# y ij = µ + α i + ɛ ij,

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 α i = 0 idetifiability costrait Plaed comparisos - sigle pairs of meas, or costraits specified i advace Differece of Meas e.g. µ i µ j : like a two-sample test -but, we have a ANOVA model ad hece the pooled variace estimate s for the commo variace σ. 00( α)%ci ȳ i ȳ j ± t α/[ν] SEȳi ȳ j, ν = a, SEȳi ȳ j = s + i j Ex: (Pea sectio data) Legth of pea sectios grow i tissue cultures [] Cotrol Glucose Fructose Gluc+ Fruc Sucrose peas=sca() pea.df=data.frame(peas,culture=as.factor(rep(:5,0))) culture=as.factor(rep(:5,0)) culture [46] Levels: pea.lm=lm(peas~culture,data=pea.df) aova(pea.lm) Respose: peas Df Sum Sq Mea Sq F value Pr(>F) culture e-6 *** Residuals Sigif. codes: 0 *** 0.00 ** 0.0 * pea.resid=residuals(pea.lm) pea.fitted=fitted(pea.lm) matrix(roud(pea.resid,),col=5,byrow=t) [,] [,] [,3] [,4] [,5] [,]

2 [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [0,] > sum(roud(pea.resid,)^) [] 45.5 matrix(roud(pea.fitted,),col=5,byrow=t) [,] [,] [,3] [,4] [,5] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [0,] % CI for µ c µ g, differece betwee the cotrol group ad the glucose group: ȳ c ȳ g = = 0.8 s = s = MS withi = 5.46 =.34 SEȳc ȳ g = =.046 dof ν = a = 50 5 = 45 ; t.975[45] =.05 C.I. = 0.8 ± (.0)(.046) = [8.69,.9] Cotrast: A liear combiatio of meas where the coefficiets sum to zero: Populatio Sample a a γ = c i µ i c i = 0 i= i= c i ȳ i i Ex: sugars vs. cotrol γ = µ c 4 (µ g + µ f + µ g+f + µ s ), coefficiets (c i ) = (, 4, 4, 4, 4 ) Typically used to compare groups of meas or certai weighted combiatios ( orthogoal cotrasts ) such as liear or quadratic effects. Variace of a sample cotrast (assumig the sample meas are idepedet) Estimated SE c, SE c = s i c i i, Var c = Var ( c i ȳ i ) = c i Var(ȳ i ) = c i 00( α)%ci C ± t α/ [ν]se c σ i

3 Ex: C = 70. ( ) = 0. 4 c i = i 0 [ ] = = c 8, i = i 8 = SE c = (.34) (0.3536) = CI for C is 0. ± (.0) (0.873) = [8.53,.87] Hypothesis Test for Cotrast: e.g. H 0 : γ = γ 0 Form a t-statistic t = C γ 0 t [ν] if H 0 is true. SE c Remark: differeces betwee meas are a speacial case of cotrasts: e.g. µ c µ g = a i= c i µ i with (C i ) = (,, 0, 0, 0). These types of ivestigatios should be doe o combiatios of factors that were determied i advace of observig the experimetal results, or else the cofidece levels are ot as specified by the procedure. Also, doig several comparisos might chage the overall cofidece level. This ca be avoided by carefully selectig cotrasts to ivestigate i advace ad makig sure that: the umber of such cotrasts does ot exceed the umber of degrees of freedom betwee the treatmets *oly orthogoal cotrasts are chose. However, there are also several powerful multiple compariso procedures we ca use after observig the experimetal results. Uplaed comparisos After lookig at the data, we may wish to assess the sigificace of, or give C.I. s for certai differeces e.g. µ g µ f (i pea legth e.g.) or cotrasts e.g. µ s (µ 3 g + µ f + µ gf ) That looked iterestig a posteriori. Viewed a priori, however, there are may differeces or cotrasts that could potetially attract attetio. We eed to adjust our sigificace levels ad P-values (larger) or our C.I. s (wider) to allow for this search over all possibilities. Subject of multiple comparisos -see books, e.g. Miller, R.G. Simulateous Statistic Iferece All pairs of differeces with a treatmets, there are ( ) a = a(a ) possible comparisos of differet meas: µ i µ j, i =,..., a; j =,..., i If we used t-itervals, would have may itervals of form I ij ȳ i ȳ j ± t α/[ν] SEȳi ȳ j But the chace that all itervals simultaeously cover all µ i µ j : P {I ij coverµ i µ j ; for all]i < j} < α To obtai a simultaeous coverage property, make itevals wider I T K ij ȳ i ȳ j ± Q α[a,ν] SEȳi ȳ j TK = Tukey Kramer Q α[a,ν] are percetage poits of studetized rage distributio. Formal defiitio: Q [a,ν] = max Z i Z j s where Z, Z,..., Z a N(0, ); νs χ (ν) ad all idepedet gives wider itervals Q α[a,ν] a=!) Ex. (pea legths) Q.95[5,45] = 4.0 =.84(>.0 = t.975[45] ) > t α/[ν] (uless 3

