Sharpening the Boundaries of the Sequential Probability Ratio Test
|
|
- Andrew Doyle
- 6 years ago
- Views:
Transcription
1 Wester Ketuck Uiversit TopSCHOLAR Masters Theses & Specialist Projects Graduate School 5-22 Sharpeig the Boudaries of the Sequetial Probabilit Ratio Test Elizabeth Kratz Follow this ad additioal works at: Part of the Statistics ad Probabilit Commos Recommeded Citatio Kratz, Elizabeth, "Sharpeig the Boudaries of the Sequetial Probabilit Ratio Test" (22). Masters Theses & Specialist Projects. Paper This Thesis is brought to ou for free ad ope access b TopSCHOLAR. It has bee accepted for iclusio i Masters Theses & Specialist Projects b a authorized admiistrator of TopSCHOLAR. For more iformatio, please cotact topscholar@wku.edu.
2
3 SHARPENING THE BOUNDARIES OF THE SEQUENTIAL PROBABILITY RATIO TEST A Thesis Preseted to The Facult of the Departmet of Mathematics ad Computer Sciece Wester Ketuck Uiversit Bowlig Gree, Ketuck I Partial Fulfillmet Of the Requiremets for the Degree Master of Sciece B Elizabeth Kratz Ma 22
4
5 ACKNOWLEDGEMENTS I would like to express m deepest gratitude ad appreciatio to Dr. Melaie Auti for guidig me i this process. If it were ot for her, I would have ever pursued a iterest i statistics ad completed this thesis. She has ivested coutless hours assistig me i ever wa, ad oe of this would have happeed without her. I am trul grateful for her as ot ol as a advisor but also as a metor. Also, I would like to express m gratitude to Dr. David Neal ad Dr. Ngoc Ngue for givig of their valuable time to read ad give correctios to this thesis. Additioall, I would like to ackowledge Dr. Do Edwards for sharig i his itellectual propert. I must also ot forget the people who urged me to pursue m dreams. I am so thakful for Cor as he has stood beside me ad provided moral support eve whe he has bee usure of what he was supportig. Also, I am thakful for m Mom, Dad, Jammie, ad Jimm for their uedig suppl of ecouragemet, support, ad faith i me. iii
6 TABLE OF CONTENTS List of Figures List of Tables Abstract v vi vii Chapter : Itroductio Chapter 2: Sequetial Probabilit Ratio Test 3 2. Wald s Sequetial Probabilit Ratio Test Migoti s Expasio of SPRT Iglewicz s Expasio of SPRT McWilliams Expasio of SPRT 2.5 Igatova, Deutsch, ad Edwards Expasio of SPRT Chapter 3: A Method for Sharpeig the Boudaries 8 3. Creatig the Bouds Results for the New Bouds 24 Chapter 4: Simulatio Studies Simulatio Method Simulatio Results 35 Chapter 5: Coclusio ad Future Work 47 Appedix 49 A. Boudaries for T = 4 49 A.2 R Code 6 A.2. check.step() 6 A.2.2 create.bouds() 6 A.2.3 alpha.bouded() 64 A.2.4 errors.bouded() 66 A.2.5 sim.sprt() 69 A.2.6 Wald.bd() 73 A.2.7 Wald.step.bd() 74 Refereces 75 iv
7 LIST OF FIGURES Figure 2. Full Pascal s Triagle Example 2 Figure 2.2 Modified Pascal s Triagle Method 3 Figure 2.3 Wald Boudar Example 7 Figure 3. Stadard Hpothesis Test 9 Figure 4. Biomial T = Simulatio with π =. ad π =.7 42 Figure 4.2 Biomial T = Simulatio with π =. ad π =.9 43 Figure 4.3 Biomial T = Simulatio with π =.2 ad π =.8 44 Figure 4.4 Biomial T = Simulatio with π =.2 ad π =.9 45 Figure 4.5 Biomial T = Simulatio with π =.3 ad π =.9 46 v
8 LIST OF TABLES Table 3. Possible Sigle Boudaries for T = 4 2 Table 3.2 Total Number of Potetial Boudaries for T 24 Table 3.3 Biomial Results for T = 26 Table 3.4 Biomial Results for T = 2 27 Table 3.5 Biomial Results for T = 3 28 Table 3.6 Biomial Results for T = 4 29 Table 3.7 Hpergeometric Results for T = 3 Table 3.8 Hpergeometric Results for T = 2 3 Table 3.9 Hpergeometric Results for T = 3 3 Table 3. Hpergeometric Results for T = 4 3 Table 4. Hpergeometric Simulatio Results for T = 36 Table 4.2 Hpergeometric Simulatio Results for T = 2 36 Table 4.3 Hpergeometric Simulatio Results for T = 3 37 Table 4.4 Hpergeometric Simulatio Results for T = 4 37 Table 4.5 Biomial Simulatio Results for T = 38 Table 4.6 Biomial Simulatio Results for T = 2 39 Table 4.7 Biomial Simulatio Results for T = 3 39 Table 4.8 Biomial Simulatio Results for T = 4 4 vi
9 SHARPENING THE BOUNDARIES OF THE SEQUENTIAL PROBABILITY RATIO TEST Elizabeth Kratz Ma Pages Directed b: Dr. Melaie Auti, Dr. David Neal, ad Dr. Ngoc Ngue Departmet of Mathematics & Computer Sciece Wester Ketuck Uiversit I this thesis, we preset a itroductio to Wald s Sequetial Probabilit Ratio Test (SPRT) for biar outcomes. Previous researchers have ivestigated was to modif the stoppig boudaries that reduce the expected sample size for the test. I this research, we ivestigate was to further improve these boudaries. For a give maximum allowable sample size, we develop a method iteded to geerate all possible sets of boudaries. We the fid the oe set of boudaries that miimizes the maximum expected sample size while still preservig the omial error rates. Oce the satisfig boudaries have bee created, we preset the results of simulatio studies coducted o these boudaries as a meas for aalzig both the expected umber of observatios ad the amout of variabilit i the sample size required to make a decisio i the test. vii
