Chisquared goodnessoffit test.


 Rebecca Kelley
 1 years ago
 Views:
Transcription
1 Sectio 1 Chisquaed goodessoffit test. Example. Let us stat with a Matlab example. Let us geeate a vecto X of 1 i.i.d. uifom adom vaiables o [, 1] : X=ad(1,1). Paametes (1, 1) hee mea that we geeate a 1 1 matix o uifom adom vaiables. Let us test if the vecto X comes fom distibutio U[, 1] usig 2 goodessoffit test: [H,P,STATS]=chi2gof(X, edges,:.2:1) The output is H =, P =.953, STATS = chi2stat: 7.9 df: 4 edges: [ ] O: [ ] E: [ ] We accept ull hypothesis H : P = U[, 1] at the default level of sigificace =.5 sice the pvalue.953 is geate tha. The meaig of othe paametes will become clea whe we explai how this test woks. Paamete cdf takes the to a fully specified c.d.f. Fo example, to test if the data comes fom N(3, 5) we would o to test Poisso distibutio (4) we would It is impotat to ote that whe we use chisquaed test to test, fo example, the ull hypothesis H : P = N(1, 2), the alteative hypothesis is H : P = N(1, 2). This is diffeet fom the settig of ttests whee we would assume that the data comes fom omal distibutio ad test H : µ = 1 vs. H : µ = 1. 62
2 Peaso s theoem. PSfag eplacemets Chisquaed goodessoffit test is based o a pobabilistic esult that we will pove i this sectio. 1 2 B 1 B 2... B p 1 p 2 p Figue 1.1: Let us coside boxes B 1,..., B ad thow balls X 1,..., X ito these boxes idepedetly of each othe with pobabilities so that Let j be a umbe of balls i the jth box: P(X i B 1 ) = p 1,..., P(X i B ) = p, p p = 1. j = #{balls X 1,..., X i the box B j } = I(X l B j ). O aveage, the umbe of balls i the jth box will be p j sice l=1 E j = EI(X l B j ) = P(X l B j ) = p j. l=1 l=1 We ca expect that a adom vaiable j should be close to p j. Fo example, we ca use a Cetal Limit Theoem to descibe pecisely how close j is to p j. The ext esult tells us how we ca descibe the closeess of j to p j simultaeously fo all boxes j. The mai difficulty i this Thoem comes fom the fact that adom vaiables j fo j ae ot idepedet because the total umbe of balls is fixed =. If we kow the couts i 1 boxes we automatically kow the cout i the last box. Theoem.(Peaso) We have that the adom vaiable ( j p j ) 2 d 2 p 1 j j=1 coveges i distibutio to 2 1distibutio with ( 1) degees of feedom. 63
3 Poof. Let us fix a box B j. The adom vaiables I(X 1 B j ),..., I(X B j ) that idicate whethe each obsevatio X i is i the box B j o ot ae i.i.d. with Beoulli distibutio B(p j ) with pobability of success ad vaiace EI(X 1 B j ) = P(X 1 B j ) = p j Va(I(X 1 B j )) = p j (1 p j ). Theefoe, by Cetal Limit Theoem the adom vaiable j p j l=1 I(X l B j ) p j = p j (1 p j ) p j (1 p j ) l=1 = I(X l B j ) E d N(, 1) Va coveges i distibutio to N(, 1). Theefoe, the adom vaiable j p j 1 p j N(, 1) = N(, 1 p j ) pj d coveges to omal distibutio with vaiace 1 p j. Let us be a little ifomal ad simply say that j p j Z j pj whee adom vaiable Z j N(, 1 p j ). We kow that each Z j has distibutio N(, 1 p j ) but, ufotuately, this does ot tell us what the distibutio of the sum 2 Z j will be, because as we metioed above.v.s j ae ot idepedet ad thei coelatio stuctue will play a impotat ole. To compute the covaiace betwee Z i ad Z j let us fist compute the covaiace betwee which is equal to i p i j p j ad pi pj i p j j p j 1 E pi pj = pi p j (E i j E i p j E j p i + 2 p i p j ) = pi p j (E i j p i p j p j p i + p i p j ) = pi p j (E i j p i p j ). To compute E i j we will use the fact that oe ball caot be iside two diffeet boxes simultaeously which meas that I(X l B i )I(X l B j ) =. (1..1) 64
