Statistik II - Exercise session &
|
|
- Susan Patterson
- 5 years ago
- Views:
Transcription
1 Statistik II - Exercise sessio & Ifo Classroom: SPA1 03 Time: Wedesdays, 16:15-17:45 i Eglish Assigmets o webpage (lvb>staff>pb) Cotact: Petra Burdejova petra.burdejova@hu-berli.de Office: SPA1 R400 (upo agreemet) Schedule: Date Week Exercises E1 4-11, 4-18, E1 4-11, 4-18, E 5-6, 5-10, 5-11, 5-14, E 5-6, 5-10, 5-11, 5-14, E3 5-16, 5-1, 5-3, E3 5-16, 5-1, 5-3, E4 6-3, 6-9, 6-13, E4 6-3, 6-9, 6-13, E5 7-3, 7-5, 7-6, E5 7-3, 7-5, 7-6, E6 8-1, 8-4, 8-7, E6 8-1, 8-4, 8-7, E7 TBA E8 Review for exam E8 Review for exam 1
2 Review week 9 & week 10 Slides: Theory of samplig (05 Stichprobetheorie) Estimatio procedures (06 Schätztheorie) 1 Sample distributio 1.1 Sample distributio of sample mea Distributio of X Populatio R.v. Distr. Coditio X i N(µ; ) kow Z = X µ / N(0, 1) t( 1) for 30 ukow T = X µ S/ N(0, 1) for > 30 Ukow distributio kow Z = X µ / ukow T = X µ S/ 1. Sample distributio of Sample proportio Sample fuctio: ˆΠ = X (Example: Smokers i Berli, X - umber of smokers i our sample.) Distributio of simple radom sample N(0, 1) for > 30 N(0, 1) for > 30 X B(; π) E(X) = π V ar(x) = π (1 π) Approximatio by ormal distributio: ( ) π(1 π) ˆΠ N π; ˆΠ = Estimatio procedures true parameter of populatio θ Estimator (fuctio) θ = g(x1,..., X ) MSE=Mea Square Error MSE = E[( θ θ) ] = E[{ θ E( θ)} ] + {E( θ) θ} }{{}}{{} =V ar( θ) =bias Example: θ = µ, θ = X = X i
3 .1 Maximum - Likelihood (ML) Method Likelihood-Fuctio L(θ) = L(θ x 1,..., x ) = f(x i θ) LogLikelihood-Fuctio log(l(θ)) = log(f(x i θ)) maximize maximize. Least Squares (LS) Method Quadratic Form Q(θ) = (x i E(X i )) = (x i g i (θ)) miimize.3 Cofidece iterval at level 1 Cofidece iterval for µ kow Cof. iterval P Estimator iterval Legth X i ormally distributed or distr. of populatio ukow, but 30 ( ) X z 1 µ X + z 1 = 1 [ X z 1 [ x z 1 ; X + z 1 ; x + z 1 ] ] z 1 l = e = z 1 with l = legth ad e = error from N(0; 1) sample size z 1 e Cofidece iterval for proportio π for Normal approximatio Approximative Cofidece iterval Estimator iterval X B(; π) ad ˆΠ = X/ is approximately ormally distributed ( X P z 1 ˆΠ π X ) + z 1 ˆΠ = 1 X X z 1 (1 ) X ; X X + z 1 (1 ) X x x z 1 (1 ) x ; x + z 1 z 1 from N(0, 1) ) x (1 x Sample size z 1 / 4 e 3
4 Exercises Exercise Lamps A supply of N = 000 lamps will be ivestigated by meas of a simple radom sample of size = 0. With a help of the radom variable X: umber of defective bulbs i the sample of size =0 aumber d of defective lamps i the supply is estimated. a) Give a ubiased estimator θ = f(x) for d ad show that E(θ) = d. b) I a give sample, the umber of defective bulbs is equal to 3. How may defective bulbs do you estimate i the delivery? Exercise Gamblig machie A gamblig machie has the followig probability distributio for the wi X per game (i EUR): x P(X = x) p p 1- p The producer of these machies hired a statisticia to perform a estimate for p to kow whether the value of p has chaged sice the gamblig machies starts up. a) The statisticia draws a sample of size = 6, i.e. plays with the machie 6 times ad writes dow the wi. The sample (X 1, X, X 3, X 4, X 5, X 6 ) had realizatio as follows: (-1,1,-1,0,1,1). Verbalize this sample result. b) Calculate the followig probabilities: P(X = 0),P(X = 1),P(X = 1). c) How would you determie the probabilities of wi X per game accordig to sample metioed above, if you have o iformatio about the probability distributio of X? d) What is the probability P {(X 1, X, X 3, X 4, X 5, X 6 ) = ( 1, 1, 1, 0, 1, 1)} based o the above probability distributio? e) Determie the maximum likelihood estimator for p i this problem. f) Estimate p by this sample result through the maximum likelihood method. g) Estimate p by this sample result through the least squared method. Exercise Dioxi emmisios It is believed that the dioxi emissios of a cosmetic factory pre miute are ormally distributed with the mea 5 ad st. deviatio 1 N (5kg; 1kg). a) What is the probability that the average of a sample of size = 9 is betwee 4 ad 6 kg. b) What is the area, where the average value will be with probability of 95%? c) How large should be the sample, so that the average dioxi emissios are exactly estimated with probability 95% ad est.error for e = 0.5 kg/mi? 4
5 d) Compute the cofidece itervals for the average dioxi emissios at the cofidece level 1. e) They measured 9 times of the dioxi emissios radomly (kg/mi): 7; 4; 5; 10; 9; 6; 8 ; 6.5; 7.5. Calculate the estimatio iterval at a cofidece level 1 = Exercise Kilometrage A) For a test 49 radomly draw car of the same type were fuelled with the same amout of fuel. With this amout of fuel the cars wet o average 50 km. Assume that st. deviatio is kow 7km. a) Give a explicit cofidece iterval [V L, V U ] for average kilometrage µ for this type of car at the cofidece level 1. b) Determie the iterval for µ whe 1 = 95%. c) What sample size is eeded, if the estimated iterval for µ at the same level shall have a width km? B) Some visitors of this test evet were radomly chose by jouralist ad asked about their membership i the ADAC (Germa automobile club). Amog 00 people 40 were ADAC members. Determie the iterval fro π whe 1 =99 %. C) Coffee machie was istalled at the tribue for this evet. It fills 0.l cup with coffee. Assume that the quatity is ormally distributed. Radom sample of size = 5 has followig values: 0.18, 0.5, 0.1, 0.0, 0.5. a) Give a explicit cofidece iterval [V L, V U ] for average quatity µ for this machie at the cofidece level 1. b) Determie the iterval for µ whe 1 = 95%. 5
Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.
STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to
More informationKurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)
Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
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 informationSTATS 200: Introduction to Statistical Inference. Lecture 1: Course introduction and polling
STATS 200: Itroductio to Statistical Iferece Lecture 1: Course itroductio ad pollig U.S. presidetial electio projectios by state (Source: fivethirtyeight.com, 25 September 2016) Pollig Let s try to uderstad
More informationRule of probability. Let A and B be two events (sets of elementary events). 11. If P (AB) = P (A)P (B), then A and B are independent.
Percetile: the αth percetile of a populatio is the value x 0, such that P (X x 0 ) α% For example the 5th is the x 0, such that P (X x 0 ) 5% 05 Rule of probability Let A ad B be two evets (sets of elemetary
More informationf(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1
Parameter Estimatio Samples from a probability distributio F () are: [,,..., ] T.Theprobabilitydistributio has a parameter vector [,,..., m ] T. Estimator: Statistic used to estimate ukow. Estimate: Observed
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
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 information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationChapter 20. Comparing Two Proportions. BPS - 5th Ed. Chapter 20 1
Chapter 0 Comparig Two Proportios BPS - 5th Ed. Chapter 0 Case Study Machie Reliability A study is performed to test of the reliability of products produced by two machies. Machie A produced 8 defective
More informationSTAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Itroductory Statistics Chapter 6: Itroductio to Iferece Lecture 18: Estimatio with Cofidece 11/14/06 Lecture 18 1 Itroductio Statistical Iferece
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
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 informationLecture 11 and 12: Basic estimation theory
Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
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 informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
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 informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationUNIT 8: INTRODUCTION TO INTERVAL ESTIMATION
STATISTICAL METHODS FOR BUSINESS UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION 8..- Itroductio to iterval estimatio 8..- Cofidece itervals. Costructio ad characteristics 8.3.- Cofidece itervals for the mea
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationChapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p
Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE Part 3: Summary of CI for µ Cofidece Iterval for a Populatio Proportio p Sectio 8-4 Summary for creatig a 100(1-α)% CI for µ: Whe σ 2 is kow ad paret
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationSOLUTIONS. 1. a) Let X and Y be random variables with. σ Y = 9. σ X 30 σ XY σ X 30 ρ σ X σ Y σ X + 24 σ XY
STAT 400 UIU Practice Problems #8 Stepaov Dalpiaz SOLUTIONS The followig are a umber of practice problems that may be helpful for completig the homework, ad will likely be very useful for studyig for exams..
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
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 informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
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 informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationAsymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week Lecture: Cocept Check Exercises Starred problems are optioal. Statistical Learig Theory. Suppose A = Y = R ad X is some other set. Furthermore, assume P X Y is a discrete
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
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 informationUnbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 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 informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
PSet ----- Stats, Cocepts I Statistics 7.3. Cofidece Iterval for a Mea i Oe Sample [MATH] The Cetral Limit Theorem. Let...,,, be idepedet, idetically distributed (i.i.d.) radom variables havig mea µ ad
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 informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationComputing Confidence Intervals for Sample Data
Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios
More informationWorking with Two Populations. Comparing Two Means
Workig with Two Populatios Comparig Two Meas Coditios for Two-Sample Iferece The data are from two radom samples from two distict idepedet populatios. Normality. Two sample t procedures are more robust
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 informationStat410 Probability and Statistics II (F16)
Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems
More informationConfidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)
Cofidece Itervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Aa Phophoem, Ph.D. aa.p@ku.ac.th Itelliget Wireless Network Group (IWING Lab) http://iwig.cpe.ku.ac.th Computer Egieerig Departmet Kasetsart Uiversity,
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
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 informationConfidence Level We want to estimate the true mean of a random variable X economically and with confidence.
Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio
More 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 informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence
PSet ----- Stats, Cocepts I Statistics Cofidece Iterval Guesswork with Cofidece VII. CONFIDENCE INTERVAL 7.1. Sigificace Level ad Cofidece Iterval (CI) The Sigificace Level The sigificace level, ofte deoted
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 informationConfidence Interval for one population mean or one population proportion, continued. 1. Sample size estimation based on the large sample C.I.
Cofidece Iterval for oe populatio mea or oe populatio proportio, cotiued 1. ample size estimatio based o the large sample C.I. for p ˆ(1 ˆ) ˆ(1 ˆ) From the iterval ˆ p p Z p ˆ, p Z p p L legh of your 100(1
More informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
More informationStatistical Inference About Means and Proportions With Two Populations
Departmet of Quatitative Methods & Iformatio Systems Itroductio to Busiess Statistics QM 220 Chapter 10 Statistical Iferece About Meas ad Proportios With Two Populatios Fall 2010 Dr. Mohammad Zaial 1 Chapter
More informationHYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018
HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible
More informationS160 #12. Review of Large Sample Result for Sample Proportion
S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Review of Large Sample Result for Sample Proportio Recall that for large sample (ie, sample size is large, say p > 5 ad (1
More informationIUT of Saint-Etienne Sales and Marketing department Mr Ferraris Prom /12/2015
IUT of Sait-Etiee Sales ad Marketig departmet Mr Ferraris Prom 2014-2016 09/12/2015 MATHEMATICS 3 rd semester, Test 2 legth : 2 hours coefficiet 2/3 The graphic calculator is allowed. Ay persoal sheet
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
More informationProbability and Statistics Estimation Chapter 7 Section 3 Estimating p in the Binomial Distribution
Probability ad Statistics Estimatio Chapter 7 Sectio 3 Estimatig p i the Biomial Distributio Essetial Questio: How are cofidece itervals used to determie the rage for the value of p? Studet Objectives:
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More informationS160 #12. Sampling Distribution of the Proportion, Part 2. JC Wang. February 25, 2016
S160 #12 Samplig Distributio of the Proportio, Part 2 JC Wag February 25, 2016 Outlie 1 Estimatig Proportio Usig Itervals Cofidece Iterval for the Populatio Proportio iclicker Questios 2 JC Wag (WMU) S160
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationBIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov
Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
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 large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
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 informationCHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More information1 Models for Matched Pairs
1 Models for Matched Pairs Matched pairs occur whe we aalyse samples such that for each measuremet i oe of the samples there is a measuremet i the other sample that directly relates to the measuremet i
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More information24.1 Confidence Intervals and Margins of Error
24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker Explore Idetifyig Likely
More informationTAMS24: Notations and Formulas
TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: =
More informationUnderstanding Dissimilarity Among Samples
Aoucemets: Midterm is Wed. Review sheet is o class webpage (i the list of lectures) ad will be covered i discussio o Moday. Two sheets of otes are allowed, same rules as for the oe sheet last time. Office
More informationClass 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 7 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 013 by D.B. Rowe 1 Ageda: Skip Recap Chapter 10.5 ad 10.6 Lecture Chapter 11.1-11. Review Chapters 9 ad 10
More informationDepartment of Statistics and Operations Research College of Science, King Saud University. Summer Semester
Departmet of Statistics ad Operatios Research College of Sciece, Kig Saud Uiversity Summer Semester 46-47 STAT 34 Probability ad Statistics for Egieers EXERCISES A Collectio of Questios Selected from Midterm
More informationStat 400: Georgios Fellouris Homework 5 Due: Friday 24 th, 2017
Stat 400: Georgios Fellouris Homework 5 Due: Friday 4 th, 017 1. A exam has multiple choice questios ad each of them has 4 possible aswers, oly oe of which is correct. A studet will aswer all questios
More informationSOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker
SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker CHAPTER 9. POINT ESTIMATION 9. Covergece i Probability. The bases of poit estimatio have already bee laid out i previous chapters. I chapter 5
More informationLecture 6: Coupon Collector s problem
Radomized Algorithms Lecture 6: Coupo Collector s problem Sotiris Nikoletseas Professor CEID - ETY Course 2017-2018 Sotiris Nikoletseas, Professor Radomized Algorithms - Lecture 6 1 / 16 Variace: key features
More informationSTAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)
STAT 515 fa 2016 Lec 15-16 Samplig distributio of the mea, part 2 cetral limit theorem Karl B. Gregory Moday, Sep 26th Cotets 1 The cetral limit theorem 1 1.1 The most importat theorem i statistics.............
More information