Confidence Intervals for the Population Mean


 Edwin Webb
 2 years ago
 Views:
Transcription
1 Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios. So if X is distributed σ σ approximately ormally, P µ 2 < X < µ if we rearrage it so µ is σ σ i the middle the P x 2 < µ < x The iterval σ σ x 2, x + 2 has a probability of 0.95 of capturig the mea. Defiitio: If X is the sample mea of a radom sample of size from a populatio with variace 2 1 α 100% cofidece iterval for µ is give by σ, a ( ) σ X Z, X + Z α/2 α/2 σ Where Z α /2 is the Z value from the ormal table with area α /2 to its right. If σ, the populatio stadard deviatio is ukow, it ca be replaced by s, the sample stadard deviatio with o serious loss of accuracy for large sample cases. If we use α = 0.05, we report that we are 95% cofidet that the populatio mea will be withi our iterval. Why are we able to say we are 95% cofidet? We kow (from the Empirical Rule) that about 95% of all possible sample meas will lie withi two stadard errors of the actual populatio mea. We hope that our sample mea is oe of these, because if it is, the our cofidece iterval will cotai the populatio mea, ad our estimate will be correct. If ot, the our iterval will be icorrect. But this oly happes 5% of the time. The term "95% cofidece" meas that if we took repeated samples, ad foud a cofidece iterval for each sample, 95% of those cofidece itervals would actually cotai the populatio mea; 5% of them would ot. Whether our ow cofidece iterval cotais the populatio mea, we will ever kow! The Empirical Rule Theorem v.s. A Cofidece Iterval The Empirical Rule Theorem ad A Cofidece Iterval for the Mea are used to aswer two differet research questios. 1
2 Cofidece Itervals Math 283 The Empirical Rule Theorem is used to aswer the questio "Most of the values for the variable fall betwee what two values?" This is a rage of values used to discuss what we kow about the idividuals i our sample or populatio. A Cofidece Iterval is used to aswer the questio "What is the mea of the populatio?" This is a rage of values used to give reasoable values for the populatio mea. The average zic cocetratio recovered from a sample of zic measuremets i 36 differet locatios i a river is foud to be 2.6 grams per milliliter. Fid the 95% ad 99% cofidece itervals for the mea zic cocetratio i the river. Assume the populatio stadard deviatio is 0.3. A importat property of plastic clays is the percet of shrikage o dryig. For a certai type of plastic clay 45 test specimes showed a average shrikage of 18.4% with a stadard deviatio of 1.2. Estimate the mea percet shrikage for this type of clay with a 90% cofidece iterval. Aother way to thik about the cofidece iterval is: x ± MOE where MOE is the σ margi of error, MOE = Zα /2. Notice the width or precisio of our cofidece iterval depeds o cofidece level 1 α, sample size, ad stadard deviatio of the populatio. The accuracy of our sample mea depeds o the sample size,, the stadard deviatio of the populatio, σ, ad bias. 2
3 Cofidece Itervals Math 283 Decisio Makig with a Cofidece Iterval The owers of Geeral Light are plaig to advertise their light bulbs i the Suday editio of the ewspaper. I the ad, they wat to report "the mea lifetime of their light bulbs." To determie the mea lifetime of their light bulbs, they took a radom sample of 40 light bulbs. For their sample, the bulbs lasted o average, hours with a stadard deviatio of 58 hours. 1. Costruct a 95% cofidece iterval for the mea lifetime of light bulbs. 2. Should Geeral Light advertise that the mea lifetime of their light bulbs is 350 hours? Why or why ot? 3. Should Geeral Light advertise that the mea lifetime of their light bulbs is 310 hours? Why or why ot? Determiig Sample Size Whe our objective is to estimate the populatio mea, µ, we should do the followig to determie our sample size: 1. Determie the largest margi of error you are willig to accept ad a cofidece level. 2. Obtai or estimate the populatio stadard deviatio. σ 3. Fid the sample size,, that makes the followig true: Your MOE = Zα /2. 4. Check the sample size agaist your budget. If ecessary, retur to step 1. You are plaig a survey of startig salaries for liberal arts major graduates from you college. From a pilot study you estimate that the stadard deviatio is about $9000. What sample size do you eed to have a margi of error equal to $400 with 95% cofidece? 3
4 Cofidece Itervals Math 283 Cautios: Data must come from a SRS No correct method from data haphazardly collected with bias of ukow size. The sample mea is ot resistat to outliers. So look at your data carefully before determiig a CI. If is small ad populatio is ot ormal, the true cofidece level may be differet from what you used. o As log as 30, CLT applies o If 15, it is ok uless there are extreme outliers i.e. quite strog skewess. Must kow σ. The Case of the Ukow σ If X is the sample mea of a radom sample of size where X, 1 X are from a ormal distributio the the radom variable X µ t = s/ has a probability distributio called the tdistributio with degrees of freedom 1. Properties of the tdistributio (or Studet s tdistributio) Bell shaped with the mea zero. ν The variace where ν is the degrees of freedom. ν 2 The limitig distributio of the t is the stadard ormal distributio as goes to ifiity. See Table attached. A ( α ) 1 100% Cofidece Iterval for µ whe σ is ukow Let x ad s be the sample mea ad stadard deviatio of a radom sample of size from a ormally distributed populatio the the cofidece iterval is give by s X t, X + t α/2 α/2 where t α /2is the value from the tdistributio with degrees of freedom 1 ad α /2 is the upper tail probability. Note, this iterval is fairly robust to oormal data. If the data is ot too skewed, the t procedure is useful whe 15 < 40. Whe 40, the t procedure ca be used eve for skewed data. s 4
5 Cofidece Itervals Math 283 The cotets of 7 similar cotaiers of sulfuric acid are 9.8, 10.2, 10.4, 9.8, 10.0, 10.2, ad 9.6 liters. Fid a 95% cofidece iterval for the mea of all such cotaiers, assumig the data are from a ormal distributio. A radom sample of 12 graduates of a certai secretarial school typed a average of 79.3 words per miute with a sample stadard deviatio of 7.8 words per miute. Assumig a ormal distributio for the umber of words typed per miute, fid a 99% cofidece iterval for the mea umber of words per miute for all graduates. Cofidece Iterval for the Populatio Proportio If X, 1 X are idepedet observatios from a populatio with probability of success, the the radom variable X = X is distributed biomial with E ( X ) = p ad ( ) ( 1 ) i= 1 V X = p p. We showed that distributio as goes to ifiity. i Z = X p p ( 1 p) approaches the stadard ormal So the samplig distributio of pˆ = X / is approximately ormal with µ ˆp = p ad ( 1 p) p σ pˆ =. 5
