Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p

Size: px
Start display at page:

Download "Chapter 8: STATISTICAL INTERVALS FOR A SINGLE SAMPLE. Part 3: Summary of CI for µ Confidence Interval for a Population Proportion p"

Transcription

1 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 populatio is ormal, use a z-value (this works for ay ). x ± z α/2 σ Whe σ 2 is ukow ad is REALLY large, use a z-value ad replace σ 2 with the observed sample variace s 2 (this works for ay paret populatio distributio because is large). x ± z α/2 s 1

2 Whe σ 2 is ukow, is relatively small, ad POPULATION IS N EARLY NORMAL, use a t-value ad the sample variace. x ± t α/2, 1 s Before you make a CI, it is a radom iterval... it depeds o the sample chose, but X will ALWAYS be at the ceter of the 2- sided CI. Example of 16 differet CI s for µ each based o a differet sample: 2

3 Example: Fuel rods i a reactor (problem 8-43 i book) A article i Nuclear Egieerig Iteratioal gives the followig measuremets o the percetage of erichmet of 12 fuel rods i a reactor i Norway: Calculate a 95% CI for the mea percetage of erichmet. Provide a ormal probability plot to verify the assumptio of ormality. ANS: This is a small sample with ukow σ 2, so we will use the procedure with the t- value. x ± t α/2, 1 s 3

4 x= s 2 = s = t α/2, 1 = t 0.025,12 1 = t 0.025,11 = % CI for µ: ± (0.0993/ 12) ± [2.8386, ] 4

5 Checkig ormality with a ormal probability plot: ormal probability plot Theoretical Quatiles Sample Quatiles Looks pretty good. The poits fall radomly aroud the diagoal lie. So, we ca believe that the data were geerated from a ormal distributio (i.e. the paret populatio was at least approximately ormal). 5

6 Populatio Proportio Parameter p Movig o to aother potetial parameter of iterest... A populatio proportio is deoted p. A sample proportio is deoted ˆp. A populatio proportio is based o a yes/o or 0/1 type variable. What proportio of the populatio favor Hillary Clito? yes/o What proportio of a maufactured good is defective? defective/ot defective 6

7 What proportio of the U.S. is republica? republica/ot republica What proportio of studets eterig college successfully complete a degree? succeed/fail To estimate a populatio proportio, we will use a sample proportio ad the bottom lie for proportios is that you just plai eed a large sample. There s sparse iformatio i each observatio (just yes/o), we do t have a ice cotiuum like i a cotiuous radom variable measuremet (like weight). 7

8 Large-Sample Cofidece Iterval for a Populatio Proportio p Sectio 8-4 The costructio of the CI for p relies o the fact that we took a large sample (large ). I will itroduce this cocept usig somethig we are familiar with, the biomial... We will let the category of iterest be called the success category (arbitrary). Let X i = 1 if observatio i falls ito the success category. Let X i = 0 if observatio i falls ito the other or fail category. X i is called a idicator variable. 8

9 Thus, i=1 X i = a cout of all idividuals i the success category. The sample proportio of idividuals fallig ito the success category is ˆP = i=1 X i = # i sample who are i success category ˆP (upper case) is a radom variable ad is the poit estimator for p. ˆp (lower case) is a realized poit estimate from a observed sample. 9

10 Note that p ad are actually the parameters for a biomial distributio (i.e. probability of a success ad umber of trials). There are trials (i.e. idepedet draws of idividuals to form the sample) The probability of gettig a success remais costat as p (assumig we have a large populatio ad is ot too large) Let Y =total umber of successes. So Y Biomial(, p) E(Y ) = p ad V (Y ) = p(1 p) I our otatio, Y = i=1 X i, so Y = i=1 X i Biomial(, p) 10

11 E( i=1 X i ) = p ad V ( i=1 X i ) = p(1 p) Thus, if is large, we have ( i=1 X i ) Z = ( i=1 X i ) p = p p(1 p) p(1 p) = ˆP p p(1 p) where Z is approximately stadard ormal. (NOTE: This is a ormal approximatio to the biomial. Ad for the approximatio to be reasoable, we should have p 5 ad (1 p) 5.) 11

12 Normal approximatio for a sample proportio ˆP If is large, the distributio of Z = ˆP p p(1 p) is approximately stadard ormal. Or similarly, ˆP N(p, p(1 p) ) This meas the samplig distributio for ˆP is ormal... See applet at: 12

