The 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


 Godfrey Garrison
 1 years ago
 Views:
Transcription
1 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 that is selected for study. Iferetial Statistics is the process of usig sample iformatio to draw ifereces or coclusios about the populatio. Cosider a populatio of 5 commuters who are all eighbors. Each commuter was asked how may miles he/she commutes to work each day. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles Fid the populatio mea of the set of data: Fid the populatio stadard deviatio: Look at a sample of two commuters ( 2 ), ad fid the mea to estimate µ. Like: C1 = 50 ad C2 = 84, the C1 ad C3, the C1 ad C4, the C1 ad C5 the C4 ad C5. How may should we have? List all possible samples of two commuters ad calculate the mea,, for each sample. Commuters Data Values Mea, The data set of all the sample meas i colum 3 is called a samplig distributio of the meas. The samplig error is the differece betwee the value of a sample mea,, ad the populatio mea, µ. Samplig error of the mea = = µ For the sample mea 67, calculate the sample error: For the sample mea 102, calculate the sample error: 1
2 Lookig back at colum 3 of our table, this data set of all the sample meas is called a samplig distributio of the meas. Calculate the mea of this data set we are calculatig the mea of the samplig distributio of the meas: The mea of all the sample meas of a samplig distributio has special otatio: mu sub bar or Calculate the stadard deviatio of this data set we are calculatig the stadard deviatio of the samplig distributio of the meas: The stadard deviatio of the all the sample meas of a samplig distributio has special otatio: sigma sub bar or The stadard deviatio is a measure of how spread out the sample meas are from µ. The mea of the samplig distributio of the meas is: = What is the populatio mea which we calculated before? = So, the mea of the samplig distributio of the meas is equal to the mea of the populatio from which the samples were selected: The stadard deviatio of the samplig distributio of the meas is: = What is the populatio stadard deviatio which we calculated before? = The stadard deviatio of the samplig distributio of the mea, also kow as the stadard error of the mea, will always be smaller tha the populatio stadard deviatio ad the formula is: The larger the stadard error of the mea is, the more dispersed the samples meas are from the populatio mea. The smaller the stadard error, the closer the sample meas are to the populatio mea. Fiite Correctio Factor: For a fiite populatio (havig a limit), the formula for the stadard deviatio of the samplig distributio of the mea, is: N N 1 2
3 8.1 The Samplig Distributio of the Mea Populatios are geerally either quite large, have a ulimited umber of data values (that is, ifiite), or partly uobtaiable. Sice a populatio ca rarely be studied completely, a sample radomly selected from the populatio serves as a coveiet ad ecoomical procedure to estimate the characteristics of a populatio. Sample iformatio is used to estimate the mea of a populatio. Whe the populatio mea is beig estimated usig sample data, the mea of the sample will probably ot be equal to the mea of the populatio. I fact, if a few samples were radomly selected from the same populatio, it is very likely that oe of these sample meas would be eactly equal to the mea of the populatio. Although there is oly oe value for the true mea of a populatio, there are may differet values for the mea whe differet samples are selected from the populatio. Therefore, the estimates of a true populatio mea will vary from sample to sample due to the data values radomly selected withi each sample, that is, by chace aloe. These variatios i the estimates of the populatio mea from sample to sample are due to chace ad are called samplig errors. Samplig error is the differece betwee the value of a sample statistic, such as the sample mea, ad the value of the correspodig populatio parameter, such as the populatio mea, μ. Thus, the samplig error for the mea is: samplig error = sample mea populatio mea assumig the sample is radom ad there are o osample errors. I symbols, samplig error of the mea = μ. Samplig error is ievitable because we are usig a chose umber of data values which are radomly selected, by chace, from the populatio. I practice, we will select oly oe sample from the populatio to estimate the mea of the populatio. A populatio ca have oly oe mea. Yet, depedig upo which sample is selected from a populatio, the mea of a sample ca vary from sample to sample as differet samples of the same size are radomly selected from the same populatio. Thus, the sample mea is a radom variable because it is depedet upo the particular data values which are radomly selected from the populatio. Eample 8.1 pg. 423 Suppose the populatio of seve college studets o the Studet Govermet Associatio (SGA) have the followig ages: 23, 19, 20, 21, 18, 19, 25 a. Compute the populatio mea age of all the studets o the SGA. b. If a radom sample of 3 studets was selected from this populatio havig ages: 19, 21, ad 18, the compute the sample mea age for this sample. c. Determie the samplig error if this sample (from part b) was used to estimate the populatio mea age. Eplai or iterpret the meaig of this samplig error. *Review parts d, e, ad f of this eample too! 3