10 Chapter : Itroductio I statistical aalsis we are cocered with data-drive decisio makig, ad oe aveue that we pursue to test cojectures is hpothesis testig. I hpothesis testig, we test H, the ull hpothesis, versus H, the alterative hpothesis. I a stadard hpothesis test, we take a sample of size, ad make a decisio based o the etire collectio of data. The sequetial probabilit ratio test (SPRT) istead requires that each item be sampled oe at a time. After each item, the researcher either makes a decisio i the hpothesis test or decides to cotiue samplig b selectig aother item. Thus, the SPRT ca result i a sigificatl smaller sample size tha a stadard hpothesis test. I practice, SPRT teds to be more efficiet tha stadard hpothesis testig ad is more cost effective i idustr. For example, SPRT has bee used i detectig Medicare fraud; istead of examiig ever case that a particular doctor turs i for Medicare to pa, the aalst looks at oe case at a time ad either makes a decisio that the case is or is ot fraudulet. From that, the ivestigator ma be able to make a decisio about the doctor without lookig at all of his cases. Thus, moe ad time are possibl saved because the testig will likel termiate before all cases have bee examied. Additioall, i qualit cotrol for idustr, istead of examiig a etire productio lie of goods, a compa could fid it more ecoomical to look at oe product at a time ad determie whether or ot the item is defective ad, i tur, make a decisio about the etire product lie before samplig the etire group of products. Thus,
11 compaies ca icrease profit b spedig less time ad moe o testig for defects i productio. The ca also more quickl idetif ad correct productio problems. For the purpose of this thesis, we are iterested i formulatig a possible method that ca reduce the expected umber of observatios ecessar to make a decisio i the sequetial probabilit ratio test. Some research i this area has bee coducted, but we set out to build upo ad improve the methods that alread exist. I Chapter 2, we first look at the origial SPRT developed b Wald ad the cotiue b lookig at expasios o SPRT b other researchers. Next, Chapter 3 will detail the process of creatig ew sets of boudaries as a exploratio i sharpeig the boudaries of the SPRT. I Chapter 4, we provide a descriptio of simulatio studies that were coducted as a wa to aalze the ew sets of boudaries. Fiall, we summarize the research i Chapter 5 ad examie some improvemets that could be made i the future. 2
12 Chapter 2: Sequetial Probabilit Ratio Test 2. Wald s Sequetial Probabilit Ratio Test I 947, Abraham Wald published a book etitled Sequetial Aalsis [6], i which he details the process for coductig the sequetial probabilit ratio test for qualit cotrol. We defie a radom variable X o the items beig ispected b lettig X = if the item observed is foud to be defective ad X = if the item observed is o-defective. Defie f(x,π) to be the probabilit distributio fuctio of the radom variable X where π is the proportio of defective items i a (possibl ifiite) collectio of size N. Whe cosiderig biar outcomes (defective vs. o-defective), the first successive observatios of x will be desigated as x, x 2,, x, where x i idicates whether the item is foud to be defective or o-defective. It is of iterest to test two competig values of π. We let H be the hpothesis that π = π ad H be the hpothesis that π = π, where π < π. Thus, f(x,π ) is the distributio fuctio of X whe the ull hpothesis is true, ad f(x,π ) is the distributio fuctio of X whe the alterative hpothesis is true. We let = x i i = be the total umber of defective items i the sampled items. For a positive value, the probabilit that a sample x,, x is i, i= observed is determied b the likelihood fuctios L(, π ) Π f ( x π ) is true ad L( π ) Π f ( x, ), i π i = = whe H = whe H is true. The sequetial probabilit ratio test (SPRT) for testig H agaist H is determied b the followig process. 3
13 Two positive costats A ad B are chose such that B < A. At each stage of the test ( th trial), compute Λ L = L (, π ) (, π ). If B Λ A, cotiue the test b takig aother observatio. If A < Λ, the test is eded, ad the ull hpothesis is rejected. If Λ < B, the test is eded, ad we fail to reject the ull hpothesis. For practicalit i computatio, l(λ ) is ofte used over Λ sice l(λ ) ca be writte as the sum of terms, i.e l( Λ ) f = l f ( x, π) ( x, π ) f l f ( x ), π ( x, π ) The i th term of the sum is deoted as z i. Thus, at each stage of the test ( th trial). z i i = the cumulative sum is computed. If l( B) zi l( A), i = cotiue the test b z i i = takig more observatios. If ( A) < l, the test is eded, ad H is rejected. If i = z i < l( B), the test is eded, ad H is ot rejected. Assumig that the test has ot termiated prior to a give step, we cosider a sample (x,, x ). We sa that the sample is of tpe if A < Λ, which leads to rejectio of H at step. We sa that the sample is of tpe 2 if Λ < B, which leads to failig to reject H at step. For a tpe sample, the probabilit of achievig such a sample uder H is at least A times as large as the probabilit of achievig such a sample uder H, as see b L, π ) A L(, ). Therefore, the probabilit that the ( π sequetial test will termiate with rejectio of H is also at least A times as large uder H as it is uder H. Whe H is true, the probabilit that the test will 4
14 termiate with rejectio of H is. This icorrect decisio is a Tpe I error, where is the Tpe I error rate. Whe H is true, the probabilit that the test will correctl termiate with rejectio of H is deoted as. Thus we obtai, L(, π ) A L(, π ) A A, (2.) which determies a upper limit for A to be. I a similar maer, a lower limit for B ca be derived. For a give tpe 2 sample (x,, x ), the probabilit of achievig such a sample uder H is at most B times as large as the probabilit of achievig such a sample uder H, L(, π) B L(, π ). Therefore, the probabilit that the sequetial test will termiate with failig to reject H is also at most B times as large uder H as it is uder H. Whe H is true, the probabilit that the test will correctl termiate without rejectio of H is deoted. Whe H is true, the probabilit that the test will icorrectl termiate with failure to reject H is. This icorrect decisio is referred to as a Tpe II error, where is the Tpe II error rate. Thus we obtai, L(, π ) B L(, π ) B( ) B, (2.2) which determies a lower limit for B to be,. Cosequetl, upper limits are obtaied for ad based o the above iequalities: 5