4 Theefoe, E i j = E I(X l B i ) I(X l B j ) = E I(X l B i )I(X l B j ) l=1 l =1 l,l = E I(X l B i )I(X l B j ) +E I(X l B i )I(X l B j ) l=l this equals to by (1..1) = ( 1)EI(X l B j )EI(X l B j ) = ( 1)p i p j. Theefoe, the covaiace above is equal to 1 2 ( 1)p i p j p i p j = p i p j. p i p j To summaize, we showed that the adom vaiable l=l (j p j ) 2 Z 2 p j j. j=1 j=1 whee omal adom vaiables Z 1,..., Z satisfy EZ 2 = 1 p i ad covaiace EZ i Z j = p i p j. i To pove the Theoem it emais to show that this covaiace stuctue of the sequece of (Z i ) implies that thei sum of squaes has 2 1distibutio. To show this we will fid a diffeet epesetatio fo 2 Z i. Let g 1,..., g be i.i.d. stadad omal adom vaiables. Coside two vectos g = (g 1,..., g ) T ad p = ( p 1,..., p ) T ad coside a vecto g (g p)p, whee g p = g 1 p g p is a scala poduct of g ad p. We will fist pove that g (g p)p has the same joit distibutio as (Z 1,..., Z ). (1..2) To show this let us coside two coodiates of the vecto g (g p)p : ad compute thei covaiace: E i th : g i g l pl pi ad j th : g j g l pl pj l=1 l=1 g i g l pl pi g j g l pl pj l=1 l=1 = p i pj p j pi + p l pi pj = 2 p i p j + p i p j = p i p j. l=1 65
5 Similaly, it is easy to compute that This poves (1..2), which povides us with aothe way to fomulate the covegece, amely, we have But this vecto has a simple geometic itepetatio. Sice vecto p is a uit vecto: vecto Vl = (p. g)p is the pojectio of vecto g o the lie alog p ad, theefoe, vecto Vz = g  (p. g)p will be the pojectio of g oto the plae othogoal to p, as show i figue 1.2. Figue 1.2: New coodiate system, Let us coside a ew othoomal coodiate system with the fist basis vecto (fist axis) equal top. I this ew coodiate system vecto g will have coodiates
6 obtaied fom g by othogoal tasfomatio V = (p, p 2,..., p ) that maps caoical basis ito this ew basis. But we poved i Lecue 4 that i that case g 1,..., g will also be i.i.d. stadad omal. Fom figue 1.2 it is obvious that vecto V 2 = g (p g)p i the ew coodiate system has coodiates ad, theefoe, (, g 2,..., g ) T V 2 2 = g (p g)p 2 = (g ) (g ) 2. 2 But this last sum, by defiitio, has 2 1 distibutio sice g 2,, g ae i.i.d. stadad omal. This fiishes the poof of Theoem. Chisquaed goodessoffit test fo simple hypothesis. Suppose that we obseve a i.i.d. sample X 1,..., X of adom vaiables that take a fiite umbe of values B 1,..., B with ukow pobabilities p 1,..., p. Coside hypotheses H : p i = p i fo all i = 1,...,, H 1 : fo some i, p i = p i. If the ull hypothesis H is tue the by Peaso s theoem T = (i p ) 2 i p i=1 i d 2 1 whee i = #{X j : X j = B i } ae the obseved couts i each categoy. O the othe had, if H 1 holds the fo some idex i, p i = p i ad the statistics T will behave diffeetly. If p i is the tue pobability P(X 1 = B i ) the by CLT i pi d N(, 1 p i ). p i If we ewite i p i = i p i + (p i p i ) pi i p i = + p i p i pi pi p i pi pi the the fist tem coveges to N(, (1 p i )p i /p i ) ad the secod tem diveges to plus o mius because p i = p i. Theefoe, ( i p ) 2 i p i + which, obviously, implies that T +. Theefoe, as sample size iceases the distibutio of T ude ull hypothesis H will appoach 2 1distibutio ad ude alteative hypothesis H 1 it will shift to +, as show i figue
7 H : T H 1 : T + PSfag eplacemets c Figue 1.3: Behavio of T ude H ad H 1. Theefoe, we defie the decisio ule H α = 1 : T c H 2 : T > c. We choose the theshold c fom the coditio that the eo of type 1 is equal to the level of sigificace : = P 1 (α = H 1 ) = P 1 (T > c) 2 1 (c, ) sice ude the ull hypothesis the distibutio of T is appoximated by 2 1 distibutio. Theefoe, we take c such that = 2 1 (c, ). This test α is called the chisquaed goodessoffit test. Example. (Motaa outlook poll.) I a 1992 poll 189 Motaa esidets wee asked (amog othe thigs) whethe thei pesoal fiacial status was wose, the same o bette tha a yea ago. Wose Same Bette Total We wat to test the hypothesis H that the udelyig distibutio is uifom, i.e. p 1 = p 2 = p 3 = 1/3. Let us take level of sigificace =.5. The the theshold c i the chisquaed 68