6 Cofidece Itervals Math 283 A ( α ) 1 100% cofidece iterval for p, the populatio proportio is give by ( 1 ) ( 1 ) ˆ ˆ ˆ ˆ pˆ Z p p ˆ /2, p Z p p α + α/2 Where Z α /2 is the Z value from the ormal table with area α /2 to its right. A survey of 1280 studet loa borrowers foud that 448 had loas totalig more tha $20,000 for their udergraduate educatio. Give a 95% cofidece iterval for the proportio of all studet loa borrowers who have loas of $20,000 or more for their uder graduate degree. Determiig Sample Size Whe our objective is to estimate the populatio proportio, p, we should do the followig to determie our sample size: 1. Determie the largest margi of error you are willig to accept ad a cofidece level. 2. Determie p from previous study or use p = Fid the sample size,, that makes the followig true: p( 1 p) Your MOE = Zα /2. 4. Check the sample size agaist your budget. If ecessary, retur to step 1. You are plaig a evaluatio of a alcohol awareess program at your college that will take place six moths after the program. How large a sample should you take if you wat the margi of error for 95% to be about 0.1? 6
7 Cofidece Itervals Math 283 Studet s t Distributio Upper tail probability d.f
8 Cofidece Itervals Math The average weight of 40 radomly selected miivas was 4150 pouds. a. Fid ad iterpret a 98% cofidece iterval for the mea weight of all miivas. The stadard deviatio is kow to be 480 pouds. b. What could we do to reduce the width of this iterval? c. What are the advatages/disadvatages of your aswers i b? 2. The weight of grapefruit follows a ormal distributio. A radom sample of 12 ew hybrid grapefruit had a mea weight of 1.7 pouds with a stadard deviatio of 0.24 pouds. Fid a 95% cofidece iterval for the mea weight of the populatio of ew hybrid grapefruits. 3. A researcher wishes to estimate, withi $25, the true average amout of postage that parets of college studets sped each year. If she wishes to be 90% cofidet, how large a sample is ecessary? The stadard deviatio is kow to be $ A survey by Brides magazie foud that 8 out of 10 brides are plaig to take the surame of their ew husbad. How large a sample is eeded to estimate the true proportio to withi 3% with 98% cofidece? 5. A researcher wishes to estimate the proportio of adult females uder 5 feet tall. He wats to be 90% cofidet that his estimate is withi 5% of the true proportio. What sample size should he use? 6. I a survey of 200 workers, 169 said they were iterrupted three or more times a hour by phoe messages, faxes, etc. Fid ad iterpret a 90% cofidece iterval of the populatio of proportio of workers who are iterrupted three or more times a hour. 7. A sample of 17 states had these cigarette taxes (i cets): 112, 120, 98, 55, 71, 35, 99, 124, 64, 150, 150, 55, 100, 132, 35, 70, 93. Fid a 98% cofidece iterval for the mea cigarette tax i all 50 states. What assumptio is ecessary? 8
Ch 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 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 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 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 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 informationKey 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 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 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 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 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 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 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
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
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 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 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 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 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 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 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 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 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 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 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 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 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 informationProbability & Statistics Chapter 9 Hypothesis Testing
I Itroductio to Probability & Statistics A statisticia s most importat job is to draw ifereces about populatios based o samples take from the populatio Methods for drawig ifereces about parameters: ) Make
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 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 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 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 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 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 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 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 informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
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 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 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 informationDiscrete Random Variables and Probability Distributions. Random Variables. Chapter 3 3.1
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationTI83, TI83 Plus or TI84 for NonBusiness Statistics
TI83, TI83 Plu or TI84 for NoBuie Statitic Chapter 3 Eterig Data Pre [STAT] the firt optio i already highlighted (:Edit) o you ca either pre [ENTER] or. Make ure the curor i i the lit, ot o the lit
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 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 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 information0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%
Sectio 10 Aswer Key: 0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% 1) A simple radom sample of New Yorkers fids that 87 are
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 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 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 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 information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
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 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 informationA Brief Study about Nonparametric Adherence Tests
A Brief Study about Noparametric Adherece Tests Viicius R. Domigues, Lua C. S. M. Ozelim Abstract The statistical study has become idispesable for various fields of kowledge. Not ay differet, i Geotechics
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 informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
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 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 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 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 informationNotes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
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 informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationASSUMPTIONS/CONDITIONS FOR HYPOTHESIS TESTS and CONFIDENCE INTERVALS
ASSUMPTIONS/CONDITIONS FOR HYPOTHESIS TESTS ad CONFIDENCE INTERVALS Oe of the importat tak whe applyig a tatitical tet (or cofidece iterval) i to check that the aumptio of the tet are ot violated. Oeample
More informationA Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:
A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio
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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
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 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 information