15 ad B. (2.3) A We must also ow cosider the opposite approach i which we are ot give the values of A ad B. Previousl, we chose a give A ad B ad determied the Tpe I ad Tpe II error rates. I practice, testig will be coducted uder predetermied maximum allowable error rates ( ad ), ad A ad B must istead be determied. We would like to test with give stregth (, ); so, deote A(, ) ad B(, ) as the values of A ad B which satisf the predetermied (, ). From the previousl metioed iequalities, (2.) implies A (, ), ad (2.2) implies (, ) B. Wald [6] proposed to set A = = a(, ), which is i fact greater tha or equal to the exact value of A(, ), ad B = b(, ) = which is less tha or equal to the exact value of B(, ). Wald discusses the cosequeces that the assiged combiatios of A ad B have o the error rates ad. Whe A is a value greater tha or equal to A(, ) ad B is equal to B(, ), the Tpe I error rate is less tha the omial, but the Tpe II error rate is greater tha. If A is equal to A(, ) ad B is a value less tha or equal to B(, ), the the Tpe II error rate is less tha the omial, but the Tpe I error rate is greater tha. If A is a value greater tha A(, ) ad B is a value less tha B(, ), the the effect o the Tpe I ad Tpe II error rates is uclear. Wald thus proposes to deote ad as the respective error rates whe A=a(, ) ad B=b(, ). 6
16 7 From above, ( ) =, ' ' a ad ( )., ' ' = b Thus, upper limits for ad ca be foud to be ' ad. ' Through some arithmetic, it ca be show that. ' ' + + Thus, we set ) ( = A ad ) ( = B. We ow cosider the case of samplig with replacemet (or samplig without replacemet from a populatio of ifiite size). Sice the probabilit of selectig a defective item is π for each item ad observatios are idepedet, each observatio ca be cosidered a Beroulli trial. Thus, Wald s boudaries ca be foud as follows. I the likelihood ratio Λ, the biomial desit ca be substituted for the likelihood fuctios, givig ( ) ( ) ( ) ( ).,, L L = = Λ π π π π π π. (2.4) Usig the criterio B Λ A preseted b Wald, ( ) ( ) ( ) ( ).,, A L L B = π π π π π π (2.5) Notice that istead of computig this ratio at each observatio, we ca solve for as follows:
17 8 ( ) ( ) ( ) ( ) ( ). ) ( ) ( l ) ( l ) l( ) ( ) ( l ) ( l ) l( l l ) ( ) ( l ) l( ) l( ) ( ) ( l ) l( + π π π π π π π π π π π π π π π π π π π π π π A B A B A B (2.6) Notice that is ow bouded above ad below b parallel lies. Thus, if the total umber of defectives at the curret positio is betwee or o these lies, is said to fall i the zoe of idifferece. Although values of ma occur here, samplig cotiues withi this regio, ad we therefore do ot refer to these results as possible outcomes of the test. We refer to values that are above the upper boudar lie as beig i the rejectio regio, for these are the outcomes that would lead to rejectio of the ull hpothesis. Though we ever actuall accept H whe values of are below the lower boud, istead of referrig to the regio as the zoe of failig to reject, for simplicit we call it the acceptace regio; values of i this regio are the outcomes that would lead to failig to reject the ull hpothesis. Wald [6] has show that the sequetial probabilit ratio test will evetuall ed i a decisio beig made with a probabilit of, but i practice, the sequetial test is trucated at a maximum allowable sample size, T. If a decisio has ot bee made prior to reachig the T observatio, samplig stops
18 with the T observatio. The a decisio is made based o the value of T. We sa that sequetial testig is a closed procedure because testig is limited to a maximum allowable size. Research o SPRT has cotiued sice its first itroductio b Wald, ad selected examples are preseted i the followig sectios of this chapter. 2.2 Migoti s Expasio of SPRT Researchers have also bee iterested i determiig the effects that the choices of the hpothesized values, π ad π, have o the required sample size. Migoti [3] discusses the sample size eeded to perform Wald s sequetial probabilit ratio test whe items are geerated b a process for which the results of the ispectios are correlated. It was show that for values of π ad π which are close together, the sample size is larger tha that which are foud to be ecessar whe usig values of π ad π that are more distat. Additioall, whe cosiderig correlatio (ρ) betwee the results of the observatios with respect to a biar respose variable, the sample size icreases as the value of the correlatio coefficiet ρ icreases. I fact, whe comparig the correspodig expected sample size for whe ρ = to whe.5 < ρ <.7, the expected sample size is three times higher; it is six times higher whe ρ > Iglewicz s Expasio of SPRT It is most commo for sequetial procedures to be compared i terms of the expected sample size, which is also kow as the average sample umber 9
19 (ASN). This tpe of compariso does ot cosider sample size variabilit ad is ofte biased i favor of tests with large sample-size variabilit. I terms of practicalit, large sample-size variabilit is costl because it ofte leads to a large umber of observatios beig required before the process termiates. Thus, Iglewicz [] proposes a alterative measure for the ASN, D( π ) N E = σ ( π ) ( π ) where N is the umber of required observatios for a fixed sample size test, π is the true value of the parameter, E( π) is the average sample umber, ad σ (π) is sample-size stadard deviatio. D(π) is a measure of the gai obtaied b usig a sequetial procedure i place of a fixed sample size procedure. Whe D(π) is large, the sequetial test is efficiet, but whe D(π) is small, the desig is ot., 2.4 McWilliams Expasio of SPRT Due to the lack of published research regardig trucatio o the SPRT, McWilliams [4] presets two case studies that ispect the ifluece that the choice of trucatio parameters have o test performace. He examies both the choice of the trucatio sample size ad the choice of the trucatio rejectio value R. Whe the test reaches the trucatio sample size, R idetifies how ma defectives are required for rejectio. Through the explaatio of the two case studies, both choices appear have a deep impact o test performace. McWilliams compares the ASN uder both the ull hpothesis, π = π, ad the alterative hpothesis, π = π, sice the value of the ASN will differ depedig