8 test α = H : T c H 1 : T > c is foud fom the coditio that 2 3 1=2(c, ) =.5 which gives c = 5.9. We compute chisquaed statistic (58 189/3) 2 (64 189/3) 2 (67 189/3) 2 T = + + =.666 < /3 189/3 189/3 which meas that we accept H at the level of sigificace.5. Goodessoffit fo cotiuous distibutio. Let X 1,..., X be a i.i.d. sample fom ukow distibutio P ad coside the followig hypotheses: H : P = P H 1 : P = P fo some paticula, possibly cotiuous distibutio P. To apply the chisquaed test above we will goup the values of Xs ito a fiite umbe of subsets. To do this, we will split a set of all possible outcomes X ito a fiite umbe of itevals I 1,..., I as show i figue p.d.f. of P PSfag eplacemets.15.1 p 2.5 p 1 p I 1 I 2 I x Figue 1.4: Discetizig cotiuous distibutio. 69
9 The ull hypothesis H, of couse, implies that fo all itevals Theefoe, we ca do chisquaed test fo P(X I j ) = P (X I j ) = p j. H : P(X I j ) = p j fo all j H 1 : othewise. Askig whethe H holds is, of couse, a weake questio that askig if H holds, because H implies H but ot the othe way aoud. Thee ae may distibutios diffeet fom P that have the same pobabilities of the itevals I 1,..., I as P. O the othe had, if we goup ito moe ad moe itevals, ou discete appoximatio of P will get close ad close to P, so i some sese H will get close to H. Howeve, we ca ot split ito too may itevals eithe, because the 2 1distibutio appoximatio fo statistic T i Peaso s theoem is asymptotic. The ule of thumb is to goup the data i such a way that the expected cout i each iteval p i = P (X I i ) 5 is at least 5. (Matlab, fo example, will give a waig if this expected umbe will be less tha five i ay iteval.) Oe appoach could be to split ito itevals of equal pobabilities = 1/ ad choose thei umbe so that p i p i = 5. Example. Let us go back to the example fom Lectue 2. Let us geeate 1 obsevatios fom Beta distibutio B(5, 2). X=betad(5,2,1,1); Let us fit omal distibutio N(µ, ν 2 ) to this data. The MLE ˆµ ad ˆν ae mea(x) =.7421, std(x,1)= Note that std(x) i Matlab will poduce the squae oot of ubiased estimato (/ 1)ˆν 2. Let us test the hypothesis that the sample has this fitted omal distibutio. [H,P,STATS]= chi2gof(x, outputs H = 1, P =.41, STATS = chi2stat: df: 7 edges: [1x9 double] O: [ ] E: [1x8 double] Ou hypothesis was ejected with pvalue of.41. Matlab split the eal lie ito 8 itevals of equal pobabilities. Notice df: 7  the degees of feedom 1 = 8 1 = 7. 7
STATISTICS: MODULE 12122. Chapter 3  Bivariate or joint probability distributions
STATISTICS: MODULE Chapte  Bivaiate o joit pobabilit distibutios I this chapte we coside the distibutio of two adom vaiables whee both adom vaiables ae discete (cosideed fist) ad pobabl moe impotatl whee
More informationLinear recurrence relations with constant coefficients
Liea ecuece elatios with costat coefficiets Recall that a liea ecuece elatio with costat coefficiets c 1, c 2,, c k c k of degee k ad with cotol tem F has the fom a c 1 a 1 + c 2 a 2 + + c k a k + F k
More information2. The Exponential Distribution
Vitual Laboatoies > 13. The Poisso Pocess > 1 2 3 4 5 6 7 2. The Expoetial Distibutio Basic Theoy The Memoyless Popety The stog eewal assumptio meas that the Poisso pocess must pobabilistically estat at
More informationTommy R. Jensen, Department of Mathematics, KNU. 1 Combinations of Sets 1. 2 Formulas for Combinations 1. 3 The Binomial Theorem 4
