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


 Phoebe Booker
 1 years ago
 Views:
Transcription
1 Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed as attempts to aswer a questio like "Does the mea blood pressure of OC users differ from that of ousers? (Are x ad y differet?)" This speculatio, that maybe there really is o differece (maybe oral cotraceptive use has o effect o blood pressure), is a hypothesis H : X Y or H : X Y The alterative to H is aother hypothesis, H1 : X Y or H1 : X Y The alterative hypothesis says that the mea blood pressure of OC users really is differet from that of ousers. (It does't say which mea is bigger, or how big the differece is, oly that there is a differece.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18
2 I this example, the parameter of iterest, x  y, has a wide rage of possible values, ad x  y =, is special because it represets "o differece" (betwee OC users ad ousers), or "o effect" (of OC use). H is called the ull hypothesis. It states that the parameter x  y equals the special value, zero. H 1, the alterative hypothesis, cosists of the complemet of H. It states that the parameter does ot equal zero. I our example we had observatios o 8 OC users ad 21 ousers. The immediate questio is What do these observatios tell us?" or Is it right to say that these data represet strog evidece that there is ideed a differece betwee the mea blood pressure levels of OC users ad ousers? or Do these observatios justify rejectig H? Authors: Blume, Greevy Bios 311 Lecture Notes Page 2 of 18
3 (Note that: we caot directly aswer the questio Which of these hypotheses is true?" because we oly have a sample of the populatio to work with.) We could aswer the previous questios usig our ew cofidece iterval tools: Costruct a 95% cofidece iterval for x  y, ad aswer "Yes, our data show that there is a differece" if the value x  y = is ot i the iterval. That is, reject the hypothesis of o differece (H ) if the value x  y = is ot i the cofidece iterval. But if the value x  y = is iside the 95% CI, our aswer is "These observatios do ot allow us to reject H." We are ot sayig that the observatios are evidece supportig H. (Remember that our best estimate of x  y was 5.42, ot.) But the data do ot represet strog eough evidece agaist H to justify its rejectio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 3 of 18
4 This procedure is essetially a classical hypothesis test: Whe H specifies the value of a parameter, (as it does i our example) (i) Costruct a 1(1)% CI for the parameter. (ii) Reject H if the hypothesized value is ot i the iterval. Recall our earlier iterpretatio of the CI as: "the values of the parameter that are cosistet with the observatios." What we're doig here is checkig to see if the value specified by H is "cosistet with the observatios". If it is ot, the we reject H. A more mathematically direct approach has prove popular may areas of sciece. Let s look at hypothesis testig more formally. Authors: Blume, Greevy Bios 311 Lecture Notes Page 4 of 18
5 A More Direct Approach To Hypothesis Testig I the blood pressure example, if the two variaces are assumed to be equal the T* X Y S m 2 P m2 X 1 1 m Y ~ t Uder our Null hypothesis (the mea pressures of OC users ad ousers are o differet), H : x  y =, m=21, =8 so T* ~ t SP 8 21 has Studet's t distributio with 27 df (uder H ). For a observed sample we ca calculate the value of this statistic. (T * is called the test statistic ) If it falls i oe of the tails of the tdistributio (say either T * > 2.52 or T * < 2.52) the we reject H. Authors: Blume, Greevy Bios 311 Lecture Notes Page 5 of 18
6 A popular explaatio for the reasoig behid this procedure is as follows: If H is true the T * will usually (95% of the time) fall betwee ad If it falls outside this iterval, the either (a) H is true, ad we just happeed to observe a relatively rare sample where T * falls this far from its expected value,, or (b) H is false; the expected value of T * is really ot. Values of T * that are far from zero (large values of T * ) are improbable uder H. If we observe such a value, we reject H. I this example we chose 2.52 as the critical value for our test, ad used the rule "Reject H if T * exceeds 2.52" This rule will sometimes lead us to reject H whe it is i fact true. It is easy to calculate the probability of makig this error: P * * T 2.52 H 1 P 2.52 T Authors: Blume, Greevy Bios 311 Lecture Notes Page 6 of 18