20 upo the true value of π i the populatio. Additioall, the author examies the amout of variabilit i required sample size ad the value of the error rates. McWilliams cocludes that adoptig a aggressive trucatio strateg results i a cosequeces i terms of the error probabilities ad the ASN. The author does ot geeralize his results but expresses that, from his fidigs, the performace of the test over various values of R eeds to be cosidered istead of a arbitrar choice of the rejectio value. 2.5 Igatova, Deutsch, ad Edwards Expasio of SPRT While cosiderig closed sequetial or multistage samplig from a lot of N items, the goal is to make coclusios about the proportio of defectives, π, i the sample. Istead of usig asmptotic results, Igatova, et. al. [2] show that exact iferece about the proportio of defectives ca be calculated usig curret statistical software. While cosiderig to be the umber of defectives at stage, a decisio is made whether to cotiue samplig or to ed testig b comparig to a set of boudaries. Sice, i practice, sequetial testig is trucated at T, the trucated SPRT is the sole focus for the authors. For a give outcome,, there are sequeces of defectives ad ( ) o-defectives that result i a total of defectives i trials. As a example, cosider the case of T =, show i Figure 2.. Each umber represeted i Figure 2. represets the umber of was to get (o the -axis) defectives i trials (o the x-axis); we
21 refer to this as a path-cout. Thus, we ca cosider the umber of paths to a specific poit (, ) o the graph. Arbitraril chose upper ad lower boudaries are show as a referece, but o modificatios i the path couts have bee made as a result of outcomes fallig outside of the boudaries. The urestricted path-couts are foud via Pascal s triagle. Figure 2.: Full Pascal s Triagle Example 2
22 The boudaries must ow be take ito accout to determie the actual umber of paths to. A observatio occurrig outside of the boudaries results i samplig to ed; thus, the paths extedig from that poit o loger exist. Based o a modified Pascal s triagle algorithm, [2] utilizes a path-cout method to determie the various paths that lead to specific termiatio poits, while staig withi the boudaries. A visual example of the path-cout method ca be see i Figure 2.2. Figure 2.2: Modified Pascal s Triagle Method 3
23 Note that the solid dots that appear i Figure 2.2 are outcomes for which a decisio is made ad samplig eds. There are ideed values of which exist betwee the lower ad upper bouds, but o decisio is made i the test ad samplig cotiues. Notice that poits (4,), (6,), (9,2), ad (,) all occur outside of the upper ad lower boudaries show. Thus, paths extedig from (4,), (6,), ad (9,2) do ot exist, ad the path-cout for successive poits are altered. For example, the path-cout to poit (5,) reduced from five i the Pascal s triagle example (Figure 2.) to four i the modified Pascal s triagle method; the path-cout to poit (7,2) reduced from 2 to 4. Additioall, otice that some of the paths to the outcomes o the trucatio boudar are also reduced. For example poits (,3) ad (,5) reduced from 2 to 62 ad 252 to 242, respectivel. Also, otice that ot all path-couts were altered. Igatova, et. al. cosider this method for both the biomial ad hpergeometric cases. The have crossed ito ew territor b examiig the hpergeometric case, as o other authors have pursued it [2]. Now cosider the case of testig without replacemet from a populatio of a fiite size N. Thus, Wald s boudaries ca be determied where is a hpergeometric radom variable. Similar to what was doe i the biomial case i (2.4), the hpergeometric desit ca be substituted for the likelihood fuctios, givig 4
24 5 = = = Λ D N D D N D N D N D N D N D D L D L ), ( ), (. (2.7) where D is the umber of defective items i the populatio uder H ad D is the umber of defectives i the populatio uder H. Usig the criterio B Λ A preseted b Wald, we have. A D N D D N D B < < (2.8) I the pursuit of trig to isolate i (2.8), we begi trig to simplif, givig ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).!!!!!!!!!!!!!!!!!!!! D N D D N A D D N D D N D D N D D N B D A D N D D N D D N D D N D B < + + < < + + < (2.9) Although it was attempted, isolatio of could ot be determied at this time. The authors created a fuctio etitled seqbi, usig the statistical software package R [5], to determie sequetial acceptace samplig for biar outcomes from a populatio of size N while samplig with or without replacemet. Based o Wald s SPRT method, a trucatio positio T ad upper ad lower boudaries are give as argumets to the fuctio. The authors determied that Wald s boudaries ca be represeted as a step boudar
