Statisticians use the word population to refer the total number of (potential) observations under consideration

Size: px
Start display at page:

Download "Statisticians use the word population to refer the total number of (potential) observations under consideration"

Transcription

1 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 (chapter 3) Therefore, a populatio may be fiite (e.g. umber of households i the US) or (effectively) ifiite (e.g. umber of stars i the uiverse)

2 e.g. questio: average umber of TV sets per household i US populatio: umber of TV sets i each household i US questio: average umber of TV sets per household i North America populatio: umber of TV sets i each household i Caada, US ad Mexico questio: probability that a star has plaets populatio: umber of plaets per star for all stars (past, preset, future) i all galaxies i the uiverse

3 I aswerig questios (e.g. what is the mea, what is the variace, what is the probability) for a give populatio, oe seldom aswers the questios usig the etire populatio. I practice the questios are aswered from a subset (a sample) of the populatio. it is importat to choose the sample i a way that does ot bias the aswers This is the subject of a area of statistics referred to as experimetal desig. (how to desig the sample such that you adequately reflect the etire populatio) e.g. i determiig the probability of gettig a pair i a poker had, you would ot sample oly poker hads that cotaied two pairs. (techically this would be a attempt to determie the probability P(pair) for the etire populatio by approximatig it by a coditioal probability P(pair two pair) e.g. To determie the average legth of logs movig o a coveyor belt at costat speed, oe might decide to measure oly the logs that pass a certai poit o the coveyor belt every 10 miutes. Upo reflectio, you realize that loger logs have a greater probability of beig at the measurig poit at the selected times, thus the sample would give a biased average legth measure that would be too large. e.g. to determie the expected lifetime of a tire, you oly test it o smooth, paved roads? e.g. to determie fuel ratig o cars, the EPA presumes that every car is drive 55 percet of the time i the city ad 45 percet of the time o the highway!?

4 Oe way to esure ubiased samplig is to esure your subset is a radom sample Suppose our sample is to cosist of observatios, x 1, x,, x. We have to select the first observatio x 1, the secod x, etc. We thik of the procedure for pickig x k as selectig a value for a radom variable X k, that is, we thik of pickig values x 1, x,, x for our sample as the process of pickig values for radom variables X 1, X,, X. Usig this thikig, we ca defie a radom sample as follows: fiite populatio: A set of observatios X 1, X,, X costitutes a radom sample of size from a fiite populatio of size N, if values for the set are chose so that each subset of of the N elemets of the populatio has the same probability of beig selected. ifiite populatio: A set of observatios X 1, X,, X costitutes a radom sample of size from the ifiite populatio described by distributio (discrete) or desity (cotiuous) f(x) if 1. each X i is a RV whose distributio/desity is give by f(x). the RVs are idepedet The phrase radom sample is applied both to the RV s X 1, X,, X ad their values x 1, x,, x

5 How to achieve a radom sample? e.g. the populatio is fiite (ad relatively small) Label each elemet of the populatio 1,,, N. Draw umbers sequetially, i groups of, from a radom digits table

6 Whe the populatio size is large or ifiite, this process ca become practically impossible, ad careful thought must be give to, at least approximate, radom samplig desig. e.g. areal samplig usig a regular grid works if uderlyig populatio (e.g. chemical cotamiat cocetratio) is relatively homogeous. Does t work if uderlyig populatio is spatially cocetrated. e.g. replicate samplig i aomalous areas

7 6. Samplig Distributio of the Mea For each sample x 1, x,, x of observatios, we ca compute a mea x. The mea value will vary with each of our samples. Thus we ca thik of the sample mea (mea value for each sample) as a radom variable X obeyig some distributio fuctio f(x ; ) The distributio f(x ; ) is referred to as the theoretical samplig distributio. We put aside for the momet the questio of the form for f(x ; ) ad ote that, i chapter 5.10, we have already computed the mea ad variace for f(x ; ) i the case of cotiuous RV s. Theorem 6.1: If a radom sample X 1, X,, X of size is take from a populatio havig mea μ ad variace σ, the X is a RV whose distributio f(x ; ) has: ifiite populatio mea value E(X) = μ ad variace Var X = σ fiite populatio mea value E(X) = μ ad variace Var X = σ N N 1 Note: The appearace of the term N for the variace of X i the fiite populatio case is N 1 uexpected based upo the calculatio i The calculatios i 5.10, whe applied to a fiite populatio, assume that N. This correctio factor, called the fiite populatio correctio (fpc) factor is icluded to accout for cases i which N. Note that the fpc factor =0 for = N. (i.e. Var X =0 whe = N). This implies that, whe oe sample is take usig the etire populatio, X exactly measures the populatio mea with o error (variace).