13 Thus, we ca ca use a z-value to form a 100(1-α)% CI for p: P ( z α/2 ˆP p p(1 p) z α/2 ) = 1 α rearragig... P ( ˆP z α/2 p(1 p) ) p ˆP p(1 p) + z α/2 = 1 α ad we have the lower ad upper bouds... Lower boud: ˆP zα/2 p(1 p) Upper boud: ˆP + zα/2 p(1 p) BUT WE DON T KNOW p, SO WE CAN T GET ACTUAL VALUES FOR THE BOUNDS! Solutio replace p with ˆP i the formulas. 13

14 Approximate 100(1-α)% CI for a populatio proportio p If ˆp is the proportio of observatios i a radom sample of size that belogs to a class of iterest, a approximate 100(1-α)% CI o the proportio of p of the populatio that belogs to this class is ˆp ± z α/2 ˆp(1 ˆp) where z α/2 is the upper α/2 percetage poit of the stadard ormal distributio. Thigs you eed for the appropriate behavior of ˆP : Populatio is large, ad you do t take too may idividuals for your sample. Maybe o more tha 10% of the total populatio. 14

15 The sample is a simple radom sample. p 5 ad (1 p) 5. This statemet meas that if you have a really rare evet, you re goig to eed a very large sample... just logical though. Example: Iterpolatio methods are used to estimate heights above sea level for locatios where direct measuremet are uavailable. After verifyig the estimates, it was foud that the iterpolatio method made large errors at 26 of 74 radom sample test locatios. Fid a 90% CI for the overall proportio of locatios at which this method will make large errors. 15

16 ANS: 16

17 Choice of sample size for estimatig p For a specified error E = p ˆP i your estimate, the previously stated behavior of ˆP suggests you should choose a sample size as: = ( zα/2 E ) 2 p(1 p) But sice we do t kow p (that s what we re tryig to estimate!), we ca t compute a sample size from this formula... uless we estimate p first. Here, we choose to err o the coservative side. It turs out the largest variace for ˆP [where V ( ˆP ) = p(1 p) ] occurs whe p = 0.5 o matter what the was: 17

18 Plot of variace of ˆP vs. p p(1 p)/ p So, if we do t kow p, we ll plug-i p = 0.5 to make sure we do t uder-estimate the variace of our estimate (usig worst case sceario idea). 18

19 Thus, the workig sample size formula for estimatig p with a 100(1 α)% CI ad a error of E is: ( ) zα/2 2 = 0.25 E We use this method because, before we collect data, we do t have ay iformatio o p, so there s o observed ˆp to plug-i. I cotrast, whe doig a CI for p, we DO have a observed estimate ˆp for p, so i that case, so use our plug-i estimate. 19

20 Example: I the iterpolatio example, how large of a sample would you eed if you wated to be at least 95% cofidet that the error i your estimate (i.e. p ˆp ) is less tha 0.08? ANS: 20

21 Example: Gallup Poll Betwee February 9-11, 2007, adults were radomly sampled (by phoe) ad asked: Would you favor or oppose Cogress takig actio to set a time-table for withdrawig all U.S. troops from Iraq by the ed of ext year. Let p = the proportio of the populatio watig a withdrawal time-table. How large is their error if they sample 1000 people ad form a a 95% CI? 21

22 ANS: See USA Today, February 12, What ca go wrog i the estimatio? Bias i selectig the sample for oe thig. Gallup idetifies flaws i 2012 electio polls 22

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

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

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

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

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

(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

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

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

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

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

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

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

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

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

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

STATS 200: Introduction to Statistical Inference. Lecture 1: Course introduction and polling

STATS 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 information

Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008

Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008 Chapter 6 Part 5 Cofidece Itervals t distributio chi square distributio October 23, 2008 The will be o help sessio o Moday, October 27. Goal: To clearly uderstad the lik betwee probability ad cofidece

More information

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

Statistical 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 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

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

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

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

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

Homework 5 Solutions

Homework 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 information

CONFIDENCE INTERVALS STUDY GUIDE

CONFIDENCE INTERVALS STUDY GUIDE CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate

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

AP Statistics Review Ch. 8

AP 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 information

Estimation of a population proportion March 23,

Estimation of a population proportion March 23, 1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes

More information

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially

More information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight) Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........