25 because as icreases b oe, the value of stas the same (if that observatio is o-defective) or icreases b oe (if that observatio is defective). Samplig cotiues if is betwee or o the boudaries ad < T. Usig the boudar-modified Pascal algorithm, the seqbi fuctio detects all possible outcomes (ad the umber of paths possible to reach that outcome) where is less tha the lower boud or greater tha the upper boud (i.e., where the test does ot stop before reachig T ) or where is o the trucatio boudar. Additioall, the seqbi fuctio calculates the umber of defectives sampled ad calculates the probabilit distributio of all possible outcomes. We ow cosider a example of Wald s trucated SPRT for testig the ull hpothesis that π =.2 versus the alterative hpothesis that π =.8, with T =, =.5, ad =.. Usig A = ( )/ ad ( ) / as defied i sectio 2., we have A = 8, ad B.53. From (2.6), the lower boudar lie is give b l , ad the upper boudar lie is give b u , idicated b the dotted lies i Figure 2.3. The blue dashed lie ad red lie represet step-boudar represetatios of the Wald upper ad lower boudaries, respectivel. The poits i Figure 2.3 are the possible outcomes; otice that poits exist i both the rejectio regio ad acceptace regio. The poits that occur at = T = represet the outcomes that occur whe samplig has ot eded before reachig the trucatio boudar. It is ecessar that each of these poits be icluded i either the rejectio regio or the acceptace regio sice a decisio eeds to be made i the test. 6
26 Figure 2.3: Wald Boudar Example 7
27 Chapter 3: A Method for Sharpeig the Boudaries 3. Creatig the Bouds Based o the work previousl completed b others, the goal of this thesis is to determie if there are better boudaries available tha the oes which have bee curretl foud i [6] ad [2]. We would like to cosider all possible sets of boudaries that exist for a give T. We will the use certai criteria to determie which of these sets of boudaries is the best. We first cosider the set of boudaries that are equivalet to the stadard hpothesis test: the lower boudar is the lie =, ad the upper boudar is the lie =. I other words, the lower boud icludes successive observatios of all o-defective items ad the upper boud icludes successive observatios of all defective items. Observatios here are alwas i the zoe of idifferece, so the sample size will alwas reach the trucatio value of T. This gives boudaries that form a triagle, as ca be see i Figure 3. for the case whe T = 4. 8
28 Figure 3.: Stadard Hpothesis Test Startig with this set of triagle boudaries, we will fid all possible sets of boudaries for a give T. We the wat to determie all sets of upper ad lower boudaries that have both the Tpe I ad Tpe II error rates bouded. The we would like to examie uder various criteria which set of boudaries is ideed the best. For a give T, we must first geerate ever possible set of boudaries. Whe a sample is take at = (i.e., ol the first item is sampled), either zero defective items ca be observed (x = = ) or oe defective item is observed (x = = ); thus, ever upper ad lower boudar is restricted to begiig at either = or =. Next, it is required that the step size is o-decreasig ad of at most oe because as the umber of observatios icreases b oe, the value of the umber of defectives ( ) ca ol either sta the same or icrease 9
29 b oe. Additioall, the upper ad lower boudaries are created such that the ca touch oe aother but ma ot cross. To create all sets of possible boudaries, we first create all possible sigle boudaries. Oce this has bee doe, we combie all possible boudaries such that we have lower ad upper boudaries that do ot cross. To create all of the sigle boudaries, we first desigate the iitial boudar as (,,,,), a boudar of T zeros (i.e., all T sampled items are o-defective). For a idex i iitiall set equal to zero, we add to the T - i positio i the boudar, as we keep the bouds that are o-decreasig with a step-size of at most. After ca o loger be added to the T - i colum due to the step-size requiremet, i is icremeted b oe, ad the process cotiues of addig to the ext positio to the left util the step-size requiremet is o loger fulfilled, ad i is agai icremeted. The process cotiues successivel util the boudar that represets a sample of T defective items, (,2,3,, T ), is obtaied. Workig backwards, we the desigate that the boud begis as (,2,3,, T ) ad for a idex i iitiall set equal to zero, we subtract from the T - i positio i the boudar as we keep the bouds that are o-decreasig with a step-size of at most. After ca o loger be subtracted from the T - i positio due to the step-size requiremet, i is icremeted b oe, ad the process cotiues of subtractig from the ext positio to the left util the step-size requiremet is o loger fulfilled, ad i is agai icremeted. The process cotiues util the (,,,,) boudar is obtaied. We the combie all bouds that have bee foud ad delete a duplicate boudaries. Table 3. displas the boudaries 2
30 geerated for T = 4 via the above described process (before deletig duplicate boudaries). Table 3.: Possible Sigle Boudaries for T = 4 Buildig Up Buildig Dow We the combie this master set of boudaries ito sets of lower ad upper boudaries, keepig ol the boudaries which do ot cross oe aother. A R [5] fuctio called create.bds has bee created to implemet this procedure (see Appedix A.2.2). A example of all possible sets of boudaries for T = 4 ca be foud i Appedix A.. Oce all of the possible sets of boudaries are created, we are left to determie which set of boudaries is the best. I practical testig, ou will be give a maximum allowace for Tpe I ad Tpe II error rates. Thus, we must reduce the possible set of boudaries b first determiig which sets of boudaries preserve, the predetermied maximum allowable Tpe I error rate. To determie if is preserved, each set of upper ad lower T boudaries is cosidered idividuall. A give set of bouds is first evaluated usig the seqbi 2
