Definition. Definition. 72 Estimating a Population Proportion. Definition. Definition


 Deborah Floyd
 1 years ago
 Views:
Transcription
1 7 stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio proportio. We ca use a sample proportio to costruct a cofidece iterval to estimate the true value of a populatio proportio, ad we should kow how to iterpret such cofidece itervals. We should kow how to fid the sample size ecessary to estimate a populatio proportio. A poit estimate is a sigle value (or poit) used to approximate a populatio parameter. The sample proportio ˆp is the best poit estimate of the populatio proportio p. 11 xample The Pew Research Ceter coducted a survey of 1007 adults ad foud that 85% of them kow what Twitter is. The best poit estimate of p, the populatio proportio, is the sample proportio: p ˆ 0.85 A cofidece iterval (or iterval estimate) is a rage (or a iterval) of values used to estimate the true value of a populatio parameter. A cofidece iterval is sometimes abbreviated as CI. A cofidece level is the probability 1 α (ofte expressed as the equivalet percetage value) that the cofidece iterval actually does cotai the populatio parameter, assumig that the estimatio process is repeated a large umber of times. (The cofidece level is also called degree of cofidece, or the cofidece coefficiet.) Most commo choices are 90%, 95%, or 99%. (α 0.10), (α 0.05), (α 0.01)
2 Iterpretig a Cofidece Iterval Cautio We must be careful to iterpret cofidece itervals correctly. There is a correct iterpretatio ad may differet ad creative icorrect iterpretatios of the cofidece iterval 0.88 < p < Kow the correct iterpretatio of a cofidece iterval. We are 95% cofidet that the iterval from 0.88 to 0.87 actually does cotai the true value of the populatio proportio p. This meas that if we were to select may differet samples of size 1007 ad costruct the correspodig cofidece itervals, 95% of them would actually cotai the value of the populatio proportio p. (Note that i this correct iterpretatio, the level of 95% refers to the success rate of the process beig used to estimate the proportio.) Cofidece itervals ca be used iformally to compare differet data sets, but the overlappig of cofidece itervals should ot be used for makig formal ad fial coclusios about equality of proportios. Usig Cofidece Itervals for Hypothesis Tests A cofidece iterval ca be used to test some claim made about a populatio proportio p. For ow, we do ot yet use a formal method of hypothesis testig, so we simply geerate a cofidece iterval ad make a iformal judgmet based o the result. Critical Values A stadard z score ca be used to distiguish betwee sample statistics that are likely to occur ad those that are ulikely to occur. Such a z score is called a critical value. Critical values are based o the followig observatios: 1. Uder certai coditios, the samplig distributio of sample proportios ca be approximated by a ormal distributio.. A z score associated with a sample proportio has a probability of α/ of fallig i the right tail. Critical Values 3. The z score separatig the righttail regio is commoly deoted by z α/ ad is referred to as a critical value because it is o the borderlie separatig z scores from sample proportios that are likely to occur from those that are ulikely to occur. A critical value is the umber o the borderlie separatig sample statistics that are likely to occur from those that are ulikely to occur. The umber z α/ is a critical value that is a z score with the property that it separates a area of α/ i the right tail of the stadard ormal distributio.
3 Fidig z α/ for a 95% Cofidece Level Commo Critical Values Cofidece Level α Critical Value, z α/ 90% % % z a / z a / Critical Values Whe data from a simple radom sample are used to estimate a populatio proportio p, the margi of error, deoted by, is the maximum likely differece (with probability 1 α, such as 0.95) betwee the observed proportio ˆp ad the true value of the populatio proportio p. 33 Margi of rror for Proportios The margi of error is also called the maximum error of the estimate ad ca be foud by multiplyig the critical value ad the stadard deviatio of the sample proportios: z α Cofidece Iterval for stimatig a Populatio Proportio p Cofidece Iterval for stimatig a Populatio Proportio p p ˆp z α/ populatio proportio sample proportio umber of sample values margi of error z score separatig a area of α/ i the right tail of the stadard ormal distributio. 1. The sample is a simple radom sample.. The coditios for the biomial distributio are satisfied: there is a fixed umber of trials, the trials are idepedet, there are two categories of outcomes, ad the probabilities remai costat for each trial. 3. There are at least 5 successes ad 5 failures.