8 e.g. For N = 1,000 ad = 10, the fpc is fpc = = Note that the results i Theorem 6.1 are idepedet of what f(x ; ) may actually be!!! Apply Chebyshev s theorem to the RV X Let ε = k σ, i.e. k = ε σ, givig P X μ > k σ < 1 k. P X μ > ε < σ ε = σ ε Therefore, for ay (arbitrarily small but) o-zero value for ε, the probability that X differs from μ ca be made arbitrarily small by makig large eough. (We eed σ ε, which meas must get very large as ε gets small). This observatio is kow as the law of large umbers (if you make the sample size large eough, a sigle sample is sufficiet to give a value for x arbitrarily close to the populatio mea.)

9 Theorem 6. Let X 1, X,, X be a radom sample, each havig the same mea value μ ad variace σ. The for ay ε > 0 P X μ > ε 0 as as the sample size gets large, the probability that the average from a sigle radom sample differs from the true mea goes to zero. Agai this result o X is idepedet of what f(x ; ) may actually be. e.g. I a experimet, evet A occurs with probability p. Repeat the experimet times ad compute relative frequecy of occurrece of A = Show that the relative frequecy of A p as umber of times A occurs i trials Cosider each trial as a idepedet RV, X 1, X,, X Each X i takes o two values, x i = 0,1 depedig o whether A does ot or does occur i experimet i. X i has mea value E X i = 0 1 p + 1 p = p ad variace Var X i = E X i E X i = 0 1 p + 1 p p = p(1 p) The X 1 + X + + X records the umber of times A occurs i trials, ad X = X 1 + X + + X is i fact the relative frequecy of occurrece of A. From Theorem 6. we have p(1 p) ε P X p > ε < 0 for ay p [0,1] as

10 Var(X) σ X = σ is referred to at the stadard error of the mea. To reduce the stadard error by a factor of two, it is ecessary to icrease 4. Thus (ufortuately) icreasig sample size decreases the stadard error at a relatively slow rate. (e.g. if goes from 5 to,500 (a factor of 100), the stadard error decreases oly by 10.) While the results i Theorems 6.1 ad 6. are idepedet of the form of the theoretical samplig distributio/desity f(x ; ), the actual form for f(x ; ) depeds o kowig the probability distributio which govers the populatio. I geeral it ca be very difficult to compute the form of f(x ; ). Two results are kow both preseted as theorems. Theorem 6.3 (cetral limit theorem) Let X be the mea of a radom sample of size take from a populatio havig mea μ ad variace σ. The the associated RV, the stadardized sample mea X μ Z σ is a RV whose distributio fuctio approaches the stadard ormal distributio as

11 The cetral limit theorem says that, as, the theoretical samplig distributio f(x ; ) a ormal distributio (i.e. X is ormally distributed) with mea μ ad variace σ The distributio f(x ; ) of X for samples of size for a populatio with expoetial distributio The distributio f(x ; ) of X for samples of size for populatio with uiform distributio I practice, the distributio for X is well approximated by a ormal distributio for as small as 5 to 30.

12 Practical use of the cetral limit theorem: You have a populatio whose mea μ ad stadard deviatio σ you assume that you kow (but whose desity fuctio f(x) you do ot kow). You sample the populatio with a sample of size. From the sample you compute a mea value x. If the sample size is sufficietly large the cetral limit theorem will tell you the probability of gettig the value x give your assumptios o the values of μ ad σ. To test your assumptio, compute the stadardized sample mea z usig the measured x ad assumed values μ ad σ. The cetral limit theorem states that the probability of gettig the value x is the same as the probability of gettig the z-score z i a stadard ormal distributio.

