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


 Jewel May Marsh
 1 years ago
 Views:
Transcription
1 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 aswer to questios as for example Discrete distributio: Give =0 childbirths ad probability of low birth weight p=0., what is the probability to observe at least 3 low birth childre? Model: X ~ bi(, p ). P( X 3 = 0, p= 0. ) Estimatio Harald Johse, Sept 00 Cotiuous distributio: Give that the birth weight X is ormally distributed with mea μ = 3750 g ad stadard deviatio σ = 500 g. What is the probability that a ewbor weighs is at least 54 grams? Model: N ~ N( μ, σ ) = N( 3750, 500 ) P( X 54 μ = 3750, σ = 500 ) = P Z 500 Populatio ad sample New questios How ca a sample (NO: utvalg) be used to estimate ukow parameters i a probability distributio? How to estimate the cetral tedecy i a distributio? Which oe is the best measure of the cetral tedecy? How to estimate variability (variace)? How to place a iterval aroud a poit estimate to idicate how sure the estimate is? A populatio (statistical populatio, target populatio) is the complete set of the possible measuremets, or the record of some qualitative trait correspodig to the etire collectio of uits for which ifereces ca be made. A sample is a limited subset of a populatio that is actually collected i the course of a ivestigatio. The objective of the process of data collectio (samplig) is to draw coclusios about the populatio. Essetial questios: Which populatio? How is the samplig doe? Give a sample, to which populatio are the coclusios valid? 3 4
2 Some commo samplig procedures A radom sample is a selectio of some members of a populatio such that members are idepedetly chose ad each member has a kow ozero probability of beig selected A simple radom sample is a radom sample i which each member has the same probability of beig selected Stratified samplig: the populatio is divided ito homogeous subsets (strata) based upo specified traits of the members(sex, age,.) ad subsequetly radom samplig withi each stratum. ad there are more Poit estimatio Give a populatio ad a represetative sample. Based o the sample, the challege is to estimate ukow quatities (parameters) i the populatio distributio. Recall that a parameter is a fixed, usually ukow umeric quatity ad accordigly oradom. Examples: I the biomial distributio X ~ bi(, p )the parameter is p I a ormal distributio X ~N( μ, σ ) the parameters are μ og σ I the Poisso distributio X~Po( λ ) the parameter is λ I the geeral case the Greek letter θ (theta) is used as symbol for a parameter. 5 6 Poit estimatio cot Suppose simple radom samplig of size from a effectively ifiite populatio with populatio mea μ ad variace σ. This gives idetically distributed radom variables X, X,..., X (ot ecessarily ormally distributed). A estimator ˆθ (theta hat)is a mathematical fuctio of the radom variables ad is used to estimate the ukow value of the parameter θ. The estimator ˆθ is a radom variable with a probability distributio. Whe a radom sample becomes available from the populatio ad ˆθ is computed from the data set, the umeric value obtaied is called a estimate of θ from the particular sample. Ulike the estimator, a estimate is oradom! The sample arithmetic mea is a ituitive or atural estimator of the populatio mea μ : μˆ X + X X = = X The properties of the estimator ˆμ = X Expected value (mea, NO: forvetig): X + X X ( ˆ ) = ( ) = = = E μ EX E μ μ Hece, the estimator is ubiased (NO: forvetigsrett).. a good property Variace (provided idepedet observatios): X + X X σ ˆ = = = = Var( μ) Var( X ) Var σ It follows that Var( μˆ ) 0whe a good property too 7 8
3 Estimator cot. For oe ad the same parameter there may exist several ubiased estimators. I symmetric distributios with oly oe mode (NO: modalverdi) the sample mea the sample media the sample mode are all ubiased estimators for the populatio mea, but their variaces may be differet Usually, the estimator havig the least variace is chose. For ormally distributed data that estimator will be the sample mea. (Sometimes a ubiased estimator is chose for the cost of a estimator with less variace.) The distributio of the mea We have show that if X,X,...,X are idepedet ad ormally distributed with meaμ ad variace σ, the σ X is ormally distributed with mea μ ad variace It ca be show that eve if X,X,...,X are ot ormally distributed, X will be approximately ormally distributed whe is sufficietly large. If the distributio of X is reasoably symmetric without too may modes, ad ot too peculiar, this setece will practically hold as early as from > The variace of the arithmetic mea Mea of oormally distributed data Fig. 3: The lower curve shows a ormal distributio with mea (expectatio)3 ad variace.44, (SD=.). The arithmetic mea of a radom sample of size 6 will be ormally distributed with mea 3 ad variace.44/6=0.09, (SD=0.3). This is the peaked, arrow distributio. Fig. 3: Travellig times from home to campus. 300 radom samples of size, 4, 9 ad 6. (Aale 998)