Pat 4 Combiatios Combiatios, Subsets ad Multisets Pited vesio of the lectue Discete Mathematics o 14. Septembe 010 Tommy R. Jese, Depatmet of Mathematics, KNU 4.1 Cotets 1 Combiatios of Sets 1 Fomulas
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 informationThe Binomial Theorem 5! ! ! 3 2 1! 1. Factorials
The Biomial Theoem Factoials The calculatios,, 6 etc. ofte appea i mathematics. They ae called factoials ad have bee give the otatio!. e.g. 6! 6!!!!! We also defie 0! Combiatoics Pemutatios ad Combiatios
More informationMeaning Formula Link to Glossary (if appropriate) a Y intercept of least a = y bx. Regression: y on x squares regression.
Alphabetial Statistial Symbols: Symbol Text Meaig Fomula Lik to Glossay a Y iteept of least a = y bx, fo lie y = a + bx Regessio: y o x squae egessio lie b Slope of least ( x x)( y y) Regessio: y o x
More informationFIRST YEAR CALCULUS W W L CHEN
FIRST YEAR CALCULUS W W L CHEN c W W L Che, 994, 2008. This chapte is available fee to all idividuals, o the udestadig that it is ot to be used fo fiacial gai, ad may be dowloaded ad/o photocopied, with
More informationBINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand
BINOMIAL THEOREM Itoductio Whe we epad ( + ) ad ( + ), we get ad ( + ) = ( + )( + ) = + + + = + + ( + ) = ( + )( + ) = ( + )( + + ) = + + + + + = + + + 4 5 espectively Howeve, whe we ty to epad ( + ) ad
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 informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem If
More information( ) ( ) ( ) ( n) ( ) ( ) ( ) ( ), ( ) ( )
Chapte 8: Pimitive Roots ad Idices 6 SECTION C Theoy of Idices By the ed of this sectio you will be able to what is meat by pimitive oot detemie the C Pimitive Roots Fist we defie what is meat by a pimitive
More informationPeriodic Review Probabilistic MultiItem Inventory System with Zero Lead Time under Constraints and Varying Order Cost
Ameica Joual of Applied Scieces (8: 37, 005 ISS 546939 005 Sciece Publicatios Peiodic Review Pobabilistic MultiItem Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue
More informationAlgebra 2 AII.2 Geometric Sequences and Series Notes. Name: Date: Block:
Algeba 2 AII.2 Geometic Sequeces ad Seies Notes Ms. Giese Name: Date: Block: Geometic Sequeces Geometic sequeces cotai a patte whee a fixed amout is multiplied fom oe tem to the ext (commo atio ) afte
More informationGEOMETRIC MEAN FOR NEGATIVE AND ZERO VALUES
IJRRAS 11 (3) Jue 01 www.apapess.com/volumes/vol11issue3/ijrras_11_3_08.pdf GEOMETRIC MEA FOR EGATIVE AD ZERO VALUES Elsayed A. E. Habib Depatmet of Mathematics ad Statistics, Faculty of Commece, Beha
More informationChapter 2 Sequences and Series
Chapte 7 Sequece ad seies Chapte Sequeces ad Seies. Itoductio: The INVENTOR of chess asked the Kig of the Kigdom that he may be ewaded i lieu of his INVENTION with oe gai of wheat fo the fist squae of
More informationGGMD Circumference of a circle
Illustative GGMD Cicumfeece of a cicle Task π Suppose we defie to be the cicumfeece of a cicle whose diamete is : > 0 π Explai why the cicumfeece of a cicle with adius is. IM Commetay π The cicumfeece
More informationTRANSMISSIBILITY MATRIX IN HARMONIC AND RANDOM PROCESSES
RANMIIBILIY MARIX IN ARMONIC AND RANDOM PROCEE Ribeio, A.M.R., Fotul, M., ilva,.m.m., Maia, N.M.M. Istituto upeio écico, Av. Rovisco Pais, 04900 Lisboa, Potugal UMMARY: he tasmissibility cocept may be
More informationLecture 6: Money Market and the LM Curve
EC01 Itemediate Macoecoomics EC01 Itemediate Macoecoomics Lectue 6: Moe Maket ad the LM Cuve Lectue Outlie:  the LM cuve, ad its elatio to the liquidit pefeece theo Essetial eadig: Makiw: Ch. 11. The
More informationExponents, Radicals and Logarithms
