I. Chisquared Distributions


 Scarlett Anderson
 1 years ago
 Views:
Transcription
1 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. These will also appear i Chapter 26 i studyig categorical variables. Notatio: N(μ, σ) will stad for the ormal distributio with mea μ ad stadard deviatio σ. The symbol ~ will idicate that a radom variable has a certai distributio. For example, Y ~ N(4, 3) is short for Y has a ormal distributio with mea 4 ad stadard deviatio 3. I. Chisquared Distributios Defiitio: The chisquared distributio with k degrees of freedom is the distributio of a radom variable that is the sum of the squares of k idepedet stadard ormal radom variables. Weʼll call this distributio χ 2 (k). Thus, if Z 1,..., Z k are all stadard ormal radom variables (i.e., each Zi ~ N(0,1)), ad if they are idepedet, the Z Z k 2 ~ χ 2 (k). For example, if we cosider takig simple radom samples (with replacemet) y 1,..., y k from some N(µ,σ) distributio, ad let Y i deote the radom variable whose value is y i, the each is stadard ormal, ad,, are idepedet, so + + ~ χ 2 (k). Notice that the phrase degrees of freedom refers to the umber of idepedet stadard ormal variables ivolved. The idea is that sice these k variables are idepedet, we ca choose them freely (i.e., idepedetly). The followig exercise should help you assimilate the defiitio of chisquared distributio, as well as get a feel for the χ 2 (1) distributio. Exercise 1: Use the defiitio of a χ 2 (1) distributio ad the rule for the stadard ormal distributio (ad/or aythig else you kow about the stadard ormal distributio) to help sketch the graph of the probability desity
2 2 fuctio of a χ 2 (1) distributio. (For example, what ca you coclude about the χ 2 (1) curve from the fact that about 68% of the area uder the stadard ormal curve lies betwee 1 ad 1? What ca you coclude about the χ 2 (1) curve from the fact that about 5% of the area uder the stadard ormal lies beyod ± 2?) For k > 1, itʼs harder to figure out what the χ 2 (k) distributio looks like just usig the defiitio, but simulatios usig the defiitio ca help. The followig diagram shows histograms of four radom samples of size 1000 from a N(0,1) distributio: These four samples were put i colums labeled st1, st2, st3, st4. Takig the sum of the squares of the first two of these colums the gives (usig the defiitio of a chisquared distributio with two degrees of freedom) a radom sample of size 1000 from a χ 2 (2) distributio. Similarly, addig the squares of the first three colums gives a radom sample from a χ 2 (3) distributio, ad formig the colum (st1) 2 +(st2) 2 + (st3) 2 +(st4) 2 yields a radom sample from a χ 2 (4) distributio. Histograms of these three samples from chisquared distributios are show below, with the sample from the χ 2 (2) distributio i the upper left, the sample from the χ 2 (3) distributio i the upper right, ad the sample from the χ 2 (4) distributio i the lower left. The histograms show the shapes of the three distributios: the χ 2 (2) has a sharp peak at the left; the χ 2 (3) distributio has a less sharp peak ot quite as far left; ad the χ 2 (4) distributio has a still lower peak still a little further to the right. All three distributios are oticeably skewed to the right.
3 3 There is a picture of a typical chisquared distributio o p. A113 of the text. Thought questio: As k gets bigger ad bigger, what type of distributio would you expect the χ 2 (k) distributio to look more ad more like? [Hit: A chisquared distributio is the sum of idepedet radom variables.] Theorem: A χ 2 (1) radom variable has mea 1 ad variace 2. The proof of the theorem is beyod the scope of this course. It requires usig a (rather messy) formula for the probability desity fuctio of a χ 2 (1) variable. Some courses i mathematical statistics iclude the proof. Exercise 2: Use the Theorem together with the defiitio of a χ 2 (k) distributio ad properties of the mea ad stadard deviatio to fid the mea ad variace of a χ 2 (k) distributio. II. t Distributios Defiitio: The t distributio with k degrees of freedom is the distributio of a Z radom variable which is of the form where U k i. Z ~ N(0,1) ii. U ~ χ 2 (k), ad iii. Z ad U are idepedet.
