is ; H 0 is rejected if. Slide 0 Miitab Output Recall 3; 0. MaWhitey est ad CI: Species, Species Species N Media 0.50 Species N 6 Media Poit estimate for EAEA is Percet CI for EAEA is (5.99,56.0) W 3.0 est of EA EA vs EA ot EA is sigificat at 0.0 he test is sigificat at (adjusted for ties) Miitab calls the the procedure the the MaWhitey U est, equivalet to to the the Wilcoxo Rak Sum est. he test statistic is is W 3 3 ad has pvalue.0. Do Do ot reject H 0 for 0 for α.05. Slide Large Sample Approximatio: Wilcoxo Rak Sum est Whe ad are large (greater tha 0 is large eough), a ormal approximatio ca be used to approximate the critical values i able.. Calculate ad. Let mi,. he statistic z σ z distributio with µ ( + + ) ad σ has a approximate sum of the raks of sample (A s). ( + + ) Slide ). ( + + ) µ

3 Some Notes Whe should you use the Wilcoxo Rak Sum test istead of the twosample t test for idepedet samples? whe the resposes ca oly be raked ad ot quatified (e.g., ordial qualitative data) whe the F test or the Rule of humb shows a problem with equality of variaces whe a ormality plot shows a violatio of the ormality assumptio he Sig est he sig test is a fairly simple procedure that ca be used to compare two populatios whe the samples cosist of paired observatios. It ca be used whe the assumptios required for the paireddifferece test are ot valid or whe the resposes ca oly be raked as oe better tha the other, but caot be quatified. Slide 3 Slide he Sig est he Sig est For each pair, measure whether the first respose say, A exceeds the secod respose say, B. he test statistic is is x, the umber of times that A exceeds B i the pairs of observatios. Oly pairs without ties are icluded i the test. Critical values for the rejectio regio or exact pvalues ca be foud usig the cumulative biomial distributio (SOCR resource olie). Slide 5 H 0 : the two populatios are idetical versus H a : oe or twotailed alterative is is equivalet to H 0 : p P(A exceeds B).5 versus H a : p (, <, or >).5 est statistic: x umber of plus sigs Rejectio regio, pvalues from Bisize, p). Slide 6 he Gourmet Chefs wo gourmet chefs each tasted ad rated eight differet meals from to 0. Does it appear that oe of the chefs teds to give higher ratigs tha the other? Use α.0. Meal Chef A Chef B Sig H 0 : 0 : the the ratig distributios are are the the same (p (p.5).5) H a : a : the the ratigs are are differet (p (p.5).5) Slide Meal Chef A Chef B Sig + + pvalue.5 0 is too large to pvalue.5 is too large to reject H 0. H 0 : 0 : p.5 0. here is is isufficiet.5 evidece to to idicate that oe H a : a : p.5.5 with (omit chef the the tied pair) teds to to rate oe meal est Statistic: x umber of of higher plus sigs tha the the other. Use able with ad p pvalue P(observe x or or somethig equally as as ulikely) P(x ) ) + P(x 5) 5) (.).5 k P(x k) Slide 8

4 Large Sample Approximatio: he Sig est Y~Bi, p) E(Y) p Var(Y) p(p) Whe 5, a ormal approximatio ca be used to approximate the critical values of Biomial distributio.. Calculate x. he statistic z distributio. umber of plus sigs. x.5 z.5 has a approximate You record the umber of accidets per day at a large maufacturig plat for both the day ad eveig shifts for 00 days. You fid that the umber of accidets per day for the eveig shift x E exceeded the For correspodig a two tailed umber test, we we of reject accidets H 00 i the day shift x D o 63 of the if if 00 z z > days..96 (5% Do these level). results provide sufficiet evidece to idicate H that 0 is more accidets ted to occur 0 is rejected. here is is evidece o oe shift tha o the other? of of a differece betwee the the day ad H 0 : 0 : the the distributios (# (# of of accidets) ight shifts. are are the the same (p (p.5).5) H a : a : the the distributios are are differet (p (p.5).5) est statistic: z x (00) Slide 9 Slide 0 Which test should you use? Which test should you use? We compare statistical tests usig Defiitio: Power β P(reject H 0 whe H a is true) he power of the test is the probability of rejectig the ull hypothesis whe it is false ad some specified alterative is true. he power is the probability that the test will do what it was desiged to do that is, detect a departure from the ull hypothesis whe a departure exists. If all parametric assumptios have bee met, the parametric test will be the most powerful. If ot, a oparametric test may be more powerful. If you ca reject H 0 with a less powerful oparametric test, you will ot have to worry about parametric assumptios. If ot, you might try more powerful oparametric test or icreasig the sample size to gai more power Slide Slide he Wilcoxo SigedRak est differet form Wilcoxo Rak Sum est he Wilcoxo SigedRak est is a more powerful oparametric procedure that ca be used to compare two populatios whe the samples cosist of paired observatios. It uses the raks of the differeces, d x x that we used i the paireddifferece test. Slide 3 he Wilcoxo SigedRak est differet form Wilcoxo Rak Sum est For each pair, calculate the differece d x x x.. Elimiate zero differeces. Rak the absolute values of of the differeces from to to.. ied observatios are assiged average of of the raks they would have gotte if if ot tied. + rak sum for positive differeces rak sum for egative differeces If the two populatios are the same, + ad should be early equal. If If either + or or is is uusually large, this provides evidece agaist the ull hypothesis. Slide