31 fuctio. Usig this, we the determie which of the outcomes are i the rejectio regio, which are i the acceptace regio, ad which are o the trucatio boudar. The, assumig that the ull hpothesis is true (π = π ), the outcomes i the rejectio regio, if the exist, are used to calculate. For each outcome i the rejectio regio, we cosider the probabilit of that outcome occurrig, P (( )) out, out, assumig π = π where out ad out deote the umber of items sampled ad the umber of defectives, respectivel, for a give outcome. (( )) P, out out is foud b takig the product of the probabilit mass fuctio ad the proportio of paths staig withi the boudaries leadig to (, out out), as show b [2]. The, is calculated b takig the sum of (( )) P, out out for each outcome i the rejectio regio. If o outcomes appear i the rejectio regio, = iitiall. It is importat to ote that at this stage, the outcomes o the trucatio boudar are ot et cosidered. Oce has iitiall bee calculated for a give set of bouds, we determie if this iitial value of is ideed bouded b the pre-specified value. All sets of boudaries which preserve the give Tpe I error rate are retaied, ad the remaiig sets of boudaries are elimiated. A R [5] fuctio called alpha.bouded has bee writte to determie the sets of boudaries for which the Tpe I error rate is preserved (see Appedix A.2.3). From the remaiig sets of boudaries, we must the determie which of those sets of bouds also maitai, the predetermied maximum allowable Tpe II error rate. To determie if is maitaied, each set of upper ad lower 22
32 -bouded boudaries for T must be cosidered. First, a give set of - bouded bouds is evaluated usig the seqbi fuctio. Usig this, we the determie which of the outcomes are i the rejectio regio, which are i the acceptace regio, ad which are o the trucatio boudar. The, assumig that the ull hpothesis is true (π = π ), the outcomes i the rejectio regio, if the exist, are used to calculate as explaied above. If o outcomes appear i the rejectio regio, = iitiall. Now the outcomes that occur o the trucatio boudar eed to be cosidered, ad a decisio i the test must be made. Begiig with the outcome o the trucatio poit closest to the rejectio regio, we add that outcome to the rejectio regio as log as remais bouded b our desired sigificace level. Each successive outcome o the trucatio poit is added to the rejectio regio util is beod our desired boud. Thus, the remaiig trucatio outcomes are the added to the acceptace regio. Assumig that the alterative hpothesis is true (π = π ), is calculated usig the sum of P( ( )) for all of the ( ) out, out outcomes i the out, out acceptace regio. All sets of boudaries which preserve the give Tpe II error rate are retaied, ad the remaiig sets of boudaries are elimiated. A R [5] fuctio called errors.bouded has bee writte to determie the sets of boudaries for which both the Tpe I ad II error rates are preserved (see Appedix A.2.4. Oce the sets of boudaries that preserve both ad have bee collected, we begi trig to discover which set of boudaries is the best for give values of T, π, ad π. Oe wa to defie which set of boudaries is the 23
33 best is to idetif which set of boudaries miimizes the expected umber of observatios required to make a decisio i the hpothesis test. Sice the true value of π is ukow, we use a fie grid of π-values from to i steps of. to assist i determiig the expected umber of total observatios required to make a decisio i the test, deoted as E(). Usig the value of at each outcome, out, ad the probabilit of out over all outcomes, we are able to determie the expected umber of observatios to be E( ) = P( ). out (3.) out all out Oce E() is calculated usig π equal to each of the values, we determie the maximum expected umber of observatios, maxe()}, for each set of - ad -bouded boudaries. We the determie the best set of boudaries to be that which miimizes maxe()}. 3.2 Results for the New Bouds The method described i sectio 3. has bee coducted for T =, 2, 3, ad 4 i both the biomial ad hpergeometric cases with all combiatios of π =.,.2, ad.3 ad π =.,.2,.3,.4,.5,.6,.7,.8, ad.9. The determied total umber of possible sets of lower ad upper boudaries (disregardig coservatio of the error rates) is show i Table 3.2. Table 3.2: Total Number of Potetial Boudaries for T T Number of Boudar Sets 4, , , ,893 24
34 I the case of the biomial distributio, for each combiatio of T, π, ad π the origial Wald boudar has also bee calculated as a compariso to determie if this ew method is ideed a improvemet. R [5] fuctios writte to fid the Wald boudaries ca be foud i Appedices A.2.6 ad A.2.7. Because the likelihood ratio i the hpergeometric case caot be simplified to have parallel upper- ad lower-boudar lies as i the biomial case, determiig a closed form for the Wald upper ad lower boudaries has limitatios as see i (2.9). Thus, the Wald boudaries for the hpergeometric case have ot bee cosidered at the preset time. I Tables 3.3 through 3., the results for the best ew boudaries appear for both the biomial ad hpergeometric distributios. I the biomial case, the correspodig Wald results also appear. For each π ad π combiatio that has bee cosidered, the value of the miimized maxe()} is give alog with the correspodig values of ad. The combiatios of π ad π which appear as a shaded box are ot possible due to the coditio that π < π. The combiatios of π ad π which are empt idicate that o boudaries where both ad are bouded were foud. 25
35 π..2.3 New Wald New Wald New Wald Table 3.3: Biomial Results for T = π
36 π..2.3 New Wald New Wald New Wald Table 3.4: Biomial Results for T = 2 π
37 π..2.3 New Wald New Wald New Wald Table 3.5: Biomial Results for T = 3 π *
38 π..2.3 New Wald New Wald New Wald Table 3.6: Biomial Results for T = 4 π *
39 π..2.3 New New New Table 3.7: Hpergeometric Results for T = π π..2.3 New New New Table 3.8: Hpergeometric Results for T = 2 π *
