# Statistical inference: example 1. Inferential Statistics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 as good quality or top quality. The agreemets require that the delivered goods comply with stadards predetermied quality. I particular, the proportio of good quality items must ot exceed 25% of the total. From a cosigmet 40 items are extracted ad 29 of these are of top quality whereas the remaiig 11 are of good quality. Statistical iferece is the brach of statistics cocered with drawig coclusios ad/or makig decisios cocerig a populatio based oly o sample data. INFERENTIAL PROBLEMS: 1. provide a estimate of π ad quatify the ucertaity associated with such estimate; 2. provide a iterval of reasoable values for π; 3. decide whether the delivered goods should be retured to the supplier Statistical iferece: example 1 Statistical iferece: example 2 Formalizatio of the problem: POPULATION: all the pieces of clothes of the cosigmet; VARIABLE OF INTEREST: good/top quality of the good biary variable; PARAMETER OF INTEREST: proportio of good quality items π; SAMPLE: 40 items extracted from the cosigmet. The value of the parameter π is ukow, but it affects the samplig values. Samplig evidece provides iformatio o the parameter value. 189 A machie i a idustrial plat of a bottlig compay fills oe-liter bottles. Whe the machie is operatig ormally the quatity of liquid iserted i a bottle has mea µ = 1 liter ad stadard deviatio σ =0.01 liters. Every workig day 10 bottles are checked ad, today, the average amout of liquid i the bottles is x = with s = INFERENTIAL PROBLEMS: 1. provide a estimate of µ ad quatify the ucertaity associated with such estimate; 2. provide a iterval of reasoable values for µ; 3. decide whether the machie should be stopped ad revised. 190

2 Formalizatio of the problem: Statistical iferece: example 2 POPULATION: all the bottles filled by the machie; VARIABLE OF INTEREST: amout of liquid i the bottles cotiuous variable; PARAMETERS OF INTEREST: mea µ ad stadard deviatio σ of the amout of liquid i the bottles; SAMPLE: 10 bottles. The values of the parameters µ ad σ are ukow, but they affect the samplig values. Samplig evidece provides iformatio o the parameter values. 191 The sample Cesus survey: attempt to gather iformatio from each ad every uit of the populatio of iterest; sample survey: gathers iformatio from oly a subset of the uits of the populatio of iterest. Why usig a sample? 1. Less time cosumig tha a cesus; 2. less costly to admiister tha a cesus; 3. measurig the variable of iterest may ivolve the destructio of the populatio uit; 4. a populatio may be ifiite. 192 Probability samplig A probability samplig scheme is oe i which every uit i the populatio has a chace greater tha zero of beig selected i the sample, ad this probability ca be accurately determied. Probabilistic descriptio of a populatio SIMPLE RANDOM SAMPLING: every uit has a equal probability of beig selected ad the selectio of a uit does ot chage the probability of selectig ay other uit. For istace: extractio with replacemet; extractio without replacemet. For large populatios compared to the sample size the differece betwee these two samplig techiques is egligible. I the followig we will always assume that samples are extracted with replacemet from the populatio of iterest. 193 Uits of the populatio; variable X measured o the populatio uits; sometimes the distributio of X is kow, for istace i X Nµ,σ 2 ; ii X Beroulliπ. 194

3 Probabilistic descriptio of a sample The observed samplig values are x 1,x 2,...,x ; BEFORE the sample is observed the samplig values are ukow ad the sample ca be writte as a sequece of radom variables Samplig distributio of a statistic 1 Suppose that the sample is used to compute a give statistic, for istace i the sample mea X; ii the sample variace S 2 ; iii the proportio P of uits with a give feature; X 1,X 2,...,X for simple radom samples with replacemet: 1. X 1,X 2,...,X are i.i.d.; 2. the distributio of X i is the same as that of X for every i = 1,...,. 195 geerically, we cosider a arbitrary statistic T = gx 1,...,X where g is a give fuctio. 196 Samplig distributio of a statistic 2 Suppose that X Nµ,σ 2 Normal populatio Oce the sample is observed, the observed value of the statistic is give by t = gx 1,...,x ; suppose that we draw all possible samples of size from the give populatio ad that we compute the statistic T for each sample; the samplig distributio of T is the distributio of the populatio of the values t of all possible samples. i this case the statistics of iterest are: i the sample mea X = 1 X i i=1 ii the sample variace S 2 = 1 X i X 2 1 i=1 the correspodig observed values are x = 1 x i ad s 2 = 1 x i x 2, i=1 1 i=1 respectively