13 Theorem (Normal populatios) Let X be the mea of a radom sample of size take from a populatio that is ormally distributed havig mea μ ad variace σ. The the stadardized sample mea X μ Z σ has the stadard ormal distributio fuctio regardless of the size of. (i.e. f(x ; ) for X is ormal desity with mea μ ad variace σ /). Practical use of this theorem: You have a populatio whose distributio is (assumed to be) ormal ad whose mea μ ad stadard deviatio σ you assume that you kow. You sample the populatio with a sample of size. From the sample you compute a mea value x. This theorem will tell you the probability of gettig the value x give your assumptios o ormality ad the values of μ ad σ. To test your assumptios, compute the stadardized sample mea z usig the measured x ad assumed values μ ad σ. This theorem states that the probability of gettig the value x is the same as the probability of gettig the z-score z i a stadard ormal distributio.

14 e.g. 1-gallo pait cas (the populatio) from a particular maufacturer cover, o average sq. ft, with a stadard deviatio of 31.5 sq. ft. What is the probability that the mea area covered by a sample of 40 1-gallo cas will lie withi to 50.0 sq. ft. Fid the stadardized sample meas for the two limits of the rage: z 1 = = 0.66, z = = Assumig the cetral limit theorem, we have from Table 3 P < X < 50.0 = P 0.66 < Z < 1.34 = F 1.34 F 0.66 = =

15 6.3 The Samplig Distributio of the Mea whe σ is ukow (usual case) I 6. we discussed aspects of the distributio of the sample mea X (it has a distributio with mea μ,variace σ (for cotiuous RVs), ad the related RV X μ Z σ the stadardized sample mea approaches the stadard ormal distributio as ). I practice σ is ot kow ad we have to deal with the values x μ t s where s is the sample stadard deviatio s = s, ad s is the sample variace s x i x = 1 Similar to X, we defie the radom variable S called the sample variace S X i X = 1 which has values s. I this sectio ad the ext, we are iterested i the behavior of t ad S thought of as radom variables.

16 Little is kow about the behavior of the distributio for t whe is small uless we are samplig from a populatio govered by the ormal distributio (a ormal populatio ) Theorem 6.4 If X is the sample mea for a radom sample of size take from a ormal populatio havig mea μ, the X μ t S is a radom variable havig the t distributio with parameter v = 1. Note: it is covetio to use small t for the RV for the t distributio (breakig the covetio to use capital letters for the RV ad small letters for its values). We will use small t to stad for both the RV ad its values.

17 The t distributio: a oe-parameter family of RVs, with values defied o (, ) desity fuctio f t; v = Γ v + 1 vπγ v 1 + t v+1 v mea value 0 (for v > 1), otherwise udefied variace v v (for v > ), for 1 < v <, otherwise udefied The t distributio is symmetric about 0, ad very close to the stadard ormal distributio. I fact the t distributio the stadard ormal distributio as v. The t distributio has heavier tails tha the stadard ormal distributio (i.e. there is higher probability i the tails of the t distributio). It is ofte referred to as studet s t distributio v v v v

18 The parameter v i the t distributio is referred to as the (umber of) degrees of freedom (df) Recall that the sum of the sample deviatios x i x is 0, hece oly 1 of the deviatios are idepedet of each other. Thus the RVs S ad, by the same reasoig, t both have 1 degrees of freedom. Similar to the z α for the stadard ormal distributio, we defie the t α for the t distributio. Because of the symmetry of the stadard ormal ad t distributios we have z 1 α = z α, t 1 α = t α Recall that Table 3 lists values of the cumulative stadard ormal distributio F(z) for various values of z I cotrast, Table 4 lists values of t α for various values of α ad v. (Recall, α is the probability i the right-had tail above t α ) By symmetry, the probability i the left-had tail below t α is also α. Note that for, t α = z α The stadard ormal distributio provides a good approximatio to the t distributio for samples of size 30 or more.

19 Practical use of theorem 6.4: You have a populatio whose distributio is ( assumed to be) ormal ad whose mea μ you assume that you kow (but whose stadard deviatio you do ot kow). You sample the populatio with a sample of size. From the sample you compute a sample mea value x ad the sample stadard deviatio s. Theorem 6.4 will tell you the probability of gettig the values x ad s give your assumptios o ormality ad the value of μ. To test your assumptio, compute the stadardized sample mea z usig the measured x ad s ad the assumed values μ. Theorem 6.4 states that the probability of gettig the value x, s is the same as the probability of gettig the value t i a t distributio with v = 1

