Matematici speciale Seminar 12
|
|
- Avis Miller
- 6 years ago
- Views:
Transcription
1 Matematici speciale Semiar 1 Mai 017
2 ii
3 Statistica este arta de a miti pri itermediul cifrelor. Wilhelm Stekel 1 Notiui de statistica Datele di dreapta arata temperaturile de racire ale uei cesti de cafea, care tocmai a fost preparata. Temperatura la care ajuge aparatul de cafea este 180 de grade Fahreheit (aproximativ 8 C). I aul 199 o femeie a dat i judecata McDoald s petru ca au servit cafeaua la temperatura 180 F si aceasta i-a cauzata arsuri serioase i mometul i care a icercat sa o bea (vezi Liebeck vs. McDoald s ). U expert adus di partea acuzarii a sustiut la proces ca lichidele care se afla la aceasta temperatura pot cauza distrugerea totala a pielii umae i doua paa la sapte secude. S-a stabilit si ca daca ar fi fost servita la 155 F (68 C) s-ar fi racit la timp si ar fi fost evitat tot icidetul. Femeia a primit i prima istata o 1
4 despagubire de.7 milioae de dolari. Ca urmare a acestui caz faimos multe restaurate servesc acum cafeaua la o temperatura de aproximativ 155 F. Cat de mult ar trebui sa astepte restauratele di mometul i care cafeaua este turata i ceasca di aparat si paa cad ea poate fi servita, petru a se asigura ca u este mai fierbite de 155 F? Determiati ecuatia uui model de regresie expoetiala petru a reprezeta datele Reprezetati grafic curba obtiuta Decideti daca ecuatia obtiuta este bua petru a reprezeta datele existete i tabel Iterpolare: Cad ajuge temperatura cafelei la 106 F? Extrapolare: Care este temperatura prezisa, de modelul gasit, peste o ora?
5 Notiui teoretice: Statistica descriptiva: populatie statistica, esatio statistic, serie statistica, frecveta abosluta, frecveta relativa, histograma, media x, mediaa m 3, amplitudiea A, dispersia σ, deviatia stadard σ, moda (modulul) m o, dispersia de selectie s, deviatia stadard de selectie s, cuartilele Q 1, Q, Q 3, idicatorul de asimetrie sk (skewess), idicatorul de aplatizare k (kurtosis) Itervale de icredere cofidece itervals are used whe we wat to estimate a populatio parameter from a sample. The parameter may be estimated by a sigle value (a poit estimate) but it is usually preferable to estimate it by a iterval which will give some idicatio of the amout of ucertaity attached to the estimate. the commo otatio for the parameter i questio is θ. Ofte, this parameter is the populatio mea μ, which is estimated through the sample mea x. the level C of a cofidece iterval gives the probability that the iterval produced by the method employed icludes the true value of the parameter. The selectio of a cofidece level for a iterval determies the probability that the cofidece iterval produced will cotai the true parameter value. Commo choices for the cofidece level C are 0.90, 0.95, ad These levels correspod to percetages of the area of the ormal desity curve. For example, a 95% cofidece iterval covers 95% of the ormal curve. The probability of observig a value outside of this area is less tha Because the ormal curve is symmetric, half of the area is i the left tail of the curve, ad the other half of the area is i the right tail of the curve. As show i the diagram, for a cofidece iterval with level C, the area i each tail of the curve is equal to (1 C)/. For a 95% cofidece iterval, the area i each tail is equal to 0.05/ = The value z * represetig the poit o the stadard ormal desity curve such that the probability of observig a value greater tha z * is equal to p 3
6 is kow as the upper p critical value of the stadard ormal distributio. For example, if p = 0.05, the value z * such that P (Z > z * ) = 0.05, or P (Z < z*) = 0.975, is equal to For a cofidece iterval with level C, the value p is equal to (1 C)/. A 95% cofidece iterval for the stadard ormal distributio is the the iterval ( 1.96, 1.96), sice 95% of the area uder the curve falls withi this iterval. Medie ecuoscuta si deviatie stadard cuoscuta Teorema: Petru o populatie cu media μ ecuoscuta si deviatie stadard σ cuoscuta, u iterval de icredere petru media populatiei, costruit pe baza uui esatio de volum, este: ( x z * σ, x + z * σ ) ude z * este valoarea critica corespuzatoare lui 1 C petru distributia ormala stadard, adica z * = Φ( 1 C ). Medie ecuoscuta si deviatie stadard ecuoscuta cad deviatia stadard σ este ecuoscuta este estimata de obicei pri s umita eroarea stadard /deviatia stadard de selectie, ude: s = (x i x) i=1 1 si este volumul selectiei. Teorema: Petru o populatie cu media ecuoscuta μ si deviatia stadard σ ecuoscuta, u iteval de icredere petru media populatiei, costruit pe baza uui esatio de volum, este: ( x t * s, x + t * s ) ude t * 1 C este valoarea critica corespuzatoare lui petru distributia t-studet cu -1 grade de libertate. Pasul fial costa i iterpretarea rezultatului: pe baza datelor avute sutem C% siguri ca adevarata medie a populatiei se afla itre valorile date de itervalul gasit De retiut valorile critice z * si t * se pot gasi i tabelul urmator z-t-table distributia t sau distributia Studet este data de catre urmatoarea desitate de probabilitate: f(t) = +1 Γ( ) πγ( ) ( ) t 4
7 ude este umarul de grade de libertate si Γ este fuctia lui Euler. Exemplu: Suppose a studet measurig the boilig temperature of a certai liquid observes the readigs (i degrees Celsius) 10.5, 101.7, 103.1, 100.9, 100.5, ad 10. o 6 differet samples of the liquid. He calculates the sample mea to be If he kows that the stadard deviatio for this procedure is 1. degrees, what is the cofidece iterval for the populatio mea at a 95% cofidece level? I other words, the studet wishes to estimate the true mea boilig temperature of the liquid usig the results of his measuremets. If the measuremets follow a ormal distributio, the the sample mea will have the distributio N(μ, σ ). Sice the sample size is 6, the stadard deviatio of the sample mea is equal to 1. 6 = The critical value for a 95% cofidece iterval is 1.96, where (1 C)/ = (1 0.95)/ = A 95% cofidece iterval for the ukow mea is: ( , ) = (100.86, 10.78) As the level of cofidece decreases, the size of the correspodig iterval will decrease. Suppose the studet was iterested i a 90% cofidece iterval for the boilig temperature. I this case, C = 0.90, ad (1 C)/ = The critical value z * for this level is equal to 1.645, so the 90% cofidece iterval is: ( , ) = (101.01, 10.63) A icrease i sample size will decrease the legth of the cofidece iterval without reducig the level of cofidece. This is because the stadard deviatio decreases as icreases. The margi of error e of a cofidece iterval is defied to be the value added or subtracted from the sample mea which determies the legth of the iterval: e = z * σ. Suppose i the example above, the studet wishes to have a margi of error equal to 0.5 with 95% cofidece. Substitutig the appropriate values ito the expressio for m ad solvig for gives the calculatio = ( /0.5) =.09. To achieve a 95% cofidece iterval for the mea boilig poit with total legth less tha 1 degree, the studet will have to take 3 measuremets. 5
8 Testarea ipotezelor statistice I a decisio-makig process maagers make hypotheses which afterwards ca be tested usig the tools of statistics. A hypothesis test examies two opposig hypotheses about a populatio: the ull hypothesis ad the alterative hypothesis. How you set up these hypotheses depeds o what you are tryig to show. Null hypothesis H 0 the ull hypothesis states that a populatio parameter is equal to a value. The ull hypothesis is ofte a iitial claim that maagers specify usig previous research or kowledge. Alterative Hypothesis H a the alterative hypothesis states that the populatio parameter is differet tha the value of the populatio parameter i the ull hypothesis. The alterative hypothesis is what you might believe to be true or hope to prove true. What are some commo hypotheses? E.g.: Hypothesis to determie whether a populatio mea μ, is equal to some target value μ 0 iclude the followig: for a big sample size or σ kow we use the z test ad compute: z calc = x μ 0 σ for a sample size < 30 ad σ ukow we use the t test ad compute: t calc = x μ 0 s Two-tailed test: H 0 : μ = μ 0 H a : μ μ 0 the critical regio/ regio of rejectio, whe we reject H 0 is give by: z calc < z * α or z calc > z * α t calc < t * α, 1 or t calc > t * α, 1 Upper-tailed test: H 0 : μ = μ 0 H a : μ > μ 0 the critical regio/ regio of rejectio, whe we reject H 0 is give by: 6