4 Cofidece Iterval for stimatig a Populatio Proportio p Cofidece Iterval for stimatig a Populatio Proportio p pˆ < p< pˆ + ˆp ± where z α ( pˆ, pˆ + ) RoudOff Rule for Cofidece Iterval stimates of p Roud the cofidece iterval limits for p to three sigificat digits. 44 Procedure for Costructig a Cofidece Iterval for p 1. Verify that the required assumptios are satisfied. (The sample is a simple radom sample, the coditios for the biomial distributio are satisfied, ad the ormal distributio ca be used to approximate the distributio of sample proportios because p 5, ad q 5 are both satisfied.). Refer to Table A ad fid the critical value z α/ that correspods to the desired cofidece level. 3. valuate the margi of error z ˆˆ α pq Procedure for Costructig a Cofidece Iterval for p  cot 4. Usig the value of the calculated margi of error ad the value of the sample proportio, ˆp, fid the values of ˆp ad ˆp+. Substitute those values i the geeral format for the cofidece iterval: pˆ < p< pˆ + 5. Roud the resultig cofidece iterval limits to three sigificat digits. xample I the Chapter Problem we oted that a Pew Research Ceter poll of 1007 radomly selected adults showed that 85% of respodets kow what Twitter is. The sample results are 1007 ad pˆ a. Fid the margi of error that correspods to a 95% cofidece level. b. Fid the 95% cofidece iterval estimate of the populatio proportio p. c. Based o the results, ca we safely coclude that more tha 75% of adults kow what Twitter is? d. Assumig that you are a ewspaper reporter, write a brief statemet that accurately describes the results ad icludes all of the relevat iformatio.
5 xample  Cotiued Requiremet check: simple radom sample; fixed umber of trials, 1007; trials are idepedet; two outcomes per trial; probability remais costat. Note: umber of successes ad failures are both at least 5. a) Use the formula to fid the margi of error. ( 085)( 015).. zα xample  Cotiued b) The 95% cofidece iterval: pˆ < p < pˆ < p < < p < xample  Cotiued c) Based o the cofidece iterval obtaied i part (b), it does appear that more tha 75% of adults kow what Twitter is. Because the limits of 0.88 ad 0.87 are likely to cotai the true populatio proportio, it appears that the populatio proportio is a value greater tha xample  Cotiued d) Here is oe statemet that summarizes the results: 85% of U.S. adults kow what Twitter is. That percetage is based o a Pew Research Ceter poll of 1007 radomly selected adults. I theory, i 95% of such polls, the percetage should differ by o more tha. percetage poits i either directio from the percetage that would be foud by iterviewig all adults i the Uited States. Aalyzig Polls Whe aalyzig polls cosider: 1. The sample should be a simple radom sample, ot a iappropriate sample (such as a volutary respose sample).. The cofidece level should be provided. (It is ofte 95%, but media reports ofte eglect to idetify it.) 3. The sample size should be provided. (It is usually provided by the media, but ot always.) 4. xcept for relatively rare cases, the quality of the poll results depeds o the samplig method ad the size of the sample, but the size of the populatio is usually ot a factor. Cautio Never follow the commo miscoceptio that poll results are ureliable if the sample size is a small percetage of the populatio size. The populatio size is usually ot a factor i determiig the reliability of a poll.
6 Sample Size Suppose we wat to collect sample data i order to estimate some populatio proportio. The questio is how may sample items must be obtaied? Determiig Sample Size z a (solve for by algebra) ( z ) a Sample Size for stimatig Proportio p Whe a estimate of ( z ) Whe o estimate of a ( z a ) 0.5 ˆp is kow: ˆp is kow: 66 RoudOff Rule for Determiig Sample Size If the computed sample size is ot a whole umber, roud the value of up to the ext larger whole umber. xample May compaies are iterested i kowig the percetage of adults who buy clothig olie. How may adults must be surveyed i order to be 95% cofidet that the sample percetage is i error by o more tha three percetage poits? a. Use a recet result from the Cesus Bureau: 66% of adults buy clothig olie. b. Assume that we have o prior iformatio suggestig a possible value of the proportio. a) Use xample  Cotiued pˆ 0.66 ad qˆ 1 pˆ 0.34 α 0.05 so zα ( zα ) ( 1.96) ( 0.66)( 0.34) ( 0.03) To be 95% cofidet that our sample percetage is withi three percetage poits of the true percetage for all adults, we should obtai a simple radom sample of 958 adults.
7 b) Use xample  Cotiued α 0.05 so z α ( zα ) 0.5 ( ) ( 0.03) To be 95% cofidet that our sample percetage is withi three percetage poits of the true percetage for all adults, we should obtai a simple radom sample of 1068 adults. Fidig the Poit stimate ad from a Cofidece Iterval Poit estimate of : (upper cofidece limit) + (lower cofidece limit) ˆp Margi of rror: (upper cofidece limit) (lower cofidece limit) 77
Key Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationUsing Excel to Construct Confidence Intervals
OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationPractice Problems for Test 3
Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT  Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationConfidence Intervals
Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more
More informationReview for Test 3. b. Construct the 90% and 95% confidence intervals for the population mean. Interpret the CIs.