4 4 Commet: Notice that this defiitio says that the otio of degrees of freedom for a tdistributio comes from the otio of degrees of freedom of a chisquared distributio: The degrees of freedom of a tdistributio are the umber of squares of idepedet ormal radom variables that go ito makig up the chisquared distributio occurrig uder the radical i the deomiator of the t radom Z variable. U k To see what a tdistributio looks like, we ca use the four stadard ormal samples of 1000 obtaied above to simulate a t distributio with 3 degrees of freedom: We use colum s1 as our sample from Z ad (st2) 2 + (st3) 2 +(st4) 2 as our Z sample from U to calculate a sample from the t distributio with 3 degrees U 3 of freedom. The resultig histogram is: Note that this histogram shows a distributio similar to the tmodel with 2 degrees of freedom show o p. 554 of the textbook: Itʼs arrower i the middle tha a ormal curve would be, but has heavier tails ote i particular the outliers that would be very uusual i a ormal distributio. The followig ormal probability plot of the simulated data draws attetio to the outliers as well as the oormality. (The plot is quite typical of a ormal probability plot for a distributio with heavy tails o both sides.)
5 5 III. Why the tstatistic itroduced o p. 553 of the textbook has a t distributio: 1. Geeral setup ad otatio: Puttig together the two parts of the defiitio of tstatistic i the box o p. 553 gives t = y µ, s where y ad s are, respectively, the mea ad sample stadard deviatio calculated from the sample y 1, y 2,, y. To talk about the distributio of the tstatistic, we eed to cosider all possible radom 1 samples of size from the populatio for Y. Weʼll use the covetio of usig capital letters for radom variables ad small letters for their values for a particular sample. I this case, we have three statistics ivolved: Y, S ad T. All three have the same associated radom process: Choose a radom sample from the populatio for Y. Their values are as follows: The value of Y is the sample mea y of the sample chose. The value of S is the sample stadard deviatio s of the sample chose. The value of T is the tstatistic t = y µ calculated for the sample chose. s The distributios of Y, S ad T are called the samplig distributios of the mea, the sample stadard deviatio, ad the tstatistic, respectively.
6 6 Note that the formula for calculatig t from the data gives the formula T = Y µ S, expressig the radom variable T as a fuctio of the radom variables Y ad S. Weʼll first discuss the tstatistic i the case where our uderlyig radom variable Y is ormal, the exted to the more geeral situatio stated i Chapter The case of Y ormal. For Y ormal, we will use the followig theorem: Theorem: If Y is ormal with mea µ ad stadard deviatio, ad if we oly cosider simple radom samples with replacemet 2, of fixed size, the a) The (samplig) distributio of Y is ormal with mea µ ad stadard deviatio, b) Y ad S are idepedet radom variables, ad c) (1) S 2 2 ~ χ2 (1) The proof of this theorem is beyod the scope of this course, but may be foud i most textbooks o mathematical statistics. Note that (a) is a special case of the Cetral Limit Theorem. We will give some discussio of the plausibility of parts (b) ad (c) i the Commets sectio below. So for ow suppose Y is a ormal radom variable with mea µ ad stadard deviatio : Y~ N(µ, ). By (a) of the Theorem, the samplig distributio of the sample mea Y (for simple radom samples with replacemet, of fixed size ) is ormal with mea µ ad stadard deviatio : Y ~ N(µ, Stadardizig Y the gives ). Y µ ~ N(0,1). (*) But we doʼt kow, so we eed to approximate it by the sample stadard deviatio s. It would be temptig to say that sice s is approximately equal to,
7 7 this substitutio (i other words, cosiderig Y µ ) should give us somethig s approximately ormal. Ufortuately, there are two problems with this: First, usig a approximatio i the deomiator of a fractio ca sometimes make a big differece i what youʼre tryig to approximate (See Footote 3 for a example.) Secod, we are usig a differet value of s for differet samples (sice s is calculated from the sample, just as the value of Y is.) This is why we eed to work with the radom variable S rather tha the idividual sample stadard deviatio s. I other words, we eed to work with the radom variable T = Y µ S To use the theorem, first apply a little algebra to to see that Y µ S = (**) Sice Y is ormal, the umerator i the right side of (**) is stadard ormal, as oted i equatio (*) above. Also, by (c) of the theorem, the deomiator of the right side of (**) is of the form U ( 1) where U = (1) S 2 2 ~ χ2 (1). Sice alterig radom variables by subtractig costats or dividig by costats does ot affect idepedece, (b) of the theorem implies that the umerator ad deomiator of the right side of (**) are idepedet. Thus for Y ormal, our test statistic T = Y µ S satisfies the defiitio of a t distributio with 1 degrees of freedom. 3. More geerally: The textbook states (pp ) assumptios ad coditios that are eeded to use the tmodel: The headig Idepedece Assumptio o p. 555 icludes a Idepedece Assumptio, a Radomizatio Coditio, ad the 10% Coditio. These three essetially say that the sample is close eough to a simple radom with replacemet to make the theorem close eough to true, still assumig ormality of Y. The headig Normal Populatio Assumptio o p. 556 cosists of the Nearly Normal Coditio, which essetially says that we ca also weake ormality somewhat ad still have the theorem close eough to true for most practical purposes. (The rough idea here is that, by the
8 8 cetral limit theorem, Y will still be close eough to ormal to make the theorem close eough to true.) The appropriateess of these coditios as good rules of thumb has bee established by a combiatio of mathematical theorems ad simulatios. 4. Commets: i. To help covice yourself of the plausibility of Part (b) of the theorem, try a simulatio as follows: Take a umber of simple radom samples from a ormal distributio ad plot the resultig values of Y vs S. Here is the result from oe such simulatio: The left plot shows y vs s for 1000 draws of a sample of size 25 from a stadard ormal distributio. The right plot shows y vs s for 1000 draws of a sample of size 25 from a skewed distributio. The left plot is elliptical i ature, which is what is expected if the two variables plotted are ideed idepedet. O the other had, the right plot shows a oticeable depedece betwee Y ad S: y icreases as s icreases, ad the coditioal variace of Y (as idicated by the scatter) also icreases as S icreases. ii. To get a little isight ito (c) of the Theorem, ote first that (1) S 2 = 2, which is ideed a sum of squares, but of squares, ot 1. However, the radom variables beig squared are ot idepedet; the depedece arises from the relatioship Y= Y. Usig this relatioship, it is possible to show