Mathematics of Fiace Expoets, Radicals ad Logaithms Defiitio 1. x = x x x fo a positive itege. Defiitio 2. x = 1 x Defiitio 3. x is the umbe whose th powe is x. Defiitio 4. x 1/ = x Defiitio 5. x m/ =
More informationBINOMIAL THEOREM. Mathematics is a most exact science and its conclusions are capable of absolute proofs. C.P. STEINMETZ
Chapte 8 BINOMIAL THEOREM Maematics is a most eact sciece ad its coclusios ae capable of absolute poofs. C.P. STEINMETZ 8. Itoductio I ealie classes, we have leat how to fid e squaes ad cubes of biomials
More informationMATH Discrete Structures
MATH 1130 1 Discete Stuctues hapte III ombiatoics Pemutatios ad ombiatios The Multiplicatio Piciple Suppose choices must be made, with m 1 ways to make choice 1, ad fo each of these ways, m 2 ways to 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 informationThe statement of the problem of factoring integer is as follows: Given an integer N, find prime numbers p i and integers e i such that.
CS 2942 Sho s Factoing Algoithm 0/5/04 Fall 2004 Lectue 9 Intoduction Now that we have talked about uantum Fouie Tansfoms and discussed some of thei popeties, let us see an application aea fo these ideas.
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 informationBINOMIAL THEOREM 1
www.sakshieducatio.com BINOMIAL THEOREM  Biomial : A epessio which cotais two tems is called a biomial Pascal Tiagle: Ide coefficiet 4 4 6 4 Theoem : I each ow st ad last elemets emaiig tems ae obtaied
More informationUnderstanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions
Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isketu tadeoff ad time value of
More informationExample: Consider the sequence {a i = i} i=1. Starting with i = 1, since a i = i,
Sequeces: Defiitio: A sequece is a fuctio whose domai is the set of atural umbers or a subset of the atural umbers. We usually use the symbol a to represet a sequece, where is a atural umber ad a is the
More informationVersion: 0.1 This is an early version. A better version would be hopefully posted in the near future.
Chapte 28 Shao s theoem Vesio: 0.1 This is a ealy vesio. A bette vesio would be hopefully posted i the ea futue. By Saiel HaPeled, Decembe 7, 2009 1 This has bee a ovel about some people who wee puished
More informationGeometric Sequences. Definition: A geometric sequence is a sequence of the form
Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece
More informationTwo degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system
wo degee of feedom systems Equatios of motio fo foced vibatio Fee vibatio aalysis of a udamped system Itoductio Systems that equie two idepedet d coodiates to descibe thei motio ae called two degee of
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More information2. The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is
Chapte 7 Seqeces ad seies No calclato:. The secod tem of a aithmetic seqece is 7. The sm of the fist fo tems of the aithmetic seqece is. Fid the fist tem, a, ad the commo diffeece, d, of the seqece. (Total
More informationBohr Model of the Atom
Sectio 4: OF THE ATOM I this sectio, we descibe the stuctue ad behaviou of the simplest type of atom cosistig of a ucleus obited by a sigle electo oly. It addesses the basic questio: How do electos emai
More informationProfessor Terje Haukaas University of British Columbia, Vancouver Load Combination
Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve www.inisk.ubc.ca Load Combination This document descibes the load coincidence method (Wen 1990) and seveal load combination ules that ae utilized
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 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 informationTRIGONOMETRIC EXPRESSIONS FOR FIBONACCI AND LUCAS NUMBERS. Introduction
