Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Size: px
Start display at page:

Download "Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample."

Transcription

1 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 by the distributio (PDF or PMF) of X. The distributio of X depeds o some ukow parameter θ. Goal: lear about θ. Sample: Radomly select a sample of size from the populatio. Sample data: X 1, X 2,, X The sample observatios are idepedet, ad the distributio of each idividual observatio is same as the distributio of the populatio. I other words, the data are idepedetly ad idetically distributed (i.i.d.). Ex: What proportio of Americas thik Presidet Bush is doig a good job? Describe the populatio radom variable ad the parameter of iterest. Modes of statistical iferece: Parameter (poit) estimatio: Estimate θ by a sigle value. Cofidece itervals: Estimate θ by a iterval of values. Hypothesis testig: Test whether data are cosistet with a particular value of θ. Model fittig: Fit a model to the data. Parameter estimatio Statistic = ay fuctio W(X 1,, X ) of sample data. Estimator of θ = ay statistic used to estimate θ deoted as ˆ θ. Desirable properties of ˆ θ : 1. Ubiasedess ˆ θ is ubiased for θ if E( ˆ θ ) = θ. Estimator is correct o average. 1

2 2. Small variace Variace = ucertaity. Larger variace = less precise. We would like to have small variace or high precisio. 3. Cosistecy ˆ θ is cosistet for θ if P ( ˆ θ θ > ) 0 as. I other words, i the limit, ˆ θ = θ. Sufficiet coditio for cosistecy if ˆ θ is ubiased: Var[ ˆ θ ] 0 as. Ex 1: Estimatio of populatio mea µ = E(X) = θ. Suppose populatio variace = σ 2. Natural estimator: Ubiased? Cosistet? Does it have the smallest variace? Ex 2: Estimatio of populatio proportio of successes p = E(X) = θ. X is Beroulli (p). Populatio cosists of 0 s (failures) ad 1 s (successes). p is also a mea. Natural estimator: Similar properties as sample mea i Ex 1. Ex 3: Estimatio of populatio variace σ 2 = Var(X) = θ. Natural estimator: Also ubiased ad cosistet. 2

3 I geeral, the choice of a estimator may ot be obvious. Two geeral methods of estimatio: Method of momets ad method of maximum likelihood. - Both produce cosistet estimators. - Maximum likelihood is more popular the best if is large. - Oly the method of momet i this course as it is simpler (ad does ot require calculus!). - But remember that the method of momet estimator may ot be the best. Method of momets: k k-th populatio momet: µ = E( X ) k-th sample momet: M k 1 = k k X i i= 1 To estimate d parameters, solve the system of d equatios, M1 Md = L = µ µ 1 d Their solutio = MOME. The # equatios should be equal to the # parameters i the model. Ex: (Presidet Bush example) Populatio: X ~ Beroulli (p), where p = proportio of Americas who approve Presidet Bush s job performace. Parameter: p Sample: X 1, X 2,, X. What is the MOME of p? Is it ubiased? Is it cosistet? 3

4 Ex: Assume that the time to failure i years, X, of a telephoe switchig system is expoetially distributed with parameter λ (> 0). What is the MOME of λ based o a radom sample of times to failure, say X 1,, X? Is it ubiased? Is it cosistet? Ex: Derive the MOMEs of mea µ ad variace σ 2 of a N(µ,σ 2 ) distributio based o a radom sample X 1,, X from this populatio. 4

5 Cofidece itervals (sectio ) Estimator ˆ θ is a sigle umber that gives a plausible value of the ukow θ. Rarely the two will be equal. So, ofte it is preferable to give a iterval of plausible values a cofidece iterval (CI), which cotais the ukow θ with high probability. 100 (1 α) % cofidece iterval for θ = A iterval [L, U], where L = L(X 1,, X ) ad U = U(X 1,, X ) are statistics such that P L θ U = 1. ( ) α L ad U are radom, so the CI is radom. Parameter θ is ot radom it is ukow but fixed. (1 α) = cofidece coefficiet or cofidece level. I practice, (1 α) = 90% or 95% (most commo) or 99%. 100 (1 α) % CI for mea µ of a Normal (µ, σ 2 ) populatio (assumig σ 2 kow): X z σ σ / 2, X + zα /, α 2 where z α / 2 is such that area uder the stadard ormal curve to the right of z α / 2 = α/2. Derivatio: Step 1: Estimate θ = µ by its estimator X 1 = X i. Step 2: Recall that X ~ Normal 2 σ, X µ =. σ / µ. Therefore, Z ~ Normal( 0,1) Step 3: Fid critical values z α / 2 Step 4: The we have Step 5: Solve for µ to get: ± such that P ( z Z ) = 1 α. α / 2 zα / 2 X µ P zα / 2 zα / 2 = 1 α σ / P X zα / 2σ zα / 2σ µ X + = 1 α 5