4 > qtukey(0.95,5,45) [] 4.0 simultaeous iterval for µ c µ g i 0.8 ± (.843)(.046) = [7.83, 3.8] Simultaeous coverage property if Model I holds, ad = =... = a ( balaced ), the P (Iij T K covers µ i µ j for all i < j) = α Remark: If the ANOVA is ubalaced (ot all i equal) the these Tukey-Kramer itervals are coservative (coverage prob α). Whe comparig the meas for the levels of a factor i a aalysis of variace, a simple compariso usig t-tests will iflate the probability of declarig a sigificat differece whe it is ot i fact preset. This because the itervals are calculated with a give coverage probability for each iterval but the iterpretatio of the coverage is usually with respect to the etire family of itervals. Joh Tukey itroduced itervals based o the rage of the sample meas rather tha the idividual differeces. The itervals retured by this fuctio are based o this Studetized rage statistics. Techically the itervals costructed i this way would oly apply to balaced desigs where there are the same umber of observatios made at each level of the factor. This fuctio icorporates a adjustmet for sample size that produces sesible itervals for mildly ubalaced desigs. >peas.aov_aov(peas~gr) >TukeyHSD(peas.aov) Tukey multiple comparisos of meas 95% family-wise cofidece level Fit: aov(formula = peas ~ gr) \$gr diff lwr upr > peas.hsd_tukeyhsd(peas.aov) > plot(peas.hsd) % family wise cofidece level Differeces i mea levels of gr The term experimet wise error rate α arises because, if H 0 is true (all µ i equal), the the chace of falsely declarig as sigificat ay of the a(a ) pair wise diffs is (at most) α : P H0 {max ȳ i ȳ i SEȳi ȳ i > Q α/[a,ν] } α ( = α if all i equal) Balaced Case ad Hoestly Sigificat Differece (HSD) if all i =, the all SEȳi ȳ i = s ȳ i ȳ i > Q α/ [a, ν] s so just fid those pairs (ȳ i ȳ i ) separated by > HSD 4

5 Overlappig itervals picture - the ± HSD itervals overl ap if ad oly if ȳ i ȳ i HSD meas (µ i, µ i ) whose HSD itervals do t overlap are sigificatly differet at experimet wise error rate α. (Warig! ȳ i ± HSD is NOT a 00( α)% Cof. iterval! ) All cotrasts: The Scheffé itervals I s ± (a )F α,[a,ν] SE c have the simultaeous coverage property (for balaced or ubalaced cases) P {I s coverforallcotrasts} = a α Sice cotrasts are more geeral tha differeces, expect Scheffé itervals to be eve wider tha Tukey-Kramer Ex: γ = µ s (µ 3 g + µ f + µ gf )c = (0,,,, ) c i i ( ) = = c SE c = s i = (.34)(.365) = (a )F α[a,ν] = 4F.95[4,45] = 4 x (.58) = % Scheffé iterval for c = x s ( x 3 g + x f + x gf ) = = 5.6 has margi effor (3.)(0.8544) =.743 CI [5.6.74, ] = [.86, 8.34] (Note that Scheffé multiplier = 3. >.84 = Qα[a,ν] = Tukey-Kramer multiplier) Remark: - There is a versio of the T-K itervals for cotrasts - these ca be better (shorter) tha the Scheffé method if a is larger ad relatively fewer c i are o-zero cotr.peas=matrix(c(4,-,-,-,-,0,-,-,3,-),col=) cotr.peas [,] [,] [,] 4 0 [,] - - [3,] - - [4,] - 3 [5,] - - cotrasts(culture)=cotr.peas 5

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### (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

### 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

### 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

### 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

### REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I liear regreio, we coider the frequecy ditributio of oe variable (Y) at each of everal level of a ecod variable (X). Y i kow a the depedet variable.