4 Iterval estimatio A poit estimate (of a ukow parameter) is a umeric value obtaied by puttig observed sample values ito the mathematical formula for the estimator. Questios of iterest: How precise it the estimate? Is it possible to calculate a iterval coverig the estimated parameter with a specified probability? The aswer is NO! o But there is a recipe tellig how such a probability iterval, cofidece iterval, ca be costructed. But as soo as observed values are used ad a umeric iterval is calculated, that iterval ca ot loger be iterpreted withi a probability framework. Costructio of cofidece itervals Suppose X,X,...,X are idepedet ad ormally distributed with mea μ ad variace σ. This gives ˆ μ μ Z = ~N(,0) σ ˆ μ μ < = P zα/< z α/ = α σ ad P( z Z z ) α/ α/ Because zα / = z α /, σ σ P ˆ μ z α/ < μ ˆ μ + z α/ = α, which is a probability statemet ad the recipe to costruct a ( α )cofidece iterval for the parameter μ 3 4 σ σ The iterval is ˆ μ z α/,z α/ + Suppose repeated samples of size. Each time we estimate a ew ˆμ = x, ad a ew cofidece iterval. Choosig α = 0.05 ad exchagig ˆμ with X, we have σ σ P X.96 < μ X +.96 = 0.05 = 0.95 We arrive at the radom iterval σ σ X.96, X +.96 with σ fixed legth.96 Factors affectig the legth? Siceμ is ad remais ukow, we will ever kow which itervals i fact do cotai the parameter!! 5 6
5 Cofidece iterval, ormal distributio with ukow variace The same argumet as above, but the populatio variace is estimated by the sample variace σ = s = (Xi X ) i= ad we arrive at the tdistributio with  degrees of freedom givig s s P X t, α/ < μ X + t, α/ = α ad get the radom iterval s s X t, α/ < μ X + t, α/ with s radom legth t, α / (What is radom.?) Approximatio to ormal distributio Biomial series of trials (discrete distributio) i) Each trial yields oe of two outcomes techically called success (A) ad failure (A*) ii) For each trial, the probability of success P(A) is the same ad is deoted p=p(a). The probability of failure is the P(A*) =  p ad is deoted p, so that p + q =. iii) Trials are idepedet. The probability of success i a trial does ot chage give ay amout of iformatio about the outcomes i other trials. iv) The umber if success, X, is observed i trials. k P( X = k ) = p ( p) k k 7 8 Cofidece iterval for p X p( p) ˆp =, E( p) ˆ = p, Var( p ˆ ) = (cosistet estimator) Stadardisig ˆp by subtractio of mea ad divisio to stadard deviatio: We defie The pˆ p pˆ p Z = = SD( p ˆ ) p( p ) atoutcome A, P( A) = p I = 0atoutcome A*, P( A*) = p= q i i= X = I = I + I I is the umber of outcomes A i trials X I + I I Note that ˆp = = is s a sum of several idepedet evets, each oe without domiace to X. The cetral limit theorem ow implies that as icreases, pˆ p pˆ p Z = = will coverge to the stadard ormal SD( p ˆ ) p( p ) distributio. The approximatio works especially well if p( ˆ p ˆ ) > 5 Whe the coditio above is met, the ( α ) cofidece iterval for p is approximately ˆp± z p( p) / α / x A umerical result is achieved by replacig p with a estimate ˆp = (ote small x, observed value of X). 9 0
6 Example 6.44, prevalece of breast cacer amog wome 50 54: Radom sample =0000, Observed umber of cacer x = 400 Poit estimate of prevalece: the target populatio) 400 ˆp = = (estimated prevalece i 0000 p( ˆ p ˆ ) = = 38.4 > 5, approximatio to ormal distributio applies cofidece iterval estimate: p ˆ z p( ˆ p) ˆ /, p ˆ + z p( ˆ p) ˆ / ( ) =( / 0000, / 0000 ) = ( , ) = ( 0.036,0.044) Suppose we kow that the prevalece i the populatio is 0.0. How to iterpret the fidigs above? Example, exercise 4.40 Sample size =00 Primary evet, A: bacteriuria, P(A) = p = 0.05 X: umber of wome havig bacteriuria Questio: What is P( X 3) Model: X bi(,p) P( X 3) = P( X < ) = ( P( X = 0) + P( X = ) + P( X = ) ) P( X = 0) = = P( X = ) = = P( X = ) = = 0.08 P( X < ) = = 0.8 P( X 3 ) = 0.8 = 0.88 Approximatio to ormal distributio p( p ) = ( 0.05 ) = 4.75, (borderlie for.a.) E X = p = = 5 [ ] [ ] Var X = p( p ) = P( X 3) P( Z =.5) = P( Z.5) = reasoably fair compared to Fial commet, small sample cases: If is ot sufficietly large for the cetral limit theorem to apply, or approximatio to the ormal distributio does t work, exact methods have to be used. 3 4
71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationStatistical Intervals for a Single Sample
3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8 Cofidece Iterval o the Mea
More informationCH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions
CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationActivity 3: Length Measurements with the FourSided Meter Stick
Activity 3: Legth Measuremets with the FourSided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a foursided meter
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationStat 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
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several wellow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationCURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:
CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH email: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationDiscrete 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
More informationThe Poisson Distribution
MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This
More informationUnbiased Estimation. February 712, 2008