4 The samplig distributio of the mea is a probability distributio which lists the sample meas from all possible samples of the same sample size selected from the same populatio alog with the probability associated with each sample mea. Notatio for the Mea of the Samplig Distributio of the Mea The mea of the samplig distributio of the mea is deoted by, read mu sub bar. Thus, = mea of all the sample meas of the samplig distributio. Notatio for the Stadard Deviatio of the Samplig Distributio of the Mea The stadard deviatio of the samplig distributio of the mea is deoted by, read sigma sub bar. Thus, = stadard deviatio of all the sample meas of the samplig distributio. 8.2 The Mea ad Stadard Deviatio of the Samplig Distributio of the Mea Mea of the Samplig Distributio of the Mea, The mea of the sample meas of all possible samples of size is called the mea of the samplig distributio of the mea, deoted by. It is equal to the mea of the populatio from which the samples were selected. I symbols, this is epressed as: Stadard Deviatio of the Samplig Distributio of the Mea or Stadard Error of the Mea, deoted by The stadard error of the mea is the stadard deviatio of the sample meas of all possible samples of size of the samplig distributio, deoted by. The stadard error of the mea is equal to the stadard deviatio of the populatio, σ, divided by the square root of the sample size. That is: stadard error of the mea populatio stadard deviatio = sample size Iterpretatio of the Stadard Error of the Mea The stadard deviatio of the samplig distributio of the mea is referred to as the stadard error of the mea because it is a measure of how much a sample mea is likely to deviate from the populatio mea, that is, a measure of the average samplig error. If the stadard error of the mea,, is a small umber, the the samplig distributio of the mea has relatively little dispersio ad the sample meas will be relatively close to the populatio mea. O the other had, if the stadard error of the mea,, is a large umber, the the samplig distributio of the mea has a relatively large dispersio ad the sample meas will be relatively far from the populatio mea. 4
5 Eample 8.3 pg. 430 Accordig to a study of TV viewig habits, the average umber of hours a teeager watches MTV per week is 17.9 hours with a stadard deviatio of 3.8 hours. If a sample of 64 teeagers is radomly selected from the populatio, the determie the mea ad stadard error of the mea of the samplig distributio of the mea. Eample 8.4 pg The registrar at a large Uiversity states that the mea grade poit average of all the studets is 2.95 with a populatio stadard deviatio of a. Determie the mea ad stadard error of the samplig distributio if the samplig distributio of the mea cosists of all possible sample meas from samples of size 25. b. Determie the mea ad stadard error of the samplig distributio if the samplig distributio of the mea cosists of all possible sample meas from samples of size 100. c. What effect did icreasig the sample size have o the mea ad stadard error of the samplig distributio? d. I which samplig distributio of the mea ( = 25 or = 100) would you have a better chace of selectig a sample mea which is closer to the populatio mea grade poit average? Review Eample 8.5 o pg
6 8.4 The Shape of the Samplig Distributio of the Mea Samplig from a Normal Populatio THEOREM The Shape of the Samplig Distributio whe Samplig from a Normal Populatio If the populatio beig sampled is a ormal distributio, the the samplig distributio of the mea is a ormal distributio regardless of the sample size,. Characteristics of the Samplig Distributio of the Mea Whe Samplig from a Normal Populatio Whose Mea is μ ad Stadard Deviatio is σ If all possible samples of size are selected from a ormal populatio, the the samplig distributio of the mea has the followig three characteristics: 1. The samplig distributio of the mea is a ormal distributio, regardless of sample size,. 2. The mea of the samplig distributio of the mea,, is equal to the mea of the populatio, μ:. 3. The stadard error of the samplig distributio of the mea,, is equal to the stadard deviatio of the populatio, σ, divided by the square root of the sample size, :. Eample 8.7 pg. 438 At a large New Eglad college, the grade poit average (GPA) of all attedig studets is ormally distributed with a mea of 2.95 ad a populatio stadard deviatio of A sample is radomly selected from this populatio ad its sample mea,, is calculated. Determie the mea,, the stadard error, idetify the shape of the samplig distributio of the mea of the samples of size: a. = 9 b. = 49 c. = 100 6
7 Samplig from a NoNormal Populatio THEOREM 8.2 The Cetral Limit Theorem For ay populatio, the samplig distributio of the mea approaches a ormal distributio as the sample size becomes large. This is true regardless of the shape of the populatio beig radomly sampled. Geeral Rule for Applyig the Cetral Limit Theorem: The Greater tha 30 Rule For most applicatios, a sample size greater tha 30 is cosidered large eough to apply the Cetral Limit Theorem. Thus, the samplig distributio of the mea ca be reasoably approimated by a ormal distributio wheever the sample size is greater tha 30. THEOREM 8.3 Characteristics of the Samplig Distributio of the Mea whe Samplig from a NoNormal Populatio If, the followig three coditios are satisfied: give ay ifiite populatio with mea, μ, ad stadard deviatio, σ, ad all possible radom samples of size are selected from the populatio to form a samplig distributio of the mea, ad the sample size,, is large (greater tha 30). the: 1. the samplig distributio of the mea is approimately ormal. 