6 Ex: Suppose that a observed sample of size 10 from a N(µ, 4) populatio gives x = Fid the 95% CI for µ. Notice that this iterval is fixed it s a umerical iterval. There is othig radom about it. Questio: Ca we say that this observed iterval cotais the true value of µ with 95% probability? So, how do we really iterpret a CI? Recall, σ σ P X zα / 2 µ X + zα / 2 = 1 α = 0.95, I other words, if we repeat a large umber of times the process of takig a radom sample of size from Normal (µ, σ 2 ) populatio ad costruct the CI usig the above formula, the roughly 95% of times the observed CI s will be correct, i.e., it will capture the true value of µ. This CI formula gives a icorrect iterval oly 5% (small) of the times. It is wrog to say that the observed iterval cotais the true value of µ with 95% probability. It either cotais the true value of µ or it does ot we do t kow what the case is. Thus, i a sese, we have 95% cofidece i the CI formula it gives the correct aswer 95% of the times. Ex: Let s use simulatio to verify this iterpretatio. I draw a radom sample of size 10 from N(5, 10) distributio ad use the observed sample to costruct a 95% CI for µ usig the above formula. I repeated this procedure 100 times. The figure below plots the costructed CI s. 6

7 Roughly, 7% of these 100 itervals do ot cover the true value of µ, which is 5 i this simulatio. But 100 is ot cosidered a large # of repetitios i simulatio. So, if I repeat this process times, the oly 5.2% of the itervals do ot cover the true value 5. I other words, roughly 94.8% of the itervals cover the true value of µ. Questio: Give a radom sample, which CI would you prefer: a 95% CI or a 99% CI? Ex: Suppose that a observed sample of size 10 from a N(µ, 4) populatio gives x = Fid the 95% level ad 99% level CI s forµ. 7

8 Note the tradeoff: The precisio of a CI is give by its width. The reliability of a CI is give its cofidece level. Higher cofidece = lower precisio (wider). The width of a 100% CI is:. It is a useless iterval: extremely reliable but extremely imprecise! Questio: What ca we do to get a arrower CI without lowerig the cofidece? σ Width = 2 zα / 2 Icrease to make CI more precise. Choosig the sample size : Let w = desired CI width for 1 α cofidece. Margi of error = 2w σ Set 2 z α / 2 = w 2 σ Solvig for gives: = 2zα / 2 w Ex: Suppose that we wish to estimate the mea CPU service time of a job ad we wish to assert with 99% cofidece that the estimated value is withi less tha 0.5 s of the true value. Suppose that the past experiece suggests that CPU service time is ormally distributed with stadard deviatio σ = 1.5 s. How may observatios should we take? 8

9 100 (1 α) % CI for mea µ of a Normal (µ, σ 2 ) populatio (assumig σ 2 ukow): Ukow σ 2 is more realistic. Estimate σ by sample variace S = ( X i X ) 1 i X µ Use the fact that Z = ow follows a t distributio with ( 1) degrees of freedom, istead S / of the Normal (0, 1) distributio. A t 1 -distributio looks like a Normal (0, 1) but it has heavier tails. A heavier tail accouts for the fact that there is there is more ucertaity i Z whe S is used i place of σ. Whe is large, a t 1 -distributio Normal (0, 1). 2 Result: CI for µ = X ± t α / 2, 1 S The t critical poits are tabulated i the Appedix. Sample size calculatio ow becomes complicated tha before because S eeds to be kow before data are collected. Oe optio is to make a itelliget guess about S ad be coservative (guess a larger value of S so that larger tha ecessary is chose). Ex: Some studies of Alzheimer s disease (AD) have show a icrease of 14 C0 2 productio i patiets with the disease. I oe such study the followig 14 C0 2 values were obtaied from 16 eocortical biopsy samples from AD patiets Fid the 95% CI for mea 14 C0 2 value assumig that 14 C0 2 values are ormally distributed. 9

10 Large Sample Cofidece Itervals Populatio mea µ: So far we assumed that the populatio was ormal. Questio: Ca we use this CI eve whe the populatio is ot ormal? Yes, provided is large. S 100(1 α)% CI for mea µ of ay populatio: X ± zα / 2 Works because of the cetral limit theorem, which says X is ormal whe is large. Rule of thumb: > 30. May ot be valid for smaller sample sizes. Ex: We wish to estimate the mea executio time of a program. The program was ru 35 times o radomly selected iputs, ad the sample mea ad the sample stadard deviatio of the executio times were evaluated as x = 230 ms ad s = 14 ms, respectively. Fid a 95% CI for the true mea executio time µ. Populatio proportio of successes p: Populatio: X ~ Beroulli (p), where p = proportio of successes i populatio. Sample data: X 1,., X. (Note: they are 0 s ad 1 s). Recall: Estimator for p = p ˆ = X = proportio of successes i the sample Questio: What is the approximate distributio of pˆ for large? p(1 p) CLT whe is large, pˆ ~ Normal p, p ˆ(1 pˆ ) Estimate it Var( pˆ ) as. 10

11 pˆ p For large, Z = pˆ(1 pˆ) ~ Normal(0,1). Result: 100(1-α)% CI for p = pˆ(1 pˆ) pˆ ± z α / 2 where p ˆ = X. Rule of thumb: pˆ 10 ad ( 1 pˆ) 10. Ex: From a large populatio of RAM chips, a radom sample of 50 is take ad a test carried out o each to see whether they perform correctly. I the test, oly 20 chips are foud to perform correctly. Fid a 95% CI for p, the true proportio of chips that perform correctly. Choosig the sample size : Width of CI = 2 z α / 2 pˆ(1 pˆ) Let w = desired CI width for 1 α cofidece. Margi of error = 2w pˆ(1 pˆ ) Set 2 z α / 2 = w 2 4zα Solvig for gives: / 2 pˆ(1 pˆ ) = 2 w This formula ivolves pˆ, which is ot kow before the experimet. Oe alterative: take pˆ = 0.5 because pˆ (1 pˆ ) is maximum whe p ˆ = This strategy will yield a coservative values of. (The sample size will be larger tha ecessary.) 11

12 Ex: Suppose we are plaig a survey to estimate the proportio of America who approve of Presidet Bush s job. We would like our estimate to be withi 3% of the true proportio with 95% cofidece. How much sample size should we take? 12

Chapter 8: Estimating with Confidence

Chapter 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 information

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population

A 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 information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 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 information

MATH/STAT 352: Lecture 15

MATH/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 information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

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

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

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Interval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),

Interval 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

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

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Agreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times

Agreement 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 information

MBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS

MBACATÓLICA. 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 information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6) STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated

More information

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9

BIOS 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 information

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates. 5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece

More information

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis

Section 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 information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 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 information

Since 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

Since 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 information

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

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

More information

Stat 421-SP2012 Interval Estimation Section

Stat 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 information

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

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give

More information

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more

More information

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso

More information

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo

More information

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234 STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6

More information

Random Variables, Sampling and Estimation

Random 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 information

A Question. Output Analysis. Example. What Are We Doing Wrong? Result from throwing a die. Let X be the random variable

A Question. Output Analysis. Example. What Are We Doing Wrong? Result from throwing a die. Let X be the random variable A Questio Output Aalysis Let X be the radom variable Result from throwig a die 5.. Questio: What is E (X? Would you throw just oce ad take the result as your aswer? Itroductio to Simulatio WS/ - L 7 /

More information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.

Inferential 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 information

Lecture 33: Bootstrap

Lecture 33: Bootstrap Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece

More information

Sampling Distributions, Z-Tests, Power

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

More information

STAT431 Review. X = n. n )

STAT431 Review. X = n. n ) STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ,

More information

STAC51: Categorical data Analysis

STAC51: Categorical data Analysis STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo

More information

Announcements. Unit 5: Inference for Categorical Data Lecture 1: Inference for a single proportion

Announcements. Unit 5: Inference for Categorical Data Lecture 1: Inference for a single proportion Housekeepig Aoucemets Uit 5: Iferece for Categorical Data Lecture 1: Iferece for a sigle proportio Statistics 101 Mie Çetikaya-Rudel PA 4 due Friday at 5pm (exteded) PS 6 due Thursday, Oct 30 October 23,

More information

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M. MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY 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 information

Estimation for Complete Data

Estimation 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 information

Lecture 3. Properties of Summary Statistics: Sampling Distribution

Lecture 3. Properties of Summary Statistics: Sampling Distribution Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

More information

HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018

HYPOTHESIS 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 information

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

Lecture 2: Monte Carlo Simulation

Lecture 2: Monte Carlo Simulation STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?