4 Expected value of the sample mea The sample mea The sample mea is a liear combiatio of the variables formig the sample ad this property ca be exploited i the computatio of the expected value of X, that is E X; the variace of X, that is Var X ; the probability distributio of X. For a simple radom sample X 1,...,X, the expected value of X is X1 +X E X = E 2 + +X = 1 EX 1 +X 2 + +X = 1 [EX 1+EX 2 + +EX ] = 1 µ = µ Variace of the sample mea For a simple radom sample X 1,...,X, the variace of X is Var X X1 +X = Var 2 + +X = 1 2VarX 1 +X 2 + +X Samplig distributio of the mea For a simple radom sample X 1,X 2,...,X, the sample mea X has expected value µ ad variace σ 2 /; if the distributio of X is ormal, the = 1 2 [VarX 1+VarX 2 + +VarX ] = 1 2 σ2 = σ2 X N µ, σ2 more geerally, the cetral limit theorem ca be applied to state that the distributio of X is APPROXIMATIVELY ormal

6 The sample proportio 2 The sample proportio is such that Estimatio EP = π ad VarP = π1 π for the cetral limit theorem, the distributio of X is approximatively ormal; sometimes the followig empirical rules are used to decide if the ormal approximatio is satisfyig: 1. π > 5 ad 1 π > 5. Parameters are specific umerical characteristics of a populatio, for istace: a proportio π; a mea µ; a variace σ 2. Whe the value of a parameter is ukow it ca be estimated o the basis of a radom sample. 2. p1 p > Poit estimatio A poit estimate is a estimate that cosists of a sigle value or poit, for istace oe ca estimate a mea µ with the sample mea x; a proportio π with a sample proportio p; Estimator vs estimate A estimator of a populatio parameter is a radom variable that depeds o sample iformatio, whose value provides a approximatio to this ukow parameter. a poit estimate is always provided with its stadard error that is a measure of the ucertaity associated with the estimatio process. A specific value of that radom variable is called a estimate

7 Estimatio ad ucertaity Poit estimatio of a mea σ 2 kow Parameter θ; the samplig statistics T = gx 1,...,X o which estimatio is based is called the estimator of θ ad we write ˆθ = T the observed value of the estimator, t, is called a estimate of θ ad we write ˆθ = t; it is fudametal to assess the ucertaity of ˆθ; a measure of ucertaity is the stadard deviatio of the estimator, that is SDT = SDˆθ. This quatity is called the STANDARD ERROR of ˆθ ad deoted by SEˆθ. Cosider the case where X 1,...,X is a simple radom sample from X Nµ,σ 2 ; Parameters: µ, ukow; assume that the value of σ 2 is kow. the sample mea ca be used as estimator of µ: µ = X; the distributio of the estimator is ormal with STANDARD ERROR µ E µ = µ ad Var µ = σ2 SE µ = σ Poit estimatio of a mea σ 2 ukow Poit estimatio of a mea with σ 2 kow: example I the bottlig compay example, assume that the quatity of liquid i the bottles is ormally distributed. The a poit estimate of µ is Typically the value of σ 2 is ot kow; i this case we estimate it as ˆσ 2 = s 2 ; this ca be used, for istace, to estimate the stadard error of ˆµ µ = ad the stadard error of this estimate is SEˆµ = σ 10 = = ŜEˆµ = ˆσ. I the bottlig compay example, if σ is ukow it ca be estimated as ŜEˆµ = =