9 z calc > z * α t calc > t * α, 1 Lower-tailed test: H 0 : μ = μ 0 H a : μ < μ 0 the critical regio/ regio of rejectio, whe we reject H 0 is give by: z calc < z * α t calc < t * α, 1 i all these examples α is the sigificace level correspodig to a cofidece level C = 1 α the critical values z * ad t * for differet cofidece itervals are show i the z-t-table Estimarea parametrilor pri metoda mometelor The method of momets is a method of estimatio of populatio parameters. The method is based o the assumptio that the sample momets are good estimates of the correspodig populatio momets. for a populatio X the momets μ k (or M k ) of order k are defied as: x k f(x)dx, μ k = M(X k ) = x k i p i, i I if X is cotiuous if X is discrete the sample momet m k of order k of a sample of size is defied as: m k = 1 The method of momets estimatio simply equates the momets of the distributio with the sample momets μ k = m k ad solves for the ukow parameters. (the distributio must have fiite momets) Method of momets: 1. we wat to estimate a parameter θ i=1. calculate low-order momets μ k as fuctios of θ 3. set up a system of equatios settig the populatio momets μ k equal to the sample momets m k, ad derive expressios for the parameter as fuctios of the sample momets m k. X k i 7
10 Let X 1, X,... X a sample from a biomial distributed populatio X Bi( 0, p) with parameters 0 ad p. Estimate these parameters usig the method of momets. Solutie: Sice M(X) = 0 p ad: M (X) = M(X ) = D (X) + M(X) = 0 p(1 p) + 0p, we ca write 0 p(1 p) = M (X) M(X ). Equatig: ( M(X) = m 1 = X ) 1 + X X ad oe ca observe: thus: Exemplu: ( M (X) = m = X 1 + X X ) 1 p = m m 1 m 1 p = m 1 + m 1 m m 1 ca be used as a estimator for the parameter p. I the same cotext: 0 = m 1 p = m 1 m 1 + m 1 m. 8
11 Aaliza regresiva pri metoda celor mai mici patrate i sectiuile aterioare am cosiderat experimete petru care am observat o sigura catitate (variabila) aleatoare, iar esatioaele respective au costat di date reprezetate de umere reale x 1, x,..., x i aceasta sectiue vom cosidera experimete î care sutem iteresati de doua catitati (variabile) aleatoare, deci esatioaele respective vor fi reprezetate de perechi de umere reale (x 1, y 1 ), (x, y ),..., (x, y ) i aaliza regresiva ua di cele doua variabile (spre exemplu X) este privita ca o variabila ce poate fi masurata (determiata) cu precizie, umita variabila idepedeta si sutem iteresati de modul cum cealalta variabila Y (umita variabila depedeta) depide de aceasta: spre exemplu sutem iteresati de modul de aportul de crestere Y al aimalelor î fuctie de catitatea zilica de hraa X. i geeral, itr-u aumit experimet alegem valorile x 1, x,..., x apoi observam valorile y 1, y,..., y ale uei variabile aleatoare Y, obtiad astfel u esatio (x 1, y 1 ), (x, y ),..., (x, y ) Se pue problema gasirii uei curbe care sa aproximeze cat mai bie datele obitute experimetal (orul de pucte) aceasta aproximare se face de obicei impuad coditia ca suma patratelor distatelor de la pucte la curba sa fie miima (metoda celor mai mici patrate) Regresia liiara estimam orul de pucte pritr-o dreapta y = f(x) = a + bx impuad coditia data de metoda celor mai mici patrate se obtie sistemul: 9
12 si: care are solutia: { a + b i=1 xi a i=1 = yi i=1 x i = i=1 xi + b b = xy x y x ( x) i=1 a = y i i=1 xiyi i=1 b x i = Y b X. Regresia parabolica estimam orul de pucte pritr-o parabola y = f(x) = a + bx + cx impuad coditia data de metoda celor mai mici patrate se obtie sistemul: a + b x + c x = y a x + b x + c x 3 = xy a x + b x 3 + c x 4 = x y Regresia hiperabolica estimam orul de pucte pritr-o hiperbola y = f(x) = a + b x impuad coditia data de metoda celor mai mici patrate se obtie sistemul: { a + b 1 x = y a 1 x + b 1 x = y x Regresia expoetiala estimam orul de pucte pritr curba y = f(x) = a b x se logaritmeaza relatia si obtiem: l y = l a + l b x care are forma uui model de regresie liiara petru datele (x i, l y i ), deci a si b se determia di: l b = x l y x l y x ( x) i = 1, si: i=1 l a = l y i i=1 l b x i. pri itermediul formulelor a = e l a l b si b = e 10