7 By choosig this critical value, we cotrol the error probability. If we use a larger critical value, say t = 2.771, the we reduce the probability that we will reject H whe it is really true: P * * T H 1 P T This is a example of aother geeral scheme for hypothesis testig: Istead of costructig a cofidece iterval, choose a observable "test statistic" (T * ) for which (i) the probability distributio is kow (at least approximately) whe H is true, ad (ii) the larger the value of the test statistic, the stroger the evidece agaist H. The we choose some critical value, ad reject H if the observed value of the test statistic exceeds the critical value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 7 of 18
8 I our OC example the test statistic was T * ~ t SP 8 21 Whe H is true, T * has a Studet's t distributio with 27 df. Sice the distributio of the test statistic whe H is true is kow, we ca choose the critical value to fix the error probability (reject H whe H is really true) at whatever value we like. Popular choices are.5 ad.1 (1/2 ad 1/1). This error probability is ofte called the "sigificace level" of the test ad ofte represet by =P(reject H H true) (Remember, we use the fact that H is true to costruct the test statistic.) Authors: Blume, Greevy Bios 311 Lecture Notes Page 8 of 18
9 Testig a Hypothesis About a Sigle Mea Assume that the data have ormal probability distributio, variace kow. The Null hypothesis is H : =. The Test statistic: T * X Notice the form of T *? * X T E X H Var X This is a appropriate test statistic for this hypothesis, because (i) its distributio whe H is true is kow (stadard ormal), ad (ii) the larger the value of the test statistic, the farther the sample mea, X, is from, ad it seems appropriate to say that the farther X is from, the stroger the evidece that is ot the expected value of X. Authors: Blume, Greevy Bios 311 Lecture Notes Page 9 of 18
10 If we choose the critical value 1.96, the we have P reject H X H P 1.96 H P Z This gives a test of H at the 5% sigificace level. To test H at the 1% sigificace level, we must use a larger critical value, P reject H X H P H P Z To reject H at the 1% level requires stroger evidece (a more extreme value of the test statistic) tha is required to reject at the 5% level. At the 5% level we reject the hypothesis that H : =E[X]= if X falls further tha 1.96 SE's away from the hypothesized value,. At the 1% level we oly reject if X falls further tha SE's away from. Authors: Blume, Greevy Bios 311 Lecture Notes Page 1 of 18
11 Example We kow (Pagao ad Gauvreau) that the distributio of serum cholesterol i adult males i the U.S. has a mea of 211 mg/1ml ad a stadard deviatio of mg/1ml. We wat to lear about me who are hypertesive smokers. Specifically, is the distributio of serum cholesterol amog such me ay differet from that i the geeral populatio? Let's suppose the stadard deviatio is the same as i the geeral populatio ( mg/1ml), ad test whether the mea, μ, is the same or ot. H : = 211 mg/1ml H 1 : 211 mg/1ml Our test statistic will be T * X 211 which, uder H, has a stadard ormal distributio. Authors: Blume, Greevy Bios 311 Lecture Notes Page 11 of 18
12 Thus to test at the 5% level, we will reject H if the absolute value of the test statistic exceeds This meas that we will reject uless X which implies that we will reject uless the 95% CI cotais the hypothesized value, 211: X X 1.96??? Authors: Blume, Greevy Bios 311 Lecture Notes Page 12 of 18
13 I Pagao's example, the sample size is = 12 ad the sample mea is 217, so the test statistic's value is T *.45 Sice this is much less tha the critical value, 1.96, we do ot come close to rejectig H. (The 95% CI is (191, 243), ad the hypothesized value, 211, is well withi the iterval.) If we take a much larger sample of hypertesive smokers, say 25 istead of 12, ad fid the same value for the sample mea, X 25 = 217, the test statistic will be T * 2.6 Sice this is greater tha the critical value, 1.96, the result would ow be "Reject H at the 5% level." Authors: Blume, Greevy Bios 311 Lecture Notes Page 13 of 18
14 Whe the observatios lead to rejectio of H at a specified level, such as 5%, they are said to be "statistically sigificat at the 5% level." (Whe they do't, they are "ot statistically sigificat.") We saw that a sample of size 12 with mea 217 is ot statistically sigificat for testig H : =211 at the 5% level ( σ = kow ). A sample of 25 with the same mea, 217, is statistically sigificat at the 5% level (but ot at the 1% level, sice the test statistic's value, 2.6, is less tha the 1% critical value, ) The 95% CI is 217 ± 1.96(/ 25 ), or ( 211.3, ), which excludes the hypothesized value, 211, ad The 99% CI is 217 ± 2.576(/ 25 ), or ( 29.5, ) which icludes that value. Authors: Blume, Greevy Bios 311 Lecture Notes Page 14 of 18