8 Poit estimatio of a proportio Parameter: π; Poit estimatio for the mea of a o-ormal populatio X 1,...,X i.i.d. with EX i = µ ad VarX i = σ 2 ; the distributio of X i is ot ormal; for the cetral limit theorem the distributio of X is approximatively ormal. the sample proportio P is used as a estimator of π π = P this estimator is approximately ormally distributed with E π = π ad Var π = π1 π the STANDARD ERROR of the estimator is π1 π SE π = ad i this case the value of stadard error is ever kow Estimatio of a proportio: example For the clothig store chai example the estimate of the proportio π of good quality items is π = = ad a ESTIMATE of the stadard error is ŜEˆπ = = Properties of estimators: ubiasedess A poit estimator ˆθ is said to be a ubiased estimator of the parameter θ if the expected value, or mea, of the samplig distributio of ˆθ is θ, formally if Eˆθ = θ Iterpretatio of ubiasedess: if the samplig process was repeated, idepedetly, a ifiite umber of times, obtaiig i this way a ifiite umber of estimates of θ, the arithmetic mea of such estimates would be equal to θ. However, ubiasedess does ot guaratees that the estimate based o oe sigle sample coicides with the value of θ

9 Poit estimator of the variace The sample variace S 2 is a ubiased estimator of the variace σ 2 of a ormally distributed radom variable ES 2 = σ 2. Bias of a estimator Let ˆθ be a estimator of θ. The bias of ˆθ, Biasˆθ, is defied as the differece betwee the expected value of ˆθ ad θ O the other had S 2 is a biased estimator of σ 2 Biasˆθ = Eˆθ θ E S 2 = 1 σ 2. The bias of a ubiased estimator is Properties of estimators: Mea Squared Error MSE For a estimator ˆθ of θ the ukow estimatio error is give by θ ˆθ The Mea Squared Error MSE is the expected value of the square of the error MSEˆθ = E[θ ˆθ 2 ] = Var ˆθ +[θ Eˆθ] 2 = Var ˆθ +Biasˆθ 2 Hece, for a ubiased estimator, the MSE is equal to the variace. 221 Most Efficiet Estimator Let ˆθ 1 ad ˆθ 2 be two estimator of θ, the the MSE ca be use to compare the two estimators; if both ˆθ 1 ad ˆθ 2 are ubiased the ˆθ 1 is said to be more efficiet tha ˆθ 2 if Var ˆθ 1 < Var ˆθ 2 ote that if ˆθ 1 is more efficiet tha ˆθ 2 the also MSEˆθ 1 < MSEˆθ 2 ad SEˆθ 1 < SEˆθ 2 ; the most efficiet estimator or the miimum variace ubiased estimator of θ is the ubiased estimator with the smallest variace. 222

10 Iterval estimatio A poit estimate cosists of a sigle value, so that if X is a poit estimator of µ the it holds that P X = µ = 0 more geerally, Pˆθ = θ = 0. Iterval estimatio is the use of sample data to calculate a iterval of possible or probable values of a ukow populatio parameter. Cofidece iterval for the mea of a ormal populatio σ kow X 1,...,X simple radom sample with X i Nµ,σ 2 ; assume σ kow; a poit estimator of µ is ˆµ = X N µ, σ2 the stadard error of the estimator is SEˆµ = σ Before the sample is extracted... The sample distributio of the estimator is completely kow but for the value of µ; the ucertaity associated with the estimate depeds o the size of the stadard error. For istace, the probability that ˆµ = X takes a value i the iterval µ±1.96 SE is 0.95 that is 95%. Cofidece iterval for µ The probability that X belogs to the iterval µ 1.96SE, µ+1.96se is 95%; this ca be also stated as: the probability that the iterval X 1.96SE, X +1.96SE area 95% cotais the parameter µ is 95% SE SE µ 3 SE µ 2 SE µ 1 SE µ µ + 1 SE µ + 2 SE µ + 3 SE µ µ P µ 1.96 SE X µ+1.96 SE =

11 Formal derivatio of the 95% cofidece iterval for µ σ kow It holds that so that X µ SE 0.95 = P N 0,1 where SE = σ 1.96 X µ SE 1.96 = P 1.96 SE X µ 1.96 SE = P X 1.96 SE µ X SE = P X 1.96 SE µ X SE Cofidece iterval for µ with σ kow: example I the bottlig compay example, if oe assumes σ = 0.01 kow, a 95% cofidece iterval for µ is that is so that ; ; ; After the sample is extracted... O the basis of the sample values the observed value of µ = x is computed. x may belog to the iterval µ±1.96 SE or ot. For istace a differet sample... A differet sample may lead to a sample mea x that, as i the example below, does ot belog to the iterval µ±1.96 SE ad, as a cosequece, also the iterval x 1.96 SE; x+1.96 SE will ot cotai µ. area 95% x area 95% x µ 3 SE µ 2 SE µ 1 SE µ µ + 1 SE µ + 2 SE µ + 3 SE ad i this case x belogs to the iterval µ±1.96 SE ad, as a cosequece, also the iterval x 1.96 SE; x+1.96 SE will cotai µ. 229 µ 3 SE µ 2 SE µ 1 SE µ µ + 1 SE µ + 2 SE µ + 3 SE The iterval x 1.96 SE; x+1.96 SE will cotai µ for the 95% of all possible samples. 230