13 Probleme rezolvate Problema 1. Calculaţi cuartilele Q 1, Q, Q 3 petru următoarea serie statistica simplă şi abaterea cuartilică. X : 1,, 5, 7, 11, 1,, 3, 9 Solutie: Facem mai îtâi observaţia că mediaa m e coicide cu cuartila Q. Deoarece seria statistică dată are u umăr impar de termei (9 mai exact), vom folosi formula corespuzătoare petru a determia cuartila Q şi avem x 9+1 = x 5 = 11 m e = Q = 11. Mai departe petru a determia prima cuartilă ţiem cot de seria statistică simplă 1,, 5, 7, 11 care are tot u umăr impar de termei şi obţiem x 5+1 = x 3 = 5 Q 1 = 5. Aalog procedăm petru a treia cuartilă ţiâd cot de seria statistică simplă 11, 1,, 3, 9 care are tot u umăr impar de termei şi rezultă x 5+1 = x 3 = Q 3 =. Atuci rezultă că abaterea cuartilică este Q = Q 3 Q 1 = 5 = 17. Problema. Fie seria statistică Determiaţi: a) amplitudiea absolută A. b) abaterea medie pătratică a (X). c) dispersia σ (X). d) deviatia stadard σ (X). e) coeficietul de variaţie cv(x). X : 1, 5, 4, 0, 3, 16. Solutie: a) Amplitudiea absolută A este A = X max X mi = 0 1 =
14 b) Abaterea medie pătratică a (X) se obţie astfel a (X) = ude media x este Atuci rezultă c) Dispersia este σ (X) = x + 5 x + 4 x + 0 x + 3 x + 16 x, 6 x = (x i x) = i=1 a (X) 6, 55. = 8, 16. = 1 ( 7, , , , , , 84 ) 6 = 51, d) deviatia stadard rezultă imediat de mai sus σ (X) = σ (X) = 51 = 7, e) Di cele de mai sus, rezultă coeficietul de variaţie cv(x) = σ (X) x 100 = 85, 78. Problema 3. Pe o perioadă de mai mulţi ai, u profesor a îregistrat rezultatele elevilor şi a obţiut ca media μ a acestor rezultate este 7 şi abaterea stadard σ = 1. Clasa de 36 de elevi pe care-i îvaţă î prezet are o medie x = 75,, iar profesorul afirmă ca ea este superioară celor de pâă acum. Îtrebarea care se pue este dacă media clasei x este u argumet suficiet petru a susţie afirmaţia profesorului la u ivelul de semificaţie dat α = 0, 05 (95% sigur). Solutie: Etapa 1: Formularea ipotezei ule H 0 H 0 : x = μ = 7 clasa u este superioară. Etapa : Formularea ipotezei alterative H a H a : x = μ > 7 clasa este superioară. Etapa 3: Metodologia de verificare a ipotezelor a) Câd î ipoteza ulă media populaţiei şi deviaţia stadard sut cuoscute, atuci folosim scorul stadard z ca şi test statistic. b) Nivelul de semificaţie este dat şi este α = 0, 05. c) Î baza teoremei limită cetrală distribuţia mediilor eşatioaelor este aproape ormală, deci pri urmare distribuţia ormală va fi folosită petru 1
15 determiarea regiuii critice. Regiuea critică este egală cu mulţimea valorilor scorului stadard z care determiă respigerea ipotezei ule şi este situată la extremitatea dreaptă a distribuţiei ormale. Regiuea critică este la dreapta deoarece valori mari ale mediei eşatioului susţi ipoteza alterativă î timp ce valori apropiate valorii 7 susţi ipoteza ulă. Valoarea critică ce desparte zoa valorilor u este superior de zoa valorilor este superior este determiată de probabilitatea α = 0, 05 de a comite o eroare de tip I (eroarea de tip I apare câd ipoteza ulă este adevărată şi tot ea este respisă). Etapa 4: Determiarea valorii testului statistic Valoarea testului statistic este dată de formula z calc = x μ 75, 7 σ = = 1, Etapa 5: Luarea uei decizii şi iterpretarea ei Dacă comparăm valoarea găsită cu valoarea critică observăm că: 1, 6 < 1, 65 Coform celor stabilite i sectiuea ipotezelor statistice respigem ipoteza H 0 daca: z calc > z * α Decizia: u putem respige ipoteza ulă! Î fial, tragem cocluzia că probele u sut suficiete petru a susţie că actuala clasă este superioară celor aterioare. Problema 4. Noua ditre studeţii uei facultati cu profil sportiv au fost selectaţi petru a da u test de alergare pe distaţă mare. Masurătorile petru acest grup au codus la u timp mediu de 1, 87 miute cu o abatere stadard s = 1, 3. Să se aproximeze, cu o probabilitate de 90%, timpul mediu pe care studetii itregii facultati il vor iregistra pe acea distata. Solutie: Deoarece u se cuoaşte dispersia populaţiei iar eşatioul are volumul mai mic dacât 30, itervalul de îcredere este dat de formula ( x s ) s t 1, α, x + t 1, α, ude x = 1, 87 ; s = 1, 3 ; = 9 ; α = 0, 10 ; iar t 1, α este valoarea critică a repartiţiei Studet (statisticiaul William Sealy Gosset folosea acest pseudoim i articolele sale ) cu 1 grade de libertate corespuzătoare valorii α = 1 C care î cazul ostru este t 9 1, 0.05 = t 8, 0,05 = 1, 860 coform tabelului z-t-table Obtiem itervalul (1.064, ) I cocluzie sutem 90% siguri ca timpul mediu iregistrat de u studet pe acea distata va fi i acest iterval! 13