20 e.g. a maufacturer s fuses (the populatio) will blow i 1.40 miutes o average whe subjected to a 0% overload. A sample of 0 fuses are subjected to a 0% overload. The sample average ad stadard deviatio were observed to be, respectively, ad.48 miutes. What is the probability of this observatio give the maufacturers claim? t = = 3.19, v = 0 1 = 19.48/ 0 From Table 4, for v = 19, we see that a t value of.861 already has oly 0.5% probability (α = 0.005) of beig exceeded. Cosequetly there is less tha a 0.5% probability that a t value smaller tha will occur. Sice the t value obtaied i our sample of 0 is 3.19, we coclude that there is less tha 0.5% probability of gettig this result. We therefore suspect that the maufacturers claim is icorrect, ad that the maufacturers fuses will blow i less tha 1.40 miutes o average whe subjected to 0% overload. If the populatio is ot ormal, studies have show that the distributio of X μ S is fairly close to that of the t distributio as log as the populatio distributio is relatively bell-shaped ad ot too skewed. This ca be checked usig a ormal scores plot o the populatio.

21 6.4 The Distributio of the Sample Variace S Theorem 6.5 Cosider a radom sample of size take from a ormal populatio havig variace σ. The the RV ( 1)S X i X σ = σ has the chi-square distributio with parameter v = 1 The chi-square distributio: a oe-parameter family of RVs, with values defied o (0, ) desity fuctio 1 f x; v = v Γ v x v 1 e v mea value v variace v The chi-square distributio is just the gamma distributio with α = v, β = Agai, the parameter v is referred to as the (umber of) degrees of freedom (df) We defie the α otatio similar to that of z α ad t α. Just as for Table 4, Table 5 lists values of α for various values of α ad v.

22 v v v v v v

23 e.g. (the populatio) glass blaks from a optical firm suitable for gridig ito leses Variace or refractive idex of glass is Radom sample of size 0 selected from ay shipmet, ad if variace of refractive idex of sample exceeds 10 4, the sample is rejected. What is probability of rejectio assumig uderlyig populatio is ormal? For the measured sample of = 30. From Table 5, for v = 19, 30. correspods to a value α = There is therefore a 5% probability of rejected a shipmet

24 Practical use of theorem 6.5: You have a populatio whose distributio is ( assumed to be) ormal ad whose variace σ you assume that you kow. You sample the populatio with a sample of size. From the sample you compute a sample variace s. Theorem 6.5 will tell you the probability of gettig the value s give your assumptios o ormality ad the value of σ. To test your assumptio, compute the chi square value usig the measured s ad the assumed value σ. Theorem 6.5 states that the probability of gettig the value s is the same as the probability of gettig the value i a chi square distributio with v = 1

25 Recap sample 1 outcomes y 1 y sample space (N outcomes if fiite) e.g. throws each of k dice sample values for RV x 1 x e.g. k-dice sums sample j Thik of each x i value as resultig from a RV X i such that 1. each X i has the same desity f(x), mea μ, ad variace σ. the X i are idepedet radom sample The populatio of outcomes i the sample space geerates values for the RVs

26 Each sample geerates a sample mea x ad a sample variace s = Thik of the sample meas ad variaces are values for the RVs X ad S What are F X, E X, Var X, F S, E S, Var S? 1 x i x Chapter 5 states: E X = μ, Var X E X = μ, Var X = σ / for a ifiite populatio = σ N N 1 for a ifiite populatio Chapter 6 addresses the questios o F X, F S Law of large umbers for a sigle sample (ad sigle value of X) Cetral limit theorem is a RV whose distributio F Z P X μ > ε < σ ε Z X μ σ stadard ormal N(0,1) as (i.e. X is a RV whosedistributio F X N(μ, σ) as )

27 If the X i are ormally distributed with mea μ ad variace σ X μ Z σ is a RV whose distributio F Z = N(0,1) for all i.e. X is a RV whose distributio F X = N(μ, σ) for all If the X i are ormally distributed with mea μ X μ t S is a RV whose distributio F t is the t-distributio with df v = 1 If the X i are ormally distributed with variace σ ( 1)S X i X σ = σ is a RV whose distributio F is the chi square distributio with df v = 1