2. the mea of the samplig distributio of the mea is equal to the mea of the populatio. This is epressed as: 3. the stadard error of the samplig distributio of the mea is equal to the stadard deviatio of the populatio divided by the square root of the sample size. This is writte as: Eample 8.8 pg. 442 GRAMHAM Bell, a telephoe compay, states that the average legth of time of logdistace telephoe calls is 21.3 miutes with a stadard deviatio of 3.8 miutes. Determie the mea ad the stadard error of the samplig distributio of the mea ad describe the shape of the samplig distributio of the mea whe the sample size is: a. = 36 b. = 100 c. Compare the samplig distributio of the mea for the sample size of = 36 ad = 100. d. If you had to estimate the mea of the populatio by either radomly selectig a sample of size = 36 or of = 100 from the populatio, the which sample size would give you a better chace of obtaiig a smaller samplig error? Eplai. 7
8 8
9 8.5 Calculatig Probabilities Usig the Samplig Distributio of the Mea Now i this chapter we are workig with the samplig distributio of the mea so the data values are sample meas,, thus, the z score formula becomes: or sample mea mea of the samplig distributio z score of a samplig mea stadard error of the samplig distributio z of OR we ca Use the TI83/84 ormalcdf fuctio: 2 d DISTR:2 ( lower sample mea value, higher sample mea value,, Remember: ad ) ENTER Eample 8.9 pg At a large public state college i Virgiia, the mea Verbal SAT score of all attedig studets was 600 with a populatio stadard deviatio of 65. If a radom sample of 100 studets is selected from a populatio of studets to determie: a. The probability that the mea Verbal SAT score of the selected sample will be less tha 615. b. The probability that the mea Verbal SAT score of the selected sample will be withi 10 poits of the populatio mea. Eample 8.10 pg. 447 The populatio mea weight of ewbor babies for a wester suburb is 7.4 lbs. with a stadard deviatio of 0.8 lbs. What is the probability that a sample of 64 ewbors selected at radom will have a mea weight greater tha 7.5 lbs.? Eample 8.11 pg. 449 The populatio of the ages of all U. S. college studets is skewed to the right with a mea age of 27.4 years ad a stadard deviatio of 5.8 years. Determie the probability that a radom sample of 49 studets selected from the populatio will have a sample mea age withi oe year of the populatio mea age? 9
Sampling 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 informationSection 73 Estimating a Population. Requirements
Sectio 73 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio
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 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 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 informationDefinition. Definition. 72 Estimating a Population Proportion. Definition. Definition
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
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 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 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 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 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 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 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 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 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 informationHypothesis Tests Applied to Means
The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with
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 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 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 informationSection 7.2 Confidence Interval for a Proportion
Sectio 7.2 Cofidece Iterval for a Proportio Before ay ifereces ca be made about a proportio, certai coditios must be satisfied: 1. The sample must be a SRS from the populatio of iterest. 2. The populatio
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 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 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 information23.3 Sampling Distributions
COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE SIC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through
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 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 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 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 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 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 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: 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 informationStatistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals
Statistics 111  Lecture 14 Itroductio to Iferece Hypothesis Tests Admiistrative Notes Sprig Break! No lectures o Tuesday, March 8 th ad Thursday March 10 th Exteded Sprig Break! There is o Stat 111 recitatio
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 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 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 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 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 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 informationConfidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with