5 he Wilcoxo SigedRak est H 0 : the two populatios are idetical versus H a : oe or twotailed alterative est statistic: mi ( + ad ) Critical values for a oe or twotailed rejectio regio ca be foud usig Wilcoxo SigedRak est able. o compare the desities of cakes usig mixes A ad B, six pairs of pas (A ad B) were baked sidebyside i six differet ove locatios. Is there evidece of a differece i desity for the two cake mixes? Locatio Cake Mix A Cake Mix B d x A x B H : 0 : the the desity distributios are are the the same 0 H a : a : the the desity distributios are are differet Slide 5 Slide 6 Locatio Cake Mix A Cake Mix B d x A x B Rak Cake Desities Do ot reject H 0.here is Do ot reject H 0.here is isufficiet..3 evidece. to to idicate.9.0. that.5there.35is is a differece.63 i i desities.0.00 for for the the.09 two cake mixes Rak Calculate: the the 6 + ad differeces, he test statistic is is mi ( ( +,, )).. without Rejectio regard regio: Use able For a twotailed test with to α to sig..05, reject H 0 if 0 if.. Large Sample Approximatio: he SigedRak est Whe 5, a ormal approximatio ca be used to approximate the critical values i able 8.. Calculate + ad. Let mi, µ. he statistic z hasa approximat e σ z distributio with + ) + )( + ) µ ad σ + ). Slide Slide 8 he KruskalWallis H est he KruskalWallis H est he KruskalWallis H est is a oparametric procedure that ca be used to compare more tha two populatios i a completely radomized desig. Noparametric equivalet to ANOVA Ftest! All k measuremets are joitly raked. We use the sums of the raks of the k samples to compare the distributios. Slide 9 Rak the total measuremets i i all k samples from to to.. ied observatios are assiged average of of the raks they would have gotte if if ot tied. Calculate i i rak sum for the ii th th sample i i,,,,k k Ad the test statistic H is is (aalog to: F MSS/MSSE) k H i 3( + ) ( + ) i i Slide 30

6 he he KruskalWallis H est H est H 0 : the k distributios are idetical versus H a : at least oe distributio is differet est statistic: KruskalWallis H Whe H 0 is true, the test statistic H has a approximate χ distributio with df k. Use a righttailed rejectio regio or pvalue based o the Chisquare distributio. Four groups of studets were radomly assiged to be taught with four differet techiques, ad their achievemet test scores were recorded. Are the distributios of test scores the same, or do they differ i locatio? Slide 3 Slide 3 i eachig Methods 3 65 (3) 5 () 59 () 9 (6) 8 (3) 69 (5) 8 (8) 89 (5) 3 (6) 83 () 6 () 80 (0) 9 (9) 8 () 6 () 88 () 3 Rak the the 6 6 H : 0 : the the distributios of of scores are are the the same 0 measuremets H a : a : the the distributios differ i i locatio from to to 6, 6, ad calculate i eststatistic: H 3( + ) the the four rak + ) i sums () () Slide eachig Methods H : 0 : the the distributios of of scores are are the the same 0 H a : a : the the distributios differ i i locatio i eststatistic: H 3( + ) + ) i () () Rejectio regio: Use able For a righttailed chisquare test with α ad df df 3, reject H 0 if 0 if H.8. Slide 3 Reject H here is is sufficiet evidece to to idicate that there is is a differece i i test scores for for the the four teachig techiques. I. Noparametric Methods. hese methods ca be used whe the data caot be measured o a quatitative scale, or whe. he umerical scale of measuremet is arbitrarily set by the researcher, or whe 3. he parametric assumptios such as ormality or costat variace are seriously violated. II. Wilcoxo Rak Sum est: Idepedet Radom Samples. Joitly rak the two samples: Desigate the smaller sample as sample. he Rak of sample ( + + ) Slide. Use to test for populatio to the left of populatio Use to test for populatio to the right of populatio. Use the smaller of ad to test for a differece i the locatios of the two populatios. 3. able of Appedix I has critical values for the rejectio of H 0.. Whe the sample sizes are large, use the ormal approximatio: µ z σ ( + + ) ( + + ) µ ad σ Slide 8