28 Assume we have two populatios. We may wish to iquire whether they have the same variace. Assume S 1 ad S are measured sample variaces for each populatio. Theorem 6.6 If S 1 ad S are measured sample variaces of idepedet radom samples of respective sizes 1 ad take from two ormal populatios havig the same variace the F = S 1 S is a RV havig the F distributio with parameters v 1 = 1 1 ad v = 1. The F distributio: a two-parameter family of RVs, with values defied o (0, ) desity fuctio mea value variace f x; v 1, v = v v for v > 1 B v 1, v v (v 1 +v ) v 1 v (v 4) for v > 4 v 1 v v 1 x v 1 1 The F distributio is similar to the beta distributio. B v 1, v B x, y = t x 1 1 t y 1 dt v 1 v x v 1+v is the beta fuctio

29 F distributio v 1 v 1 v 1 v v v v 1 v 1 v v

30 The parameter v 1 is referred to as the umerator degrees of freedom (df of uerator) The parameter v is referred to as the deomiator degrees of freedom (df of deomiator) As with z α, t α, etc we defie F α. Values of F α are give i Table 6 for various values of v 1 ad v for α = 0.05 (Table 6(a)) ad α = 0.01(Table 6(b)) Practical use of theorem 6.6: You have two populatio whose distributio are ( assumed to be) ormal ad whose variaces you assume to be equal. You sample populatio 1 with a sample of size 1 ad populatio with a sample of size. From each sample you compute sample variaces s 1 ad s. Theorem 6.6 will tell you the probability of gettig the ratio s 1 s give your assumptios o ormality ad equality of variace. To test your assumptio, compute the value F. Theorem 6.6 states that the probability of gettig the ratio s 1 s is the same as the probability of gettig the value F i a F distributio with v 1 = 1 1, v = 1.

31 e.g. Two radom samples of size 1 = 7 ad = 13 are take from the same ormal populatio. What is the probability that the variace of the first sample will be at least 3 times that of the secod. For v 1 = 6 ad v = 1, Table 6(a) shows a F value of 3.00 for α = Therefore there is a 5% probability that the variace of the first sample will be at least 3 times that of the secod.

32 6.5 Represetatios of ormal distributios Defiig ew radom variables i terms of others is referred to as a represetatio chi-square Let Z 1, Z,, Z v be idepedet stadard ormal RVs. Defie the RV v v = Z i The v has a chi square distributio with v df Thus we also see that the square of a stadard ormal RV is a chi-square RV Let v 1 1 = Z i v 1 +v ad = Z i i=v 1 +1 where the Z i are idepedet stadard ormal RVs (ad thus 1 ad are idepedet of each other). The 1 + has a chi square distributio with v 1 + v df. Thus we see that the sum of two idepedet chi square RVs is also a chi square RV with the sum of the idividual df

33 t distributio Let Z be a stadard ormal RV ad be a chi-square RV with v df. Assume Z ad are idepedet. The t Z has a t distributio with v df v F distributio Let 1 ad be chi-square RVs with df v 1 ad v respectively. Assume 1 ad are idepedet. The has a F distributio with v 1, v df F v1,v 1 v 1 v Thus we see that is a RV with a F 1,v distributio t Z 1 v

34 e.g. Let X 1, X,, X be idepedet ormal RVs all havig mea μ ad stadard deviatio σ. The Z i = X i μ σ is a stadard ormal RV for each i. The Z 1 is also a stadard ormal RV. Cosider i.e. Z i Z Z i = Z i = 1 Z Z i Z i = Z i Z X i μ σ = X μ σ/ + Z = Z i + Z Z Note that the LHS is chi square distributio with df. The last term o the RHS is chi square with 1 df. This implies that the first term o the RHS is chi-square with 1df. Thus we see that ( 1)S X i X σ = σ = Z i Z has a chi square distributio with 1df (as claimed i Theorem 6.5)

35 Let X i be N(μ i, σ i ) for i = 1,, be idepedet ormal RVs The is ormal with E X = u i A sum of ormal RVs is a ormal RV X = X i, Var X = σ i Let X i be a chi-square RV with df= v i for i = 1,, ; assume the X i are idepedet The X = is a chi-square RV with df v = A sum of chi-square RVs is chi-square X i v i

36 Let X i be a Poisso RV with parameter λ i for i = 1,, ; assume the X i are idepedet The X = is a Poisso RV with parameter λ = A sum of Poisso RVs is Poisso X i λ i

Chapter 6 Sampling Distributions

Chapter 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 information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. 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

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete 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 information

Sampling Distributions, Z-Tests, Power

Sampling Distributions, Z-Tests, Power Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace

More information

Random Variables, Sampling and Estimation

Random 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 information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

Module 1 Fundamentals in statistics

Module 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

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

Statistics 511 Additional Materials

Statistics 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 information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical 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 information

Topic 9: Sampling Distributions of Estimators

Topic 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 information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 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 information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,

More information

Topic 10: Introduction to Estimation

Topic 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 information

Expectation and Variance of a random variable

Expectation 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 information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,

More information

Binomial Distribution

Binomial Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, 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 information

Frequentist Inference

Frequentist 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

IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.

IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes. IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 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 information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. 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 information

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

(6) Fundamental Sampling Distribution and Data Discription

(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 information

Economics Spring 2015

Economics Spring 2015 1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures

More information

Distribution of Random Samples & Limit theorems

Distribution of Random Samples & Limit theorems STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to

More information

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9

BIOS 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 information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments: Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal

More information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 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 information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 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 information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to

More information

Basics of Probability Theory (for Theory of Computation courses)

Basics of Probability Theory (for Theory of Computation courses) Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.

More information

Understanding Samples

Understanding Samples 1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We

More information

Basis for simulation techniques

Basis for simulation techniques Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios

More information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6) STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This 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 information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variance Formula: A Detailed Study of an Old Controversy The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

More information

Estimation of a population proportion March 23,

Estimation 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 information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9 Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 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

More information

Advanced Stochastic Processes.

Advanced Stochastic Processes. Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.

More information

Probability and statistics: basic terms

Probability and statistics: basic terms Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample

More information

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes. Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

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 information

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: 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 information

Computing Confidence Intervals for Sample Data

Computing 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 information

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 Chapter Goals After completig this chapter, you should

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

Lecture 2: Monte Carlo Simulation

Lecture 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 information

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates. 5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece

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

- 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 information

AMS570 Lecture Notes #2

AMS570 Lecture Notes #2 AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)

More information

This section is optional.

This section is optional. 4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore

More information

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests

More information

CS284A: Representations and Algorithms in Molecular Biology

CS284A: Representations and Algorithms in Molecular Biology CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 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 information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL PRACTICE PROBLEMS FOR THE FINAL Math 36Q Sprig 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2

More information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

More information

Lecture 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators

Lecture 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success Poisso Distributio Note

More information

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS 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 information

Stat 319 Theory of Statistics (2) Exercises

Stat 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 information

Hypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance

Hypothesis 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 information

Lecture 3. Properties of Summary Statistics: Sampling Distribution

Lecture 3. Properties of Summary Statistics: Sampling Distribution Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST 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 information

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam. Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the

More information

STAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)

STAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem) STAT 515 fa 2016 Lec 15-16 Samplig distributio of the mea, part 2 cetral limit theorem Karl B. Gregory Moday, Sep 26th Cotets 1 The cetral limit theorem 1 1.1 The most importat theorem i statistics.............

More information

MATH/STAT 352: Lecture 15

MATH/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 information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests 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 information

f(x)dx = 1 and f(x) 0 for all x.

f(x)dx = 1 and f(x) 0 for all x. OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

SDS 321: Introduction to Probability and Statistics

SDS 321: Introduction to Probability and Statistics SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables- Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/

More information

32 estimating the cumulative distribution function

32 estimating the cumulative distribution function 32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio

More information

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 15

Discrete 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 information

Topic 18: Composite Hypotheses

Topic 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 information

Axioms of Measure Theory

Axioms of Measure Theory MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that

More information

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency

Math 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 information

Limit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).

Limit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p). Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.

More information

Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 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 information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. 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

NOTES ON DISTRIBUTIONS

NOTES ON DISTRIBUTIONS NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which

More information

Lecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS

Lecture 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 information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 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 information

An Introduction to Randomized Algorithms

An Introduction to Randomized Algorithms A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis

More information

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

AP Statistics Review Ch. 8

AP Statistics Review Ch. 8 AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.

More information

Lecture 1 Probability and Statistics

Lecture 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 information

5. INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY

5. INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY IA Probability Let Term 5 INEQUALITIES, LIMIT THEOREMS AND GEOMETRIC PROBABILITY 51 Iequalities Suppose that X 0 is a radom variable takig o-egative values ad that c > 0 is a costat The P X c E X, c is

More information

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder

More information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: 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 information

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234

Exam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234 STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6

More information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

More information