Statistical inference: example 1. Inferential Statistics
|
|
- Ella Fleming
- 7 years ago
- Views:
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
5 The sample variace The chi-squared distributio The sample variace is defied as S 2 = 1 X i X 2 1 i=1 if X i Nµ,σ 2 the 1S 2 σ 2 χ 2 1 Let Z 1,...,Z r be i.i.d. radom variables with distributio N0;1; the radom variable X = Z Z2 r is said to follow a CHI-SQUARED distributio with r degrees of freedom d.f.; we write X χ 2 r; X ad S 2 are idepedet. EX = r ad VarX = 2r Coutig problems The variable X is biary, i.e. it takes oly two possible values; for istace success ad failure ; the radom variable X takes values 1 success ad 0 failure; the parameter of iterest is π, the proportio of uits i the populatio with value 1 ; Formally, X Beroulliπ so that EX = π ad VarX = π1 π. The sample proportio 1 Simple radom sample X 1,...,X ; the variable X takes two values: 0 ad 1, ad the sample proportio is a special case of sample mea i=1 X i P = the observed value of P is p
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 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: p-value < α = reject H 0 p-value α = oreject H 0 z test for µ with σ kow X 1,...,X i.i.d. with distributio Nµ,σ 2 ; Hypotheses: { H0 : µ = µ 0 H 1 :... for the example cosidered p-value=pz PZ = 0.04; test statistic: Z = X µ 0 σ/ p-value 5% = statistically sigificat result. 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 z 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
One-sample test of proportions
Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More informationZ-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about
More informationPractice Problems for Test 3
Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationSTATISTICAL METHODS FOR BUSINESS
STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING 7.1.- Distributios associated with the samplig process. 7.2.- Iferetial processes ad relevat distributios.
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationBASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)
BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationChapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationConfidence Intervals
Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationConfidence intervals and hypothesis tests
Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More information15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011
15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationTopic 5: Confidence Intervals (Chapter 9)
Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all
More informationSTA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error
STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More informationA Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:
A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio
More informationResearch Method (I) --Knowledge on Sampling (Simple Random Sampling)
Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationQuadrat Sampling in Population Ecology
Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More information, a Wishart distribution with n -1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationMann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationConfidence Intervals for Linear Regression Slope
Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationLECTURE 13: Cross-validation
LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationUnit 8: Inference for Proportions. Chapters 8 & 9 in IPS
Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationTHE TWO-VARIABLE LINEAR REGRESSION MODEL
THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationMulti-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu
Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio
More informationCOMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationOur aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationLecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationPresent Values, Investment Returns and Discount Rates
Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationCONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION
www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha
More informationParametric (theoretical) probability distributions. (Wilks, Ch. 4) Discrete distributions: (e.g., yes/no; above normal, normal, below normal)
6 Parametric (theoretical) probability distributios. (Wilks, Ch. 4) Note: parametric: assume a theoretical distributio (e.g., Gauss) No-parametric: o assumptio made about the distributio Advatages of assumig
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationOMG! Excessive Texting Tied to Risky Teen Behaviors
BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationhp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationPlug-in martingales for testing exchangeability on-line
Plug-i martigales for testig exchageability o-lie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information