Acta Math Uiv Comeiaae Vol LXXIX, 22010, pp 199 208 199 TRIGONOMETRIC EXPRESSIONS FOR FIBONACCI AND LUCAS NUMBERS B SURY Itoductio The amout of liteatue beas witess to the ubiquity of the Fiboacci umbes
More informationFIITJEE AIEEE 2004 (MATHEMATICS)
FIITJEE AIEEE 4 (MATHEMATICS) Impotat Istuctios: i) The test is of hous duatio. ii) The test cosists of 75 questios. iii) The maimum maks ae 5. iv) Fo each coect aswe you will get maks ad fo a wog aswe
More informationLecture 24: Tensor Product States
Lectue 4: Teso oduct States hy85 Fall 009 Basis sets fo a paticle i 3D Clealy the Hilbet space of a paticle i thee diesios is ot the sae as the Hilbet space fo a paticle i oediesio I oe diesio, X ad ae
More informationSection 3.3: Geometric Sequences and Series
ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some
More informationUnit 25 Hypothesis Tests about Proportions
Uit 25 Hypothesis Tests about Proportios Objectives: To perform a hypothesis test comparig two populatio proportios Now that we have discussed hypothesis tests to compare meas we wat to discuss a hypothesis
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 informationABSTRACT KEYWORDS. Proportional hazard premium principle, subexponential distributions, bootstrap, 1. INTRODUCTION
APPLYING THE PROPORTIONAL HAZARD PREMIUM CALCULATION PRINCIPLE BY MARIA DE LOURDES CENTENO AND JOAO ANDRADE E SILVA ABSTRACT I this pape we discuss the applicatio of the popotioal hazad pemium calculatio
More informationBinomial Theorem MODULE  I Algebra BINOMIAL THEOREM
Biomial Theoem 8 BINOMIAL THEOREM Suppose ou eed to calculate the amout of iteest ou will get afte eas o a sum of moe that ou have ivested at the ate of % compoud iteest pe ea. O suppose we eed to fid
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 informationHow to Think Like a Mathematician Solutions to Exercises
How to Think Like a Mathematician Solutions to Execises Septembe 17, 2009 The following ae solutions to execises in my book How to Think Like a Mathematician. Chapte 1 Execises 1.10 (i) 5 (ii) 3 (iii)
More informationMATH Testing a Series for Convergence
MATH 0  Testig a Series for Covergece Dr. Philippe B. Laval Keesaw State Uiversity October 4, 008 Abstract This hadout is a summary of the most commoly used tests which are used to determie if a series
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 informationiff s = r and y i = x i for i = 1,, r.
Combiatoics Coutig A Oveview Itoductoy Example What to Cout Lists Pemutatios Combiatios. The Basic Piciple Coutig Fomulas The Biomial Theoem. Example As I was goig to St. Ives I met a ma with seve wives
More informationSTRAND: ALGEBRA Unit 5 Sequences and Series
CMM Subject Suppot Stad: ALGEBRA Uit 5 Sequeces ad Seies: Text STRAND: ALGEBRA Uit 5 Sequeces ad Seies TEXT Cotets Sectio 5. Geometical Sequeces 5. NeveEdig Sums 5. Aithmetic Seies 5.4 Sigma Notatio 5.5
More informationSection 8.2 Markov and Chebyshev Inequalities and the Weak Law of Large Numbers
Sectio 8. Markov ad Chebyshev Iequalities ad the Weak Law of Large Numbers THEOREM (Markov s Iequality): Suppose that X is a radom variable takig oly oegative values. The, for ay a > 0 we have X a} E[X]
More informationLearning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)
Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 21 22 Leaig
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 information1.7 Conditional Probabiliity
page 3.7 Conditional Pobabiliity.7. Definition Let the event B F be such that P(B) > 0. Fo any event A F the conditional pobability of A given that B has occued, denoted by P(A B), is defined as P(A B)