15 Textbooks ofte stress the importace of selectig the sigificace level (choosig the value for the error probability, α) before `lookig at the data. We will see later why they stress this. I practice it is very commo ot to select a fixed α at all. Istead, what is ofte doe is that for the value of the test statistic that is actually observed, the probability of observig a value that large or larger (uder H ) is calculated ad reported. This quatity is called the pvalue. It is a measure of how extreme the test statistic is, ad it is used as a measure of how strog the evidece agaist H is. We've bee cosiderig a test procedure that chooses a test statistic, T *, ad a fixed error probability,. The it fids the critical value, say t, for which P(T * > t H )=. This procedure rejects H if the observed value of the test statistic, say T obs *, exceeds the critical value, t. This is equivalet to fidig the pvalue, P(T * > T obs * H ), ad the rejectig H if this probability is less tha. Authors: Blume, Greevy Bios 311 Lecture Notes Page 15 of 18
16 For the hypertesive smokers' cholesterol levels (whe = 25) the test statistic was T * 25 X ad the observed value of the test statistic was T * obs The probability (uder H ) of observig a value of the test statistic this extreme is 25 X 25 P(T * > T * obs H )= P 2.6 H PZ Sice this is greater tha.1, but smaller tha.5, it shows that our observatios represet evidece strog eough to justify rejectio at the 5% sigificace level, but ot strog eough for the 1% level. Authors: Blume, Greevy Bios 311 Lecture Notes Page 16 of 18
17 A ote o the Pvalue The pvalue P(T * > T obs * H ), is somethig very familiar. It is just a probability statemet about the sample average. P(T * > T * obs H )= P P X P X X H Which says that the probability of observig a sample mea greater tha or equal to the 217 that was observed is oly.39 if the true mea is really 211 (the ull hypothesis is true). Authors: Blume, Greevy Bios 311 Lecture Notes Page 17 of 18
18 Two Types of Errors We have see that for testig the hypothesis that the mea of a ormal probability distributio has a specified value, H : = (agaist the alterative hypothesis that the mea is ot ), we reject H at sigificace level =.5 if T * X 1.96 This procedure ca lead us to reject the hypothesis H whe it is really true. The probability of this error is.5, ad is ofte called the Type I error, sigificace level, or the size of the test. But what if the hypothesis H is false ad we fail to reject H? This error is called the Type II error ad represet by the symbol. Type I error : =P(reject H H true) Type II error: =P(fail to reject H H false) Power: 1= P(reject H H false) (Loose defiitio we ll be more specific later) Authors: Blume, Greevy Bios 311 Lecture Notes Page 18 of 18
University of California, Los Angeles Department of Statistics. Hypothesis testing
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationStat 200 Testing Summary Page 1
Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationSuccessful HE applicants. Information sheet A Number of applicants. Gender Applicants Accepts Applicants Accepts. Age. Domicile
Successful HE applicats Sigificace tests use data from samples to test hypotheses. You will use data o successful applicatios for courses i higher educatio to aswer questios about proportios, for example,
More information1 Constructing and Interpreting a Confidence Interval
Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew CraeDroesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad
More informationChapter 22: What is a Test of Significance?
Chapter 22: What is a Test of Sigificace? Thought Questio Assume that the statemet If it s Saturday, the it s the weeked is true. followig statemets will also be true? Which of the If it s the weeked,
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe ad TwoSample Tests of Hypotheses 0 Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationStatistics 20: Final Exam Solutions Summer Session 2007
1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hardcopy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationWorksheet 23 ( ) Introduction to Simple Linear Regression (continued)
Worksheet 3 ( 11.511.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationEcon 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.
Eco 371 Exam #1 Multiple Choice (5 poits each): For each of the followig, select the sigle most appropriate optio to complete the statemet 1) The probability of a outcome a) is the umber of times that
More informationy ij = µ + α i + ɛ ij,
STAT 4 ANOVA Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
More informationChapter 4 Tests of Hypothesis
Dr. Moa Elwakeel [ 5 TAT] Chapter 4 Tests of Hypothesis 4. statistical hypothesis more. A statistical hypothesis is a statemet cocerig oe populatio or 4.. The Null ad The Alterative Hypothesis: The structure
More informationf(x)dx = 1 and f(x) 0 for all x.
OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet
More informationTo make comparisons for two populations, consider whether the samples are independent or dependent.