### 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

### 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

### 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

### 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

### 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

### Analysis of Algorithms -Quicksort-

Aalysis of Algorithms -- Adreas Ermedahl MRTC (Mälardales Real-Time Research Ceter) adreas.ermedahl@mdh.se Autum 2004 Proposed by C.A.R. Hoare i 962 Worst- case ruig time: Θ( 2 ) Expected ruig time: Θ(

### 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,

### 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

### Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

### Output Analysis and Run-Length Control

IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

### 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

### Lesson 10: Limits and Continuity

www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

### 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

### 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

### 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

### 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

### 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

### Probability, Expectation Value and Uncertainty

Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

### w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For

### CHAPTER III RESEARCH METHODOLOGY

CHAPTER III RESEARCH METHODOLOGY A. Method of the Research I this research the writer used the experimetal method. The experimetal research was aimed to kow if there were effect or ot for the populatio

### CS / MCS 401 Homework 3 grader solutions

CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of

### 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 ] / (

### Grouping 2: Spectral and Agglomerative Clustering. CS 510 Lecture #16 April 2 nd, 2014

Groupig 2: Spectral ad Agglomerative Clusterig CS 510 Lecture #16 April 2 d, 2014 Groupig (review) Goal: Detect local image features (SIFT) Describe image patches aroud features SIFT, SURF, HoG, LBP, Group

### 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

### 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,

### 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.

### 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

### MA131 - Analysis 1. Workbook 2 Sequences I

MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

### 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

### 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

### (a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

### 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

### 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

### 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

### 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

### 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

### Chapter Objectives. Bivariate Data. Terminology. Lurking Variable. Types of Relations. Chapter 3 Linear Regression and Correlation

Chapter Objectives Chapter 3 Liear Regressio ad Correlatio Descriptive Aalysis & Presetatio of Two Quatitative Data To be able to preset two-variables data i tabular ad graphic form Display the relatioship

### 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

### Analysis of Algorithms. Introduction. Contents

Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We

### Commutativity in Permutation Groups

Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are

### 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

### 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

### 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

### 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

### Correlation and Covariance

Correlatio ad Covariace Tom Ilveto FREC 9 What is Next? Correlatio ad Regressio Regressio We specify a depedet variable as a liear fuctio of oe or more idepedet variables, based o co-variace Regressio

### Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University

Dottig The Dot Map, Revisited A. Jo Kimerlig Dept. of Geoscieces Orego State Uiversity Dot maps show the geographic distributio of features i a area by placig dots represetig a certai quatity of features

### 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

### Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

### Section 5.1 The Basics of Counting

1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of

### 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

### 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

### 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

### 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,

### Overdispersion study of poisson and zero-inflated poisson regression for some characteristics of the data on lamda, n, p

Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: 2442-6571 140 Overdispersio study of poisso ad zero-iflated poisso regressio for some characteristics of the data o lamda,, p Lili Puspita Rahayu

### Math 609/597: Cryptography 1

Math 609/597: Cryptography 1 The Solovay-Strasse Primality Test 12 October, 1993 Burt Roseberg Revised: 6 October, 2000 1 Itroductio We describe the Solovay-Strasse primality test. There is quite a bit

### 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

### sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

### 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

### 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

### (b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

### HOMEWORK 2 SOLUTIONS

HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k

### SALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS

SALES AND MARKETING Departmet MATHEMATICS d Semester Bivariate statistics LESSONS Olie documet: http://jff-dut-tc.weebly.com sectio DUT Maths S. IUT de Sait-Etiee Départemet TC J.F.Ferraris Math S StatVar

### 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

### Complex Numbers Solutions

Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i

### 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

### ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t

ARIMA Models Da Sauders I will discuss models with a depedet variable y t, a potetially edogeous error term ɛ t, ad a exogeous error term η t, each with a subscript t deotig time. With just these three

### 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

### 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,

### Statistical Inference Procedures

Statitical Iferece Procedure Cofidece Iterval Hypothei Tet Statitical iferece produce awer to pecific quetio about the populatio of iteret baed o the iformatio i a ample. Iferece procedure mut iclude a

### 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

### 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

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

### 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

### Zeros of Polynomials

Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

### 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

### Recurrence Relations

Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

### 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

### 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,