12 Iterpretatio of cofidece itervals Probability is associated with the procedure that leads to the derivatio of a cofidece iterval, ot with the iterval itself. A specific iterval either will cotai or will ot cotai the true parameter, ad o probability ivolved i a specific iterval. Cofidece iterval: defiitio A cofidece iterval for a parameter is a iterval costructed usig a procedure that will cotai the parameter a specified proportio of the times, typically 95% of the times. Cofidece itervals for five differet samples of size = 25, extracted from a ormal populatio with µ = 368 ad σ = 15. A cofidece iterval estimate is made up of two quatities: iterval: set of scores that represet the estimate for the parameter; cofidece level: percetage of the itervals that will iclude the ukow populatio parameter A wider cofidece iterval for µ Sice it also holds that P µ 2.58 SE X µ+2.58 SE = 0.99 Cofidece level The cofidece level is the percetage associated with the iterval. A larger value of the cofidece level will typically lead to a icrease of the iterval width. The most commoly used cofidece levels are area 99% x area 99% x 68% associated with the iterval X ±1SE; 95% associated with the iterval X ±1.96SE; 99% associated with the iterval X ±2.58SE. µ 3 SE µ 2 SE µ 1 SE µ µ + 1 SE µ + 2 SE µ + 3 SE µ 3 SE µ 2 SE µ 1 SE µ µ + 1 SE µ + 2 SE µ + 3 SE the the probability that X 2.58SE, X+2.58SE cotais µ is 99%. Where the values 1, 1.96 ad 2.58 are derived from the stadard ormal distributio tables

13 Z N0,1; Notatio: stadard ormal distributio tables α value betwee zero ad oe; z α value such that the area uder the Z pdf betwee z α ad + is equal to α; formally furthermore PZ > z α = α ad PZ < z α = 1 α P z α/2 < Z < z α/2 = 1 α 235 Cofidece iterval for µ with σ kow: formal derivatio 1 It holds that so that X µ SE or, equivaletly, N 0,1 where SE = σ P µ z α/2 SE X µ+z α/2 SE = 1 α P z α/2 X µ SE z α/2 = 1 α 236 Cofidece iterval at the level 1 α for µ with σ kow Cofidece iterval for µ with σ kow: formal derivatio 2 1 α = P z α/2 X µ SE z α/2 = P z α/2 SE X µ z α/2 SE = P X z α/2 SE µ X +z α/2 SE = P X z α/2 SE µ X +z α/2 SE A cofidece iterval at the cofidece level 1 α, or 1 α%, for µ is give by Sice SE = σ the X z α/2 SE; X +z α/2 SE X z α/2 σ ; X +z α/2 σ

14 Margi of error The cofidece iterval Reducig the margi of error x±z α/2 σ ca also be writte as x±me where ME = z α/2 σ ME = z α/2 σ is called the margi of error. The margi of error ca be reduced, without chagig the accuracy of the estimate, by icreasig the sample size. the iterval width is equal to twice the margi of error Cofidece iterval for µ with σ ukow The Studet s t distributio 1 X N µ, σ2 ; X µ SE N0,1; i this case the stadard error is ukow ad eeds to be estimated. For Z N0;1 ad X χ 2 r, idepedet; the radom variable T = Z X/r is said to follow a Studet s t distributio with r degrees of freedom; the pdf of the t distributio differs from that of the stadard ormal distributio because it has heavier tails. SEˆµ = σ is estimated by ŜEˆµ = ˆσ where ˆσ = S Studet s t ormal ad it holds that X µ ŜE t t 1 ad N0;1 compariso. 242