More informationUniform convergence and power series
Uiform covergece ad power series Review of what we already kow about power series A power series cetered at a R is a expressio of the form c (x a) The umbers c are called the coefficiets of the power series
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 informationValuation formulas. Future Value Formulas. The value one period from now of an amount invested at rate r:
Valuatio fomulas Give peset value of oe o moe futue cash flows umbe of peiods face o omial valueof cash ' ' peiods fom ow A pe  peiod ateof etu peset value of a auity whose duatio is ' ' peiods Futue
More information4 Singular Value Decomposition (SVD)
4 Sigula Value Decompositio (SVD) The sigula value decompositio of a matix A is the factoizatio of A ito the poduct of thee matices A UDV T whee the colums of U ad V ae othoomal ad the matix D is diagoal
More informationLecture 14 November 10
STATS 300A: Theory of Statistics Fall 015 Lecture 14 November 10 Lecturer: Lester Mackey Scribe: Ju Ya, Matteo Sesia Warig: These otes may cotai factual ad/or typographic errors. 14.1 Overview 14.1.1 Hypothesis
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 informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More information5.3 The Integral Test and Estimates of Sums
5.3 The Itegral Test ad Estimates of Sums Bria E. Veitch 5.3 The Itegral Test ad Estimates of Sums The ext few sectios we lear techiques that help determie if a series coverges. I the last sectio we were
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 informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationAnnuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments
8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise
More informationBernstein Polynomials
7 Bestei Polyomials 7.1 Itoductio This chapte is coceed with sequeces of polyomials amed afte thei ceato S. N. Bestei. Give a fuctio f o [0, 1, we defie the Bestei polyomial B (f; x = ( f =0 ( x (1 x (7.1
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 informationECE 340 Lecture 13 : Optical Absorption and Luminescence 2/19/14 ( ) Class Outline: Band Bending Optical Absorption
/9/4 ECE 34 Lectue 3 : Optical Absoptio ad Lumiescece Class Outlie: Thigs you should kow whe you leave Key Questios How do I calculate kietic ad potetial eegy fom the bads? What is diect ecombiatio? How
More informationINFERENCE ABOUT A POPULATION PROPORTION
CHAPTER 19 INFERENCE ABOUT A POPULATION PROPORTION OVERVIEW I this chapter, we cosider iferece about a populatio proportio p based o the sample proportio cout of successes i the sample p ˆ = cout of observatios
More informationSEQUENCE AND SERIES. Syllabus :
Eistei Clsses Uit No. 0 0 Vdhm Rig Rod Plz Viks Pui Ext. Oute Rig Rod New Delhi 0 08 Ph. : 96905 857 MSS SEQUENCE AND SERIES Syllbus : Aithmetic d Geometic pogessio isetio of ithmetic geometic mes betwee
More informationSo we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)
Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing
More informationA.1 Statistical Inference and Estimation of Population Parameters
Statistics from Simulatios A.1 Statistical Iferece ad Estimatio of Populatio Parameters Whe the goal is to estimate the parameter of some populatio (e.g., mea or variace), the usual procedure is to desig
More information50 MATHCOUNTS LECTURES (24) COMBINATOTICS. A permutation is an arrangement or a listing of things in which order is important.
50 MATHCOUNTS LECTURES (4) COMBINATOTICS BASIC KNOWLEDGE. Tems A pemutatio is a aagemet o a listig of thigs i which ode is impotat. A combiatio is a aagemet o a listig of thigs i which ode is ot impotat.