7 III. Sig est for a Paired Experimet. Fid x, the umber of times that observatio A exceeds observatio B for a give pair.. o test for a differece i two populatios, test H 0 : p 0.5 versus a oe or twotailed alterative. 3. Use able of Appedix I to calculate the pvalue for the test.. Whe the sample sizes are large, use the ormal approximatio: x.5 z.5 IV. Wilcoxo SigedRak est: Paired Experimet. Calculate the differeces i the paired observatios. Rak the absolute values of the differeces. Calculate the rak sums ad + for the positive ad egative differeces, respectively. he test statistic is the smaller of the two rak sums.. able 8 of Appedix I has critical values for the rejectio of for both oe ad twotailed tests. 3. Whe the samplig sizes are large, use the ormal approximatio: [ + ) ] z [ + )( + ) ] Slide 9 Slide 50 V. KruskalWallis H est: Completely Radomized Desig. Joitly rak the observatios i the k samples. Calculate the rak sums, i rak sum of sample i, ad the test statistic i H 3( + ) + ) i. If the ull hypothesis of equality of distributios is false, H will be uusually large, resultig i a oetailed test. 3. For sample sizes of five or greater, the rejectio regio for H is based o the chisquare distributio with (k ) degrees of freedom. VI. he Friedma F r est: Radomized Block Desig. Rak the resposes withi each block from to k. Calculate the rak sums,,, k, ad the test statistic Fr i 3b( k + ) bk( k + ). If the ull hypothesis of equality of treatmet distributios is false, F r will be uusually large, resultig i a oetailed test. 3. For block sizes of five or greater, the rejectio regio for F r is based o the chisquare distributio with (k ) degrees of freedom. Slide 5 Slide 5 VII. Spearma's Rak Correlatio Coefficiet. Rak the resposes for the two variables from smallest to largest.. Calculate the correlatio coefficiet for the raked observatios: r s S S S xx xy yy or r s 6 di ) 3. able 9 i Appedix I gives critical values for rak correlatios sigificatly differet from 0.. he rak correlatio coefficiet detects ot oly sigificat liear correlatio but also ay other mootoic relatioship betwee the two variables. Slide 53 if thereareo ties Sesitivity vs. Specificity Sesitivity is a measure of the fractio of gold stadard kow examples that are correctly classified/idetified. Sesitivity P/(P+FN) Specificity is a measure of the fractio of egative examples that are correctly classified: Specificity N/(N+FP) P rue Positives H o : o effects (µ0) rue Reality FN False Negatives H o true H o false N rue Negatives Ca t reject N FN FP False Positives Reject H o FP P est Results Slide 5

8 he ROC Curve Receiver Operatig Characteristic (ROC) curve.0 Fractio rue Positives Sesitivity Better est Worse est Bigger area higher test accuracy ad better discrimiatio i.e. the ability of the test to correctly classify those with ad without the disease. Fractio False Positives Specificity Slide 55.0 ReceiverOperatig Characteristic curve ROC curve demostrates several thigs: It shows the tradeoff betwee sesitivity ad specificity (ay icrease i sesitivity will be accompaied by a decrease i specificity). he closer the curve follows the left border ad the the top border of the ROC space, the more accurate the test. he closer the curve comes to the 5degree diagoal of the ROC space, the less accurate the test. he slope of the taget lie at a cutpoit gives the likelihood ratio (LR) for that value of the test. You ca check this out o the graph above. he area uder the curve is a measure of test accuracy. Slide 56

### 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,

### 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

### Stat 200 -Testing Summary Page 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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 ] / (

### 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

### 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

### Paired Data and Linear Correlation

Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0

### 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

### 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

### 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

### SIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS

SIMPLE LINEAR REGRESSION AND CORRELATION ANALSIS INTRODUCTION There are lot of statistical ivestigatio to kow whether there is a relatioship amog variables Two aalyses: (1) regressio aalysis; () correlatio

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

### Chapter 4 - Summarizing Numerical Data

Chapter 4 - Summarizig Numerical Data 15.075 Cythia Rudi Here are some ways we ca summarize data umerically. Sample Mea: i=1 x i x :=. Note: i this class we will work with both the populatio mea µ ad the

### 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

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### 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.,

### 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

### 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

### 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

### ORF 245 Fundamentals of Engineering Statistics. Midterm Exam 2

Priceto Uiversit Departmet of Operatios Research ad Fiacial Egieerig ORF 45 Fudametals of Egieerig Statistics Midterm Eam April 17, 009 :00am-:50am PLEASE DO NOT TURN THIS PAGE AND START THE EXAM UNTIL