16 Probleme propuse Problema 1. Fiid date seriile statistice simple X : 1, 5, 7, 8, 10, Y : 1, 6, 100, 135 determiaţi mediaa î ambele cazuri. Problema. Îtr-o colectivitate s-au ales date statistice umerice obţiâdu-se X : 4, 1, 1, 5, 6, 3,, 1, Y : 100, 90, 40, 80, 70, 50, 100, 70. Aflaţi după care di variabilele de mai sus, colectivitatea este mai omogeă. Problema 3. Diagrama Herzsprug-Russell arata depedeta ditre magitudiile absolute si temperaturile efective de la suprafata stelelor: Petru u grup de stele di sirul pricipal al diagramei astroomii au iregistrat cu ajutorul telescopului Keck urmatoarele date: (+5, 5000 K), (+10, 3000 K), (0, K), ( 5, 5000 K), (+6, 7500 K) Cautati u model de regresie adecvat petru aceste date. 14
17 Problema 4. The operatios maager of a large productio plat would like to estimate the mea amout of time a worker takes to assemble a ew electroic compoet. Assume that the stadard deviatio of this assembly time is 3.6 miutes. a) After observig 10 workers assemblig similar devices, the maager oticed that their average time was 16. miutes. Costruct a 95% cofidece iterval for the mea assembly time. b) How may workers should be ivolved i this study i order to have the mea assembly time estimated up to ±15 secods with 95% cofidece? Problema 5. I order to esure efficiet usage of a server, it is ecessary to estimate the mea umber of cocurret users. Accordig to records, the sample mea ad sample stadard deviatio of umber of cocurret users at 100 radomly selected times is 37.7 ad 9., respectively. Costruct a 90% cofidece iterval for the mea umber of cocurret users. Problema 6. Let X 1, X,..., X be ormal radom variables with mea m ad variace σ. What are the method of momets estimators of the mea m ad variace σ? Problema 7. A cosumer group, cocered about the mea fat cotet of a certai steakburger submits to a idepedet laboratory a radom sample of 1 steakburgers for aalysis. The percetage of fat i each of the steakburgers is as follows: The maufacturer claims that the mea fat cotet of this steakburger is aroud 0%. Assumig percetage fat cotet to be ormally distributed with a stadard deviatio of 3, carry out a hypothesis test, with sigificace level α = 0.05, i order to advise the comsumer group as to the validity of maufacturer s claim. Problema 8. Durig a particular week, 13 babies were bor i a materity uit. Part of the stadard procedure is to measure the legth of the baby. Give below is a list of the legths, i cetimetres, of the babies bor i this particular week Assumig that this sample came from a uderlyig ormal populatio, test, at the 5% sigificace level, the hypothesis that the populatio mea legth is 50 cm. Problema 9. X 1, X,... X represets a selectio from a populatio X with expoetial distributio, i.e. the probability desity fuctio is: { λe λx, if x 0, f(x) = 0, otherwise Estimate the parameter λ usig the method of momets. Problema 10. X 1, X,... X represets a selectio from a populatio X with Poisso distributio, i.e. the probability mass fuctio is: λ λk { e P (X = k) = k!, if k = 0, 1,... 0, otherwise 15
18 Estimate the parameter λ usig the method of momets. 16
Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More information- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion
1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
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 informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationHYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018
HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationLecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.
STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationSTAT 155 Introductory Statistics Chapter 6: Introduction to Inference. Lecture 18: Estimation with Confidence
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Itroductory Statistics Chapter 6: Itroductio to Iferece Lecture 18: Estimatio with Cofidece 11/14/06 Lecture 18 1 Itroductio Statistical Iferece
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationCommon Large/Small Sample Tests 1/55
Commo Large/Small Sample Tests 1/55 Test of Hypothesis for the Mea (σ Kow) Covert sample result ( x) to a z value Hypothesis Tests for µ Cosider the test H :μ = μ H 1 :μ > μ σ Kow (Assume the populatio
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More information[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:
PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationComputing Confidence Intervals for Sample Data
Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationUniversity of California, Los Angeles Department of Statistics. Hypothesis testing
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationLESSON 20: HYPOTHESIS TESTING
LESSN 20: YPTESIS TESTING utlie ypothesis testig Tests for the mea Tests for the proportio 1 YPTESIS TESTING TE CNTEXT Example 1: supervisor of a productio lie wats to determie if the productio time of
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationIntroduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso
More informationSample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D.
ample ie Estimatio i the Proportioal Haards Model for K-sample or Regressio ettigs cott. Emerso, M.D., Ph.D. ample ie Formula for a Normally Distributed tatistic uppose a statistic is kow to be ormally
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More information5. A formulae page and two tables are provided at the end of Part A of the examination PART A
Istructios: 1. You have bee provided with: (a) this questio paper (Part A ad Part B) (b) a multiple choice aswer sheet (for Part A) (c) Log Aswer Sheet(s) (for Part B) (d) a booklet of tables. (a) I PART
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationMA238 Assignment 4 Solutions (part a)
(i) Sigle sample tests. Questio. MA38 Assigmet 4 Solutios (part a) (a) (b) (c) H 0 : = 50 sq. ft H A : < 50 sq. ft H 0 : = 3 mpg H A : > 3 mpg H 0 : = 5 mm H A : 5mm Questio. (i) What are the ull ad alterative
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationContinuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised
Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for
More informationStat 200 -Testing Summary Page 1
Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationFinal Review. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech
Fial Review Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech 1 Radom samplig model radom samples populatio radom samples: x 1,..., x
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationSTATISTICAL INFERENCE
STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More information6 Sample Size Calculations
6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig
More informationChapter 4 Tests of Hypothesis
Dr. Moa Elwakeel [ 5 TAT] Chapter 4 Tests of Hypothesis 4. statistical hypothesis more. A statistical hypothesis is a statemet cocerig oe populatio or 4.. The Null ad The Alterative Hypothesis: The structure
More informationChapter 13: Tests of Hypothesis Section 13.1 Introduction
Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationSTAT431 Review. X = n. n )
STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ,
More informationINSTRUCTIONS (A) 1.22 (B) 0.74 (C) 4.93 (D) 1.18 (E) 2.43
PAPER NO.: 444, 445 PAGE NO.: Page 1 of 1 INSTRUCTIONS I. You have bee provided with: a) the examiatio paper i two parts (PART A ad PART B), b) a multiple choice aswer sheet (for PART A), c) selected formulae
More informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationTest de Departajare pentru MofM 2014 (Bucureşti) Enunţuri & Soluţii
Test de Departajare petru MofM 04 Bucureşti Euţuri & Soluţii Problem. Give + distict real umbers i the iterval [0,], prove there exist two of them a b, such that ab a b < Solutio. Idex the umbers 0 a 0
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
More informationMath 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency
Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationSTAC51: Categorical data Analysis
STAC51: Categorical data Aalysis Mahida Samarakoo Jauary 28, 2016 Mahida Samarakoo STAC51: Categorical data Aalysis 1 / 35 Table of cotets Iferece for Proportios 1 Iferece for Proportios Mahida Samarakoo
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationLecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS
Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationKurskod: TAMS11 Provkod: TENB 21 March 2015, 14:00-18:00. English Version (no Swedish Version)
Kurskod: TAMS Provkod: TENB 2 March 205, 4:00-8:00 Examier: Xiagfeg Yag (Tel: 070 2234765). Please aswer i ENGLISH if you ca. a. You are allowed to use: a calculator; formel -och tabellsamlig i matematisk
More informationClass 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 7 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 013 by D.B. Rowe 1 Ageda: Skip Recap Chapter 10.5 ad 10.6 Lecture Chapter 11.1-11. Review Chapters 9 ad 10
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad o-users, x - y. Such studies are sometimes viewed
More informationBIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov
Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More information2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2
Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:
More information