40 π..2.3 New New New Table 3.9: Hpergeometric Results for T = 3 π * π..2.3 New New New Table 3.: Hpergeometric Results for T = 4 π *
41 I Tables 3.3 through 3., results marked with a asterisk (*) have multiple sets of boudaries with the same maxe()} ad value whe testig π vs. π. Thus, the result preseted i the table is the set of boudaries that has the smallest value of. Takig ote from the results preseted i Tables 3.3 through 3., it ca be see that for T =, the ew boudaries foud for testig π =. vs. π =.7, π =. vs. π =.9, π =.2 vs. π =.8, π =.2 vs. π =.9, ad π =.3 vs. π =.9 all reduced the miimized maxe()} whe compared to the Wald boudaries for the biomial case. For T =, the ew boudaries foud for testig π =. vs. π =.7 reduces the expected umber of observatios b.6875, which was greatest differece observed. It was ot util close to the ed of coductig this research that the Wald boudaries were calculated ad aalzed. Ol after aalzig this summar iformatio did we otice a discrepac betwee the Wald boudaries ad the ew boudaries. Because we set out to determie all sets of possible boudaries for a give T, the Wald boudaries should have bee icluded i our possible set of boudaries with both ad bouded. Thus, the ew best boudaries should be o worse tha the Wald boudaries. Whe aalzig the results, we should have see that if the miimized maxe()} of the ew boudaries was ot smaller tha the maxe()} of the Wald boudaries, the the miimized maxe()} should have at most bee equal to that of the Wald boudaries. 32
42 After retracig our steps, it was discovered that although our R fuctios correctl reduce a give collectio of lower/upper boudaries to those for which both ad are bouded, the process i which we geerated all possible boudaries was ot exhaustive for all T. The idea of creatig all possible bouds for T uder the coditios that the lower ad upper bouds must both begi at zero or oe, the lower ad upper bouds must have a o-decreasig step size of at most oe, ad the lower ad upper bouds ma touch oe aother but ot cross is correct. Thus, we discovered that ot all possible boudaries uder the three coditios were geerated usig the algorithm that we created; ol a subset of the possible boudaries was geerated. Ufortuatel, as T grows larger, it appears that more ad more boudaries were missed. Whe creatig the method for geeratig all possible boudaries for T, we tested the method b had for several small T values ad icorrectl assumed it would work for all T values. Sice T = is the smallest T value icluded i this research, we were able to fid ew boudaries with smaller miimized maxe()} tha their correspodig Wald boudaries because eough of the possible boudaries were geerated. Thus, we are optimistic that this method actuall would sharpe ma more of the boudaries for the SPRT if all possible boudaries are geerated. 33
43 Chapter 4: Simulatio Studies 4. Simulatio Method To test the performace of the boudaries, a simulatio stud was performed for each set of ew T -boudaries that miimized the maxe()} for a specified π so that we could observe the value of the required sample size ad the decisio made i each simulatio. Because we have two values beig tested of the kow ukow parameter π, π = π ad π = π, we performed each simulatio stud for a set of bouds uder both assumptios idividuall, similar to what was doe i [4]. For each set of lower ad upper bouds which miimize the maxe()}, the rejectio ad acceptace regios are determied ad decisios are made regardig the outcomes o the trucatio boudar. Whe samplig from a ifiite populatio (or samplig with replacemet), sample data is created with the specified probabilit of success, π = π or π = π, b usig the biomial radom variable geerator i R with T trials [5]. The total umber of defectives at stage is calculated for T. Whe samplig without replacemet, a populatio with a specified size of, is created with the correct proportio of defective ad o-defective items. From the created populatio we radoml sample T items from the populatio without replacemet (resultig i a hpergeometric radom variable), ad the umber of defectives at stage is calculated for T. The SPRT is the performig usig this geerated data ad the give set of boudaries. For each simulatio settig (a give value of π, 34
44 a give set of boudaries, ad a give distributio),, samples were simulated ad aalzed with SPRT. For each set of ew boudaries that were determied to miimized the maxe()}, simulatios were ru for both π = π ad π = π. We also performed simulatios for the boudaries where the maxe()} was withi oe of the miimized maxe()}. Additioall, whe samplig with replacemet, the Wald boudaries were also studied via simulatio. Sice the simulatios were coducted before the results were recogized as icomplete, 53 Wald boudaries, 26 miimized maxe()} boudaries, ad,26 boudaries with a maxe()} withi oe of the miimized maxe()} were aalzed. The R [5] fuctio created for the simulatio stud ca be foud i Appedix A Simulatio Results Although we kow that we do ot have all possible boudaries, we also do ot have the Wald boudaries with which to compare i the hpergeometric case. Thus, for each combiatio of π ad π we preset maxe()} (calculated as described i sectio 3.), the simulated average required umber of observatios whe π = π, deoted as Ê ( ) umber of observatios whe π = π, deoted as ˆ ( ), ad the simulated average required E. These results ca be see i Tables 4. through 4.4. For maxe()} we otice that for π =., as π icreases, the value of maxe()} either stas the same or decreases; this is cosistet with the results foud i [3]. Cosiderig π =.2 ad the lowest satisfig π value, maxe()} starts at a higher value tha the lowest 35