More information

Sample Size Determination (Two or More Samples)

Sample Size Determination (Two or More Samples) Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie

More information

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

STAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence

STAT 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 information

Computing Confidence Intervals for Sample Data

Computing 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 information

Chapter 6 Sampling Distributions

Chapter 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 information

Statistics 300: Elementary Statistics

Statistics 300: Elementary Statistics Statistics 300: Elemetary Statistics Sectios 7-, 7-3, 7-4, 7-5 Parameter Estimatio Poit Estimate Best sigle value to use Questio What is the probability this estimate is the correct value? Parameter Estimatio

More information

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE

April 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 information

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber

More information

Frequentist Inference

Frequentist Inference Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for

More information

Statistical Intervals for a Single Sample

Statistical 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 information

(7 One- and Two-Sample Estimation Problem )

(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 information

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

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y. Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad o-users, x - y. Such studies are sometimes viewed

More information

Basis for simulation techniques

Basis 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 information

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

More information

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic

More information

Final Examination Solutions 17/6/2010

Final 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 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:

More information

5. Likelihood Ratio Tests

5. 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 information

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:

Mathacle. 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 information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: 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 information

Topic 10: Introduction to Estimation

Topic 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 information

Stat 200 -Testing Summary Page 1

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

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01

ENGI 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 information

Properties and Hypothesis Testing

Properties 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. Cross-sectioal data. 2. Time series data.

More information

Statistics 20: Final Exam Solutions Summer Session 2007

Statistics 20: Final Exam Solutions Summer Session 2007 1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets

More information

f(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

f(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 information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

Department of Mathematics

Department of Mathematics Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets

More information

Common Large/Small Sample Tests 1/55

Common 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 information

Estimation of the Mean and the ACVF

Estimation of the Mean and the ACVF Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio

More information

Understanding Samples

Understanding Samples 1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We

More information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples

More information

Introductory statistics

Introductory 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 information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

More information

Output Analysis and Run-Length Control

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%

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments: Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal

More information

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

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

October 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1

October 25, 2018 BIM 105 Probability and Statistics for Biomedical Engineers 1 October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 1 Populatio parameters ad Sample Statistics October 25, 2018 BIM 105 Probability ad Statistics for Biomedical Egieers 2 Ifereces

More information

Logit regression Logit regression

Logit regression Logit regression Logit regressio Logit regressio models the probability of Y= as the cumulative stadard logistic distributio fuctio, evaluated at z = β 0 + β X: Pr(Y = X) = F(β 0 + β X) F is the cumulative logistic distributio

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

More information

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2 AMS 216 Stochastic Differetial Equatios Lecture 02 Copyright by Hogyu Wag, UCSC Review of probability theory (Cotiued) Variace: var X We obtai: = E X E( X ) 2 = E( X 2 ) 2E ( X )E X var( X ) = E X 2 Stadard

More information

Summary. Recap ... Last Lecture. Summary. Theorem

Summary. Recap ... Last Lecture. Summary. Theorem Last Lecture Biostatistics 602 - Statistical Iferece Lecture 23 Hyu Mi Kag April 11th, 2013 What is p-value? What is the advatage of p-value compared to hypothesis testig procedure with size α? How ca

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This 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 hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class

More information

Stat 319 Theory of Statistics (2) Exercises

Stat 319 Theory of Statistics (2) Exercises Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.

More information

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

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10 DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set

More information

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: 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 information

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2 Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Confidence Intervals for the Population Proportion p

Confidence Intervals for the Population Proportion p Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:

More information

Confidence Intervals รศ.ดร. อน นต ผลเพ ม Assoc.Prof. Anan Phonphoem, Ph.D. Intelligent Wireless Network Group (IWING Lab)

Confidence 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 information