### Discrete probability distributions

Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

### 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

### REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

### Sequences I. Chapter Introduction

Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

### Closed book and notes. No calculators. 60 minutes, but essentially unlimited time.

IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of

### 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

### OBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:

OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet

### 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

### (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

### 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

### Matrix Representation of Data in Experiment

Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y

### 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

### Measures of Spread: Variance and Standard Deviation

Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample

### WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still

### Elementary Statistics

Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

### Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

REGRESSION WITH QUADRATIC LOSS MAXIM RAGINSKY Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X, Y ), where, as before, X is a R d

### A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

### 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

### 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

### Monte Carlo Integration

Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

### 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.

### 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,

### 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,

### 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: >

### 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,

### 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

### Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

### Stat 400, section 5.4 supplement: The Central Limit Theorem

Stat, sectio 5. supplemet: The Cetral Limit Theorem otes by Tim Pilachowski Table of Cotets 1. Backgroud 1. Theoretical. Practical. The Cetral Limit Theorem 5. Homework Exercises 7 1. Backgroud Gatherig

### 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

### 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

### 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

### EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics

### Singular Continuous Measures by Michael Pejic 5/14/10

Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

### 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

### 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

### Introduction to Computational Molecular Biology. Gibbs Sampling

18.417 Itroductio to Computatioal Molecular Biology Lecture 19: November 16, 2004 Scribe: Tushara C. Karuarata Lecturer: Ross Lippert Editor: Tushara C. Karuarata Gibbs Samplig Itroductio Let s first recall

### 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

### MA 575, Linear Models : Homework 3

MA 575, Liear Models : Homework 3 Questio 1 RSS( ˆβ 0, ˆβ 1 ) (ŷ i y i ) Problem.7 Questio.7.1 ( ˆβ 0 + ˆβ 1 x i y i ) (ȳ SXY SXY x + SXX SXX x i y i ) ((ȳ y i ) + SXY SXX (x i x)) (ȳ y i ) SXY SXX SY

### 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

### Formulas FROM LECTURE 01 TO 22 W X. d n. fx f. Arslan Latif (mt ) & Mohsin Ali (mc ) Mean: Weighted Mean: Mean Deviation: Ungroup Data

1 Formulas FROM LECTURE 01 TO Mea: fx f Weighted Mea: X w W X i i Wi Mea Deviatio: Ugroup Data d M. D Group Data fi di M. D f d ( X X ) Coefficiet of Mea Deviatio: M. D Co-efficiet of M. D(for mea) Mea

### 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

### 1 Hash tables. 1.1 Implementation

Lecture 8 Hash Tables, Uiversal Hash Fuctios, Balls ad Bis Scribes: Luke Johsto, Moses Charikar, G. Valiat Date: Oct 18, 2017 Adapted From Virgiia Williams lecture otes 1 Hash tables A hash table is a

### IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

### 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

### Introduction to Artificial Intelligence CAP 4601 Summer 2013 Midterm Exam

Itroductio to Artificial Itelligece CAP 601 Summer 013 Midterm Exam 1. Termiology (7 Poits). Give the followig task eviromets, eter their properties/characteristics. The properties/characteristics of the

### 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

### 1. By using truth tables prove that, for all statements P and Q, the statement

Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

### Posted-Price, Sealed-Bid Auctions

Posted-Price, Sealed-Bid Auctios Professors Greewald ad Oyakawa 207-02-08 We itroduce the posted-price, sealed-bid auctio. This auctio format itroduces the idea of approximatios. We describe how well this

### Median and IQR The median is the value which divides the ordered data values in half.

STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

### 7: Sampling Distributions

7: Samplig Distributios 7.1 You ca select a simple radom sample of size = 2 usig Table 1 i Appedix I. First choose a startig poit ad cosider the first three digits i each umber. Sice the experimetal uits

### Hashing and Amortization

Lecture Hashig ad Amortizatio Supplemetal readig i CLRS: Chapter ; Chapter 7 itro; Sectio 7.. Arrays ad Hashig Arrays are very useful. The items i a array are statically addressed, so that isertig, deletig,

Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,

### 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

### 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

### Sample Correlation. Mathematics 47: Lecture 5. Dan Sloughter. Furman University. March 10, 2006

Sample Correlatio Mathematics 47: Lecture 5 Da Sloughter Furma Uiversity March 10, 2006 Da Sloughter (Furma Uiversity) Sample Correlatio March 10, 2006 1 / 8 Defiitio If X ad Y are radom variables with

### True Nature of Potential Energy of a Hydrogen Atom

True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial

### 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

### 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

### 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

### 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