Review for Test 3 1 From a radom sample of 36 days i a recet year, the closig stock prices of Hasbro had a mea of $1931 From past studies we kow that the populatio stadard deviatio is $237 a Should you
More informationCh 7.1 pg. 364 #11, 13, 15, 17, 19, 21, 23, 25
Math 7 Elemetary Statistics: A Brief Versio, 5/e Bluma Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 1, 3, 5 11. Readig Scores: A sample of the readig scores of 35 fifthgraders has a mea of 8. The stadard deviatio
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationReview for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
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 informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationSTA 2023 Practice Questions Exam 2 Chapter 7 sec 9.2. Case parameter estimator standard error Estimate of standard error
STA 2023 Practice Questios Exam 2 Chapter 7 sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (1) oe p ( 1 p) CI: prop.
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationChapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing
Chapter 10 Hypothesis Tests Regardig a Parameter A secod type of statistical iferece is hypothesis testig. Here, rather tha use either a poit (or iterval) estimate from a simple radom sample to approximate
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information1 Hypothesis testing for a single mean
BST 140.65 Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationOnesample test of proportions
Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationStatistical Methods. Chapter 1: Overview and Descriptive Statistics
Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics
More informationx : X bar Mean (i.e. Average) of a sample
A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationHypothesis testing: one sample
Hypothesis testig: oe sample Describig iformatios Flowchart for QMS 202 Drawig coclusios Forecastig Improve busiess processes Data Collectio Probability & Probability Distributio Regressio Aalysis Timeseries
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationSTATISTICAL METHODS FOR BUSINESS
STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING 7.1. Distributios associated with the samplig process. 7.2. Iferetial processes ad relevat distributios.
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationConfidence intervals and hypothesis tests
Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationStatistical Inference: Hypothesis Testing for Single Populations
Chapter 9 Statistical Iferece: Hypothesis Testig for Sigle Populatios A foremost statistical mechaism for decisio makig is the hypothesis test. The cocept of hypothesis testig lies at the heart of iferetial
More informationChapter 10 Student Lecture Notes 101
Chapter 0 tudet Lecture Notes 0 Basic Busiess tatistics (9 th Editio) Chapter 0 Twoample Tests with Numerical Data 004 PreticeHall, Ic. Chap 0 Chapter Topics Comparig Two Idepedet amples Z test for
More informationOMG! Excessive Texting Tied to Risky Teen Behaviors
BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9
More informationQuadrat Sampling in Population Ecology
Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may
More informationCorrelation. example 2
Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8 correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationMeasures of Central Tendency
Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More information15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011
15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes highdefiitio
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationMannWhitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
NoParametric ivariate Statistics: WilcoxoMaWhitey 2 Sample Test 1 MaWhitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo) MaWhitey (WMW) test is the oparametric equivalet of a pooled
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationThis is arithmetic average of the x values and is usually referred to simply as the mean.
prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity http://www.humboldt.edu/~geodept/geology51/51_hadouts/statistical_aalysis.pdf STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout
More informationSimple Linear Regression
Simple Liear Regressio We have bee itroduced to the otio that a categorical variable could deped o differet levels of aother variable whe we discussed cotigecy tables. We ll exted this idea to the case
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationResearch Method (I) Knowledge on Sampling (Simple Random Sampling)
Research Method (I) Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact
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 informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationHypothesis testing using complex survey data
Hypotesis testig usig complex survey data A Sort Course preseted by Peter Ly, Uiversity of Essex i associatio wit te coferece of te Europea Survey Researc Associatio Prague, 5 Jue 007 1 1. Objective: Simple
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationExample Consider the following set of data, showing the number of times a sample of 5 students check their per day:
Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day
More informationStat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.
Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:
More informationLecture 10: Hypothesis testing and confidence intervals
Eco 514: Probability ad Statistics Lecture 10: Hypothesis testig ad cofidece itervals Types of reasoig Deductive reasoig: Start with statemets that are assumed to be true ad use rules of logic to esure
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationCovariance and correlation
Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative
More informationTopic 5: Confidence Intervals (Chapter 9)
Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with
More informationSample size for clinical trials
Outcome variables for trials British Stadards Istitutio Study Day Sample size for cliical trials Marti Blad Prof. of Health Statistics Uiversity of York http://martiblad.co.uk A outcome variable is oe
More informationDUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8. Probability Based Learning: Introduction to Probability REVISION QUESTIONS *** SOLUTIONS ***
DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 Probability Based Learig: Itroductio to Probability REVISION QUESTIONS *** *** MACHINE LEARNING AT DIT Dr. Joh Kelleher Dr. Bria Mac Namee *** ***
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationUnit 8: Inference for Proportions. Chapters 8 & 9 in IPS
Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationChapter 5 Discrete Probability Distributions
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More information, a Wishart distribution with n 1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematiskstatistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationConfidence Intervals for Linear Regression Slope
Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationMultiserver Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu
Multiserver Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio coectio
More informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationJoint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial
More information