Sociology 54 Testig for differeces betwee two samle meas Cocetually, comarig meas from two differet samles is the same as what we ve doe i oesamle tests, ecet that ow the hyotheses focus o the arameters
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationConfidence Intervals QMET103
Cofidece Itervals QMET103 Library, Teachig ad Learig CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their
More informationLecture 9: Independent Groups & Repeated Measures ttest
Brittay s ote 4/6/207 Lecture 9: Idepedet s & Repeated Measures ttest Review: Sigle Sample ztest Populatio (otreatmet) Sample (treatmet) Need to kow mea ad stadard deviatio Problem with this? Sigle
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationIssues in Study Design
Power ad Sample Size: Issues i Study Desig Joh McGready Departmet of Biostatistics, Bloomberg School Lecture Topics Revisit cocept of statistical power Factors ifluecig power Sample size determiatio whe
More informationAsymptotic distribution of the firststage Fstatistic under weak IVs
November 6 Eco 59A WEAK INSTRUMENTS III Testig for Weak Istrumets From the results discussed i Weak Istrumets II we kow that at least i the case of a sigle edogeous regressor there are weakidetificatiorobust
More informationY i n. i=1. = 1 [number of successes] number of successes = n
Eco 371 Problem Set # Aswer Sheet 3. I this questio, you are asked to cosider a Beroulli radom variable Y, with a success probability P ry 1 p. You are told that you have draws from this distributio ad
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso productmomet correlatio
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationStatistics. Chapter 10 TwoSample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 101
Statistics Chapter 0 TwoSample Tests Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0 Learig Objectives I this chapter, you lear How to use hypothesis testig for comparig the differece
More informationStatistical Intervals for a Single Sample
3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8 Cofidece Iterval o the Mea
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationDay 83. Prakash Balachandran Department of Mathematics & Statistics Boston University. Friday, October 28, 2011
Day 83 Prakash Balachadra Departmet of Mathematics & Statistics Bosto Uiversity Friday, October 8, 011 Sectio 5.: Hypothesis Tests forµ Aoucemets: Tutorig: Math office i 111 Cummigto 1st floor, Rich Hall.
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSOLUTIONS y n. n 1 = 605, y 1 = 351. y1. p y n. n 2 = 195, y 2 = 41. y p H 0 : p 1 = p 2 vs. H 1 : p 1 p 2.
STAT 400 UIUC Practice Problems # SOLUTIONS Stepaov Dalpiaz The followig are a umber of practice problems that may be helpful for completig the homework, ad will likely be very useful for studyig for exams..
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationCH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions
CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample
More informationMedian and IQR The median is the value which divides the ordered data values in half.
STA 666 Fall 2007 Webbased Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5umber summary mea ad stadard deviatio Media
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA SA 0B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology eachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More information18. Twosample problems for population means (σ unknown)
8. Twosamle roblems for oulatio meas (σ ukow) The Practice of Statistics i the Life Scieces Third Editio 04 W.H. Freema ad Comay Objectives (PSLS Chater 8) Comarig two meas (σ ukow) Twosamle situatios
More informationCTL.SC0x Supply Chain Analytics
CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationMA131  Analysis 1. Workbook 2 Sequences I
MA3  Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationCentral Limit Theorem the Meaning and the Usage
Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationSampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should
More informationORF 245 Fundamentals of Engineering Statistics. Midterm Exam 2
Priceto Uiversit Departmet of Operatios Research ad Fiacial Egieerig ORF 45 Fudametals of Egieerig Statistics Midterm Eam April 17, 009 :00am:50am PLEASE DO NOT TURN THIS PAGE AND START THE EXAM UNTIL
More informationAssessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions
Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe, ad two, sided tolerace itervals based o either a distributiofree (oparametric) method or a method based o a ormality assumptio (parametric). A twosided
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information( µ /σ)ζ/(ζ+1) µ /σ ( µ /σ)ζ/(ζ 1)
A eective CI for the mea with samples of size 1 ad Melaie Wall James Boe ad Richard Tweedie 1 Abstract It is couterituitive that with a sample of oly oe value from a ormal distributio oe ca costruct a
More informationSampling, Sampling Distribution and Normality
4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,
More informationSTAT 203 Chapter 18 Sampling Distribution Models
STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences
UCLA STAT 13 Itroductio to Statistical Methods for the Life ad Health Scieces Istructor: Ivo Diov, Asst. Prof. of Statistics ad Neurolog Sample Size Calculatios & Cofidece Itervals for Proportios Teachig
More informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationTable 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 informationGeneral IxJ Contingency Tables
page1 Geeral x Cotigecy Tables We ow geeralize our previous results from the prospective, retrospective ad crosssectioal studies ad the Poisso samplig case to x cotigecy tables. For such tables, the test
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationStat 400, section 5.4 supplement: The Central Limit Theorem
Stat, sectio 5. supplemet: The Cetral Limit Theorem otes by Tim Pilachowski Table of Cotets 1. Backgroud 1. Theoretical. Practical. The Cetral Limit Theorem 5. Homework Exercises 7 1. Backgroud Gatherig
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationConfidence Level We want to estimate the true mean of a random variable X economically and with confidence.
Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio
More informationActivity 3: Length Measurements with the FourSided Meter Stick
Activity 3: Legth Measuremets with the FourSided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a foursided meter
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationSALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS
SALES AND MARKETING Departmet MATHEMATICS d Semester Bivariate statistics LESSONS Olie documet: http://jffduttc.weebly.com sectio DUT Maths S. IUT de SaitEtiee Départemet TC J.F.Ferraris Math S StatVar
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More information