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 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 informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationCase Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph
Case Study Cotigecy Tables Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 4 6, 2011 Case Study Example 9.3 begiig o page 213 of the text describes a experimet i which
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 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 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 informationConfidence Intervals for the Mean of Nonnormal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Cofidece Itervals for the Mea of Noormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio
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 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 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 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 informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationCHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions
CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because
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 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 informationCompare Multiple Response Variables
Compare Multiple Respose Variables STATGRAPHICS Mobile Rev. 4/7/006 This procedure compares the data cotaied i three or more Respose colums. It performs a oeway aalysis of variace to determie whether
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationConfidence Intervals and Sample Size
8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGrawHill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 71 Cofidece Itervals for the
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 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 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 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 informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
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 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 informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More information7818 Interval estimation and hypothesis testing  Set
7 7818 Iterval estimatio ad hypothesis testig  Set revised Nov 9, 010 You might wat to read some of the chapter i MGB o Parametric Iterval Estimatio. There are subtle di ereces across questios. uderstad
More informationEconomics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is
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 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 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 informationME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision  A measure of agreement between repeated measurements (repeatability).
INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie
More informationInstitute for the Advancement of University Learning & Department of Statistics
Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios)
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 informationTIEE Teaching Issues and Experiments in Ecology  Volume 1, January 2004
TIEE Teachig Issues ad Experimets i Ecology  Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013
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 informationConfidence Intervals for the Population Mean
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.
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
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 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 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 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 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 informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
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 informationConfidence Intervals
1 Cofidece Itervals Recall: Iferetial statistics are used to make predictios ad decisios about a populatio based o iformatio from a sample. The two major applicatios of iferetial statistics ivolve the
More informationExplore Identifying Likely Population Proportions
COMMON CORE Locker LESSON Cofidece Itervals ad Margis of Error Commo Core Math Stadards The studet is expected to: COMMON CORE SIC.B.4 Use data from a sample survey to estimate a populatio mea or proportio;
More informationAP * Statistics Review. Inference
AP * Statistics Review Iferece Teacher Packet AP* is a trademark of the College Etrace Examiatio Board. The College Etrace Examiatio Board was ot ivolved i the productio of this material. Copyright 009
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
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 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 informationText&Tests5. Project Maths SUPPLEMENT. Frances O Regan O. D. Morris. Leaving Certificate Higher Level Maths
Project Maths SUPPLEMENT Text&Tests5 Leavig Certificate Higher Level Maths Cotais all the Deferred Material ad Cetral Limit Theorem Fraces O Rega O. D. Morris O.D. Morris, Fraces O Rega, 2014 All rights
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 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 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 informationHypothesis testing in a Nutshell
Hypothesis testig i a Nutshell Summary by Pamela Peterso Drake Itroductio The purpose of this readig is to discuss aother aspect of statistical iferece, testig. A is a statemet about the value of a populatio
More information