More informationGet Solution of These Packages & Learn by Video Tutorials on STUDY PACKAGE
Get Solutio of These Packages & Lea by Video Tutoials o wwwmathsbysuhagcom FREE Dowload Study Package fom website: wwwtekoclassescom & wwwmathsbysuhagcom fo/u fopkj Hkh# tu] ugha vkjehks dke] foif s[k
More informationChapter 7. Inference for Population Proportions
Lecture otes, Lag Wu, UBC 1 Chapter 7. Iferece for Populatio Proportios 7.1. Itroductio I the previous chapter, we have discussed the basic ideas of statistical iferece. To illustrate the basic ideas,
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationIV. BASIC STATISTICS
IV. BASI STATISTIS A STATE OF STATISTIAL ONTROL IS NOT A NATURAL STATE FOR A MANUFATURING PROESS. IT IS INSTEAD AN AHIEVEMENT, ARRIVED AT BY ELIMINATING ONE BY ONE, BY DETERMINED EFFORT, THE SPEIAL AUSES
More informationHomework #4  Answers. The ISLM Model Due Mar 18
Winte Tem 2004 Alan Deadoff Homewok #4  Answes Page 1 of 12 Homewok #4  Answes The  Model Due Ma 18 1. Fun with the Keynesian Coss: a. Use the geomety of the Keynesian Coss diagam shown at the ight
More informationDerivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)
Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of
More informationACCELERATED LIFE TESTING WITH AN UNDERLYING THREEPARAMETER WEIBULL MODEL
ACCELERATED LIFE TESTING WITH AN UNDERLYING THREEPARAMETER WEIBULL MODEL Daiel I. De Souza J. Kamalesh Somai 2. Abstact: The mai objective of life testig is to obtai ifomatio coceig failue. This ifomatio
More informationHomework for 2/24 Due 3/12
Name: ID: Homework for /4 Due 3/1 1. Let X 1,..., X be a radom sample from a ormal distributio Nµ, σ, where µ is the ukow parameter ad σ is assumed to be kow. The hypotheses are H : µ = µ v.s. H A : µ
More information5 The Binomial and Poisson Distributions
5 The Biomial ad Poisso Distributios 5.1 The Biomial distributio Cosider the followig circumstaces (biomial sceario): 1. There are trials. 2. The trials are idepedet. 3. O each trial, oly two thigs ca
More informationImplementing Transformations. Why Transforms? Primitive Transformations: Translation. Affine Transform: Definition M = Q x Q y Q z
Why Tasfoms? Wat to aimate objects ad camea Taslatios Rotatios Sheas Ad moe.. Wat to be able to use pojectio tasfoms Implemetig Tasfomatios We use affie (liea) tasfoms Why? Ca be epeseted usig matices
More informationthe proportion of voters who intend on voting for the incumbent prime minister in the next election.
1 Iferece for Populatio Proportios So far we were iterested i estimatig ad aswerig questios for populatio meas. I these cases our parameters of iterest were either a populatio mea µ or a differece betwee
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 informationMATH 140A  HW 3 SOLUTIONS
MATH 40A  HW 3 SOLUTIONS Problem (WR Ch 2 #2). Let K R cosist of 0 ad the umbers / for = 23... Prove that K is compact directly from the defiitio (without usig the HeieBorel theorem). Solutio. Let {G
More information8 The Poisson Distribution
8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each
More informationLecture 8 : Hydraulic Design of Sewers and Storm Water Drains (Contd.)
1 P age Module 7 : Hydaulic Desig of Sewes ad Stom Wate Dais Lectue 8 : Hydaulic Desig of Sewes ad Stom Wate Dais (Cotd.) 2 P age 7.9 Hydaulic Chaacteistics of Cicula Sewe Ruig Full o Patially Full D α
More informationKeywords: Process capability indices, fuzzy set theory, mean, variance, specification limits
Iteatioal Joual of Ats ad Scieces 3(9): 0817 (010) DROM. ISSN: 19446934 IteatioalJoual.og Fuzzy Estimatios of the Idices ad İhsa Kaya, Istabul Techical Uivesity, Tukey egiz Kahama, Istabul Techical
More informationEstimating Surface Normals in Noisy Point Cloud Data
Estiatig Suface Noals i Noisy Poit Cloud Data Niloy J. Mita Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 iloy@stafod.edu A Nguye Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 aguye@cs.stafod.edu
More informationMaking Pi. Then we automatically get the formula C = 2πr, which enables us to evaluate C whenever we know the value of r.
Maths 1 Extension Notes #3.b Not Examinable Making Pi 1 Intoduction 1.1 Definition of Pi Conside any cicle with adius and cicumfeence C. In the following sections, we show that the atio C 2 is just a numbe
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 informationOn Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.
C.Candan EE3/53METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof
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 informationStatistics for Finance
Statistics for Fiace. Lecture 3:Estimatio ad Likelihood. Oe of the cetral themes i mathematical statistics is the theme of parameter estimatio. This relates to the fittig of probability laws to data. May
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationCounting Poker Hands
Counting Poke Hands Geoge Ballinge In a standad deck of cads thee ae kinds of cads: ce (),,,,,,,,,, ack (), ueen () and ing (). Each of these kinds comes in fou suits: Spade (), Heat (), Diamond () and
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More information