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

Mathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Confidence Interval Guesswork with Confidence

Mathacle. 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 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

Understanding Dissimilarity Among Samples

Understanding 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 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

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

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.

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 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

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

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

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

Expectation and Variance of a random variable

Expectation 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 information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

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

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

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

Math 140 Introductory Statistics

Math 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 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

Simple Random Sampling!

Simple Random Sampling! Simple Radom Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig:! 3/26/13 Scheaffer et al. chapter 4! 1 Goals for this Lecture! Defie simple radom samplig (SRS) ad discuss

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

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

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

Chapter 18 Summary Sampling Distribution Models

Chapter 18 Summary Sampling Distribution Models Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig

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

Chapter 20. Comparing Two Proportions. BPS - 5th Ed. Chapter 20 1

Chapter 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 information

(6) Fundamental Sampling Distribution and Data Discription

(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 information

CHAPTER 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 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 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

STAT 203 Chapter 18 Sampling Distribution Models

STAT 203 Chapter 18 Sampling Distribution Models STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset

More information

Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion

- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion 1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter

More information

Lesson 7: Estimation 7.3 Estimation of Population Proportio. 1-PropZInterval

Lesson 7: Estimation 7.3 Estimation of Population Proportio. 1-PropZInterval Lesso 7: Estimatio 7.3 Estimatio of Populatio Proportio p 1-PropZIterval October 18 Goals Preview Compute a cofidece iterval of the populatio proportio p. Cofidece Iterval for p We would estimate the populatio

More information

S160 #12. Review of Large Sample Result for Sample Proportion

S160 #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 information

1036: Probability & Statistics

1036: Probability & Statistics 036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso

More information

Solutions. Discussion D1. a. The middle 95% of a sampling distribution for a binomial proportion ˆp is cut off by the two points

Solutions. Discussion D1. a. The middle 95% of a sampling distribution for a binomial proportion ˆp is cut off by the two points formulas at the AP Exam. Studets should be able to derive it from the expressio for margi of error E i the formula for the cofidece iterval, amely, E z p(1 p) Modelig Good Aswers Oce studets have completed

More information

Chapter 6. Sampling and Estimation

Chapter 6. Sampling and Estimation Samplig ad Estimatio - 34 Chapter 6. Samplig ad Estimatio 6.. Itroductio Frequetly the egieer is uable to completely characterize the etire populatio. She/he must be satisfied with examiig some subset

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

S160 #12. Sampling Distribution of the Proportion, Part 2. JC Wang. February 25, 2016

S160 #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 information

There is no straightforward approach for choosing the warmup period l.

There is no straightforward approach for choosing the warmup period l. B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.

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

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

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9 Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I

More information

Read through these prior to coming to the test and follow them when you take your test.

Read through these prior to coming to the test and follow them when you take your test. Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1

More information

Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab

Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet

More information

Topic 6 Sampling, hypothesis testing, and the central limit theorem

Topic 6 Sampling, hypothesis testing, and the central limit theorem CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio

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

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

STAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)

STAT 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

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

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

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

Y i n. i=1. = 1 [number of successes] number of successes = n

Y 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 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

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

Background Information

Background Information Egieerig 33 Gayheart 5-63 1 Beautif ul Homework 5-63 Suppose t h a t whe t h e ph of a c e r t a i c h e m i c a l compoud i s 5.00, t h e ph measured by a radomly s e l e c t e d begiig chemistry studet

More information

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance

Hypothesis 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 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

Rule 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.

Rule 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 information

Confidence Intervals

Confidence Intervals Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio

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

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

Linear Regression Models

Linear Regression Models Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect

More information

Chapter 23: Inferences About Means

Chapter 23: Inferences About Means Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For

More information

Lecture 7: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 7: Non-parametric Comparison of Location. GENOME 560, Spring 2016 Doug Fowler, GS Lecture 7: No-parametric Compariso of Locatio GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review How ca we set a cofidece iterval o a proportio? 2 Review How ca we set a cofidece iterval

More information

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example

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

Mathematical Notation Math Introduction to Applied Statistics

Mathematical Notation Math Introduction to Applied Statistics Mathematical Notatio Math 113 - Itroductio to Applied Statistics Name : Use Word or WordPerfect to recreate the followig documets. Each article is worth 10 poits ad ca be prited ad give to the istructor

More information

Discrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15

Discrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15 CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio

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

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