15 Cofidece iterval for µ with σ ukow The Studet s t distributio 2 For r + the Studet s t distributio coverges to the stadard ormal distributio t 25 ad N0;1 compariso.. 1 α = P t 1,α/2 X µ ŜE t 1,α/2 = P t 1,α/2 ŜE X µ t 1,α/2 ŜE = P X t 1,α/2 ŜE µ X +t 1,α/2 ŜE = P X t 1,α/2 ŜE µ X +t 1,α/2 ŜE where t 1,α/2 is the value such that the area uder the t pdf, with 1 d.f. betwee t 1,α/2 ad + is equal to α/2. Hece, a cofidece iterval at the level 1 α for µ is X t 1,α/2 S ; X +t 1,α/2 S Cofidece iterval for µ with σ ukow: example For the bottlig compay example, if the value of σ is ot kow, the s = e t 9;0.025 = ad a 95% cofidece iterval for µ is that is so that ; ; Cofidece iterval for the mea of a o-ormal populatio X 1,...,X i.i.d. with EX i = µ ad VarX i = σ 2 ; the distributio of X i is ot ormal; for the cetral limit theorem the distributio of X is approximatively ormal; if oe uses the procedures described above to costruct a cofidece iterval for µ the omial cofidece level of the iterval is oly a approximatio of the true cofidece level ;

16 For the cetral limit theorem so that P π SE 1 α P Cofidece iterval for π N 0,1 where SE = z α/2 P π SE z α/2 π1 π = P z α/2 SE P π z α/2 SE = P P z α/2 SE π P +z α/2 SE = P P z α/2 SE π P +z α/2 SE Cofidece iterval for π: example For the clothig store chai example, a 95% cofidece iterval for π is SEˆπ; SEˆπ 40 so that π1 π SEˆπ = is estimated by ŜEˆπ = ˆπ1 ˆπ where ˆπ = x = ad oe obtais ; so that Sice π is always ukow, it is always ecessary to estimate the stadard error ; Example of decisio problem Problem: i the example of the bottlig compay, the quality cotrol departmet has to decide whether to stop the productio i order to revise the machie. Hypothesis: the expected mea quatity of liquid i the bottles is equal to oe liter. The stadard deviatio is assumed kow ad equal to σ = The decisio is based o a simple radom sample of = 10 bottles. Statistical hypotheses A decisioal problem i expressed by meas of two statistical hypotheses: the ull hypothesis H 0 the alterative hypothesis H 1 the two hypotheses cocer the value of a ukow populatio parameter, for istace µ, { H0 : µ = µ 0 H 1 : µ µ

17 Distributio of X uder H 0 If H 0 is true that is uder H 0 the distributio of the sample mea X has expected value equal to µ 0 = 1; has stadard error equal to SE = σ/ 10 = if X 1,...,X 10 is a ormally distributed i.i.d. sample tha also X follows a ormal distributio, otherwise the distributio of X is oly approximatively ormal by the cetral limit theorem. Observed value of the sample mea The observed value of the sample mea is x. x is almost surely differet form µ 0 = 1. uder H 0, the expected value of X is equal to µ 0 ad the differece betwee µ 0 ad x is uiquely due to the samplig error. HENCE THE SAMPLING ERROR IS x µ 0 that is observed value mius expected value Decisio rule The space of all possible sample meas is partitioed ito a Outcomes ad probabilities There are two possible states of the world ad two possible decisios. This leads to four possible outcomes. rejectio regio also said critical regio; orejectio regio. H 0 IS REJECTED H 0 TRUE H 0 FALSE Type I error α OK H 0 IS NOT REJECTED OK Type II error β The probability of the type I error is said sigificace level of the test ad ca be arbitrarily fixed typically 5%

18 Test statistic A test statistic is a fuctio of the sample, that ca be used to perform a hypothesis test. for the example cosidered, X is a valid test statistics, which is equivalet to the, more commo, z test statistic Hypothesis testig: example 5% sigificace level arbitrarily fixed; Z = X N0,1 Z = X µ 0 σ/ N0,1 the observed value of Z is z = = 2.055; the empirical evidece leads to the rejectio of H p-value approach to testig z test for µ with σ kow the p-value, also called observed level of sigificace is the probability of obtaiig a value of the test statistic more extreme tha the observed sample value, uder H 0. decisio rule: compare the p-value with α: X 1,...,X i.i.d. with distributio Nµ,σ 2 ; the value of σ is kow. p-value < α = reject H 0 p-value α = oreject H 0 for the example cosidered p-value=pz PZ = 0.04; p-value < 5% = statistically sigificat result. Hypotheses: test statistic: { H0 : µ = µ 0 H 1 :... Z = X µ 0 σ/ p-value < 1% = highly sigificat result. uder H 0 the test statistic Z has distributio N0;