9 9 that (1) is ideed the sum of 1 idepedet, stadard ormal radom variables. Although the geeral proof is somewhat ivolved, the idea is fairly easy to see whe = 2: First, a little algebra shows that (for = 2) Y  Y = ad Y  Y =. Pluggig these ito the formula for S 2 (with = 2) the gives S 2 2 (1) = 2 = (***) Sice Y 1 ad Y 2 are idepedet ad both are ormal, Y 1  Y 2 is also ormal (by a theorem from probability). Sice Y 1 ad Y 2 have the same distributio, E(Y 1  Y 2 ) = E(Y 1 )  E(Y 2 ) = 0 Usig idepedece of Y 1 ad Y 2, we ca also calculate Var(Y 1  Y 2 ) = Var(Y 1 ) + Var(Y 2 ) = 2σ Stadardizig Y 1  Y 2 the shows that is stadard ormal, so S 2 2 equatio (***) shows that (1) ~ χ 2 (1) whe = 2. Foototes 1. Radom is admittedly a little vague here. I sectio 2, iterpret it to mea simple radom sample with replacemet. (See also Footote 2). I sectio 3, iterpret radom to mea Fittig the coditios ad assumptios for the t model. 2. Techically, the requiremets are that the radom variables Y 1, Y 2,, Y represetig the first, secod, etc. values i the sample are idepedet ad idetically distributed (abbreviated as iid), which meas they are idepedet ad have the same distributio (i.e., the same probability desity fuctio). 3. Cosider, for example, usig as a approximatio of 0.01 whe estimatig 1/0.01. Although differs from 0.01 by oly 0.001, whe we use the approximatio i the deomiator, we get 1/0.011 = , which differs by more tha 9 from 1/0.01 = 100 a differece almost 3 orders of magitude greater tha the differece betwee 0.01 ad
Case 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
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 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 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  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 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 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 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 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 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 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 informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
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 informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
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 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 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 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 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 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 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 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 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 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 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 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 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 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 information3. Continuous Random Variables
Statistics ad probability: 31 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay
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 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 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 informationUSING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR
USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..
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 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 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 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 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 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 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 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 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 information428 CHAPTER 12 MULTIPLE LINEAR REGRESSION
48 CHAPTER 1 MULTIPLE LINEAR REGRESSION Table 18 Team Wis Pts GF GA PPG PPcT SHG PPGA PKPcT SHGA Chicago 47 104 338 68 86 7. 4 71 76.6 6 Miesota 40 96 31 90 91 6.4 17 67 80.7 0 Toroto 8 68 3 330 79.3
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationDerivation of the Poisson distribution
Gle Cowa RHUL Physics 1 December, 29 Derivatio of the Poisso distributio I this ote we derive the fuctioal form of the Poisso distributio ad ivestigate some of its properties. Cosider a time t i which
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 informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
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 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 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 informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
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 informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationRobust and Resistant Regression
Chapter 13 Robust ad Resistat Regressio Whe the errors are ormal, least squares regressio is clearly best but whe the errors are oormal, other methods may be cosidered. A particular cocer is logtailed
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
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 informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
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 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 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 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 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 informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationsum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by
Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are
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 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 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 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 informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
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 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 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 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 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 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 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 information