Ubiased Estimatio February 72, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationY i n. i=1. = 1 [number of successes] number of successes = n
Eco 371 Problem Set # Aswer Sheet 3. I this questio, you are asked to cosider a Beroulli radom variable Y, with a success probability P ry 1 p. You are told that you have draws from this distributio ad
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationExam 2 Instructions not multiple versions
Exam 2 Istructios Remove this sheet of istructios from your exam. You may use the back of this sheet for scratch work. This is a closed book, closed otes exam. You are ot allowed to use ay materials other
More informationEksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>.
Eco 43 Eksame 6 H Utsatt SENSORVEILEDNING Settet består av 9 delspørsmål som alle abefales å telle likt. Svar er gitt i . Problem a. Let the radom variable (rv.) X be expoetially distributed with
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationProbability, 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
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe ad TwoSample Tests of Hypotheses 0 Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationSampling, Sampling Distribution and Normality
4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationFirst Year Quantitative Comp Exam Spring, Part I  203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationChapter 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 information4.1 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
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationLecture 10 October Minimaxity and least favorable prior sequences
STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least
More informationHOMEWORK 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
More informationDistribution of Sample Proportions
Distributio of Samle Proortios Probability ad statistics Aswers & Teacher Notes TINsire Ivestigatio Studet 90 mi 7 8 9 10 11 12 Itroductio From revious activity: This activity assumes kowledge of the
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationStatistics 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 informationSampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Lecture 16
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Lecture 16 Variace Questio: Let us retur oce agai to the questio of how may heads i a typical sequece of coi flips. Recall that we
More informationModeling and Performance Analysis with DiscreteEvent Simulation
Simulatio Modelig ad Performace Aalysis with DiscreteEvet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More information7: 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
More informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationConfidence Intervals QMET103
Cofidece Itervals QMET103 Library, Teachig ad Learig CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their
More informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oedimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationJacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3
NoParametric Techiques Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3 Parametric vs. NoParametric Parametric Based o Fuctios (e.g Normal Distributio) Uimodal Oly oe peak Ulikely real data cofies
More informationStat 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
More informationB Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets
B671672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x
More informationThe coalescent coalescence theory
The coalescet coalescece theory Peter Beerli September 1, 009 Historical ote Up to 198 most developmet i populatio geetics was prospective ad developed expectatios based o situatios of today. Most work
More informationFormulas 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 Coefficiet of M. D(for mea) Mea
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationElementary 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
More informationMA131  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..........................
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,
More information( µ /σ)ζ/(ζ+1) µ /σ ( µ /σ)ζ/(ζ 1)
A eective CI for the mea with samples of size 1 ad Melaie Wall James Boe ad Richard Tweedie 1 Abstract It is couterituitive that with a sample of oly oe value from a ormal distributio oe ca costruct a
More informationLecture 9: September 19
36700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio BiasVariace
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Cotets 1. A few useful discrete radom variables 2. Joit, margial, ad
More information20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE
20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE If the populatio tadard deviatio σ i ukow, a it uually will be i practice, we will have to etimate it by the ample tadard deviatio. Sice σ i ukow,
More informationLesson 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
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationClosed 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
More information