19 z test: two-sided hypothesis z test: oe-sided hypothesis right H 1 : µ µ 0 H 1 : µ > µ 0 i this case i this case p value = PZ > z p value = PZ > z 3.5 z z z z test: oe-sided hypothesis left t test for µ with σ ukow H 1 : µ < µ 0 Hypotheses: i this case { H0 : µ = µ 0 H 1 : µ µ 0 p value = PZ < z test statistic: t = X µ 0 S/ 3.5 z p-value: PT 1 > t where T 1 follows a Studet s t distributio with 1 degrees of freedom

20 z test for π Test for a proportio Hypotheses: Null hypotheses: H 0 : π = π 0 ; Uder H 0 the samplig distributio of P is approximately ormal with expected value EP = π 0 ad stadard error π SEP = 0 1 π 0 Note that uder H 0 there are o ukow parameters. test statistic: { H0 : π = π 0 H 1 : π π 0 P π Z = 0 π 0 1 π 0 / P-value: PZ > z z test for π: example For the clothig store chai example, the hypotheses are { H0 : π = 0.25 H 1 : π > 0.25 Hece, uder H 0 the stadard error is SE = = so that z = = ad the p-value is PZ 0.37 = 0.36 ad the ull hypothesis caot be rejected. 265

### 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

### 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

### STATISTICAL 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

### 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

### 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.,

### Recall 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

### 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

### Chapter 1 (Definitions)

FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple

### 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

### 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

### 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

### Chapter 13, Part A Analysis of Variance and Experimental Design

Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

### Stat 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

### Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter

### 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

### Statistics 20: Final Exam Solutions Summer Session 2007

1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets

### 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

### STA 4032 Final Exam Formula Sheet

Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace

### Element sampling: Part 2

Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

### 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

### Confidence Level We want to estimate the true mean of a random variable X economically and with confidence.

Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio

### 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

### Statisticians 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

### Chapter 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

### y ij = µ + α i + ɛ ij,

STAT 4 ANOVA -Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i

### Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab

Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet

### Chapter 22: What is a Test of Significance?

Chapter 22: What is a Test of Sigificace? Thought Questio Assume that the statemet If it s Saturday, the it s the weeked is true. followig statemets will also be true? Which of the If it s the weeked,

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Asymptotic distribution of the first-stage F-statistic under weak IVs

November 6 Eco 59A WEAK INSTRUMENTS III Testig for Weak Istrumets From the results discussed i Weak Istrumets II we kow that at least i the case of a sigle edogeous regressor there are weak-idetificatio-robust

### Testing Statistical Hypotheses for Compare. Means with Vague Data

Iteratioal Mathematical Forum 5 o. 3 65-6 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach

### It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig

### (all terms are scalars).the minimization is clearer in sum notation:

7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1

### Singular Continuous Measures by Michael Pejic 5/14/10

Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

### Probability and Statistics

ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,

### EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics

### Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

### R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State

Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com

### Asymptotic Results for the Linear Regression Model

Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is

### Statistics. Chapter 10 Two-Sample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 10-1

Statistics Chapter 0 Two-Sample Tests Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0- Learig Objectives I this chapter, you lear How to use hypothesis testig for comparig the differece

### 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

### CH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions

CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample

### A LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!

A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

### Discrete probability distributions

Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

### G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

### Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3

No-Parametric Techiques Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3 Parametric vs. No-Parametric Parametric Based o Fuctios (e.g Normal Distributio) Uimodal Oly oe peak Ulikely real data cofies

### Topic 6 Sampling, hypothesis testing, and the central limit theorem

CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio

### Median and IQR The median is the value which divides the ordered data values in half.

STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

### NCSS Statistical Software. Tolerance Intervals

Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

### DISTRIBUTION LAW Okunev I.V.

1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated

### 3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear

### Section 14. Simple linear regression.

Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo

### Probability, Expectation Value and Uncertainty

Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

### PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

### Lesson 10: Limits and Continuity

www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

### STAT 203 Chapter 18 Sampling Distribution Models

STAT 203 Chapter 18 Samplig Distributio Models Populatio vs. sample, parameter vs. statistic Recall that a populatio cotais the etire collectio of idividuals that oe wats to study, ad a sample is a subset

### Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

### A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

### 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

### Introducing Sample Proportions

Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

### Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 70504-1010,

### Lecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables

CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze

### Output Analysis and Run-Length Control

IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

### Simple Linear Regression

Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio

### A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,

### 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

### Econ 371 Exam #1. Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement.

Eco 371 Exam #1 Multiple Choice (5 poits each): For each of the followig, select the sigle most appropriate optio to complete the statemet 1) The probability of a outcome a) is the umber of times that

### CTL.SC0x Supply Chain Analytics

CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos

### MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE

Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:

### 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

### Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

### A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)

Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig

### UCLA STAT 110B Applied Statistics for Engineering and the Sciences

UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,

### 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 ] / (

### Elementary Statistics

Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

### HOMEWORK 2 SOLUTIONS

HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k

### Solution of Final Exam : / Machine Learning

Solutio of Fial Exam : 10-701/15-781 Machie Learig Fall 2004 Dec. 12th 2004 Your Adrew ID i capital letters: Your full ame: There are 9 questios. Some of them are easy ad some are more difficult. So, if

### Maximum Likelihood Methods (Hogg Chapter Six)

Maximum Likelihood Methods Hogg Chapter ix TAT 406-0: Mathematical tatistics II prig emester 06 Cotets 0 Admiistrata Maximum Likelihood Estimatio. Maximum Likelihood Estimates............ Motivatio....................

### Advanced Engineering Mathematics Exercises on Module 4: Probability and Statistics

Advaced Egieerig Mathematics Eercises o Module 4: Probability ad Statistics. A survey of people i give regio showed that 5% drak regularly. The probability of death due to liver disease, give that a perso

### Lecture 9: Hierarchy Theorems

IAS/PCMI Summer Sessio 2000 Clay Mathematics Udergraduate Program Basic Course o Computatioal Complexity Lecture 9: Hierarchy Theorems David Mix Barrigto ad Alexis Maciel July 27, 2000 Most of this lecture

### ON POINTWISE BINOMIAL APPROXIMATION

Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

### Testing Statistical Hypotheses with Fuzzy Data

Iteratioal Joural of Statistics ad Systems ISS 973-675 Volume 6, umber 4 (), pp. 44-449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui

### Confidence 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:

### Sequences I. Chapter Introduction

Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

### MA131 - Analysis 1. Workbook 2 Sequences I

MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

### First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

### The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters

### Lecture 9: September 19

36-700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio Bias-Variace

### ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy

### Lecture 9: Independent Groups & Repeated Measures t-test

Brittay s ote 4/6/207 Lecture 9: Idepedet s & Repeated Measures t-test Review: Sigle Sample z-test Populatio (o-treatmet) Sample (treatmet) Need to kow mea ad stadard deviatio Problem with this? Sigle

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties

### Matrix Representation of Data in Experiment

Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y

### Lecture 10 October Minimaxity and least favorable prior sequences

STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least

### SALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS

SALES AND MARKETING Departmet MATHEMATICS d Semester Bivariate statistics LESSONS Olie documet: http://jff-dut-tc.weebly.com sectio DUT Maths S. IUT de Sait-Etiee Départemet TC J.F.Ferraris Math S StatVar

### Lecture 3 The Lebesgue Integral

Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

### Formulas FROM LECTURE 01 TO 22 W X. d n. fx f. Arslan Latif (mt ) & Mohsin Ali (mc ) Mean: Weighted Mean: Mean Deviation: Ungroup Data

1 Formulas FROM LECTURE 01 TO Mea: fx f Weighted Mea: X w W X i i Wi Mea Deviatio: Ugroup Data d M. D Group Data fi di M. D f d ( X X ) Coefficiet of Mea Deviatio: M. D Co-efficiet of M. D(for mea) Mea

### Notes on iteration and Newton s method. Iteration

Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

### IP Reference guide for integer programming formulations.

IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more