45 combiatio of π =. ad π, but the patter does cotiue that as π ad π become more distat, the value of maxe()} either decreases or stas the same. However, i simulatio, this patter also came ver close to holdig true for Ê ( ) ad ˆ ( ) E. If we had icreased the umber of simulatios, it is possible that the patter would have alwas held for Ê ( ) ad ( ) Ê. Table 4.: Hpergeometric Simulatio Results for T = E } Ê π π max ( ) Ê ( ) ( ) Table 4.2: Hpergeometric Simulatio Results for T = 2 E } Ê π π max ( ) Ê ( ) ( ) *
46 Table 4.3: Hpergeometric Simulatio Results for T = 3 E } Ê π π max ( ) Ê ( ) ( ) * Table 4.4: Hpergeometric Simulatio Results for T = 4 E } Ê π π max ( ) Ê ( ) ( ) *
47 I Tables 4.2 through 4.4, results marked with a asterisk (*) i the maxe()} colum have multiple sets of boudaries with the same maxe()} whe testig π vs. π. Thus, the result preseted i the table is for oe such set of boudaries. Sice we had the available Wald boudaries for compariso i the biomial case, we also cosidered maxe()}, Ê ( ), ad Ê ( ) for each combiatio of π ad π usig the Wald boudaries, as see i Tables 4.5 through 4.8. I Tables 4.7 ad 4.8, results marked with a asterisk (*) i the maxe()} colum have multiple sets of boudaries with the same maxe()} whe testig π vs. π. Thus, the result preseted i the table is for oe such set of boudaries. Table 4.5: Biomial Simulatio Results for T = Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald Ê ( ) 38
48 Table 4.6: Biomial Simulatio Results for T = 2 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald Ê ( ) Table 4.7: Biomial Simulatio Results for T = 3 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald * Ê ( ) 39
49 Table 4.8: Biomial Simulatio Results for T = 4 Wald Wald E } Ê Ê ( ) max E π π max ( ) ( )..2.3 ( )} Ê ( ) Wald * Ê ( ) We observed that the same patter described i [3] was exhibited i the biomial case for both the ew boudaries ad the Wald boudaries, as it was i the hpergeometric case. After closer observatio, we also foud that for a few of the ew boudaries, Ê ( ), ad Ê ( ) were smaller tha that of their Wald couterparts. These istaces are highlighted i Tables 4.5 through 4.8. To be exact, Ê ( ) was foud to be better o four occurreces, ad Ê ( ) was foud to be better o three occurreces, all occurrig whe T =. These istaces correspod to the ew boudaries that were foud to be better tha the Wald boudaries, described i sectio 3.2. For a additioal qualificatio of what could be cosidered as the best boudar, we have also cosidered the boudaries that i simulatio miimize 4
50 the variabilit i required sample size, similar to []. Sice ol five ew sets of boudaries were foud to have miimized maxe()} i compariso to the Wald boudaries, we ol preset further results for the five cases where a improvemet was foud. However, ma more simulatio results are available from the author. Figures 4. through 4.5 detail the results for the combiatios of π ad π for which a improvemet i maximum expected sample size was foud (with the ew boudaries labeled as Mi ). For each set of boudaries, the horizotal dotted lie exteds oe stadard deviatio above ad below Ê ( ), which is idicated b the ope circle. Similarl, the solid horizotal lie exteds oe stadard deviatio above ad below Ê ( ), which is idicated b the ope circle. Additioall, the maxe()} is preseted as a solid circle. Sice we also coducted simulatios for the sets of boudaries whose maxe()} is withi oe of the miimized maxe()} value, we preset results for those observatios as well. These boudaries are labeled as Mi + i the figures, ad the subscript serves as a sigifig idex for each cosidered set of boudaries for the hpotheses beig tested. 4
51 Figure 4.: Biomial T = Simulatio with π =. ad π =.7 I Figure 4., we otice that uder the ull hpothesis (π =.), based upo the simulatio performed, all ew boudaries cosidered ted to require a smaller sample size to make a decisio i the test tha Wald s boudaries; however, less variatio is observed i the ew boudaries uder the alterative hpothesis (π =.7). To the cotrar, uder the alterative hpothesis, the use of Wald s boudaries teds to require a smaller sample size tha the ew boudaries (except for the ew best boudar), ad the ull case exhibits less 42
52 variatio for the Wald boudaries. Additioall, all ew maxe()} (except for Mi + ) are less tha maxe()} for Wald. Figure 4.2 demostrates that although maxe()} has bee reduced with the ew boudaries, uder simulatio Wald s boudaries result i more cosistec i that it has small variatio uder both the ull (π =.) ad alterative hpotheses (π =.9). I simulatio usig the ew best bouds ad the bouds whose maxe()} are withi oe, uder the ull hpothesis all but oe (Mi + 7 ) has less variatio tha uder Wald s simulatio. Figure 4.2: Biomial T = Simulatio with π =. ad π =.9 43
53 Figure 4.3 shows that although maxe()} is reduced, Wald appears agai to be more cosistet uder both the ull (π =.2) ad alterative (π =.8) hpotheses. Also, maxe()} for the ew bouds is the ol maxe()} value which is less tha Wald s. Figures 4.4 ad 4.5 demostrate that uder the alterative hpothesis (π =.9), much more variabilit is show i the ew best bouds tha i Wald s boudaries. Figure 4.3: Biomial T = Simulatio with π =.2 ad π =.8 44
A statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9
Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I
More information6 Sample Size Calculations
6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig
More informationPower and Type II Error
Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error
More informationRecursive Computations for Discrete Random Variables
Recursive Computatios for Discrete Radom Variables Ofte times, oe sees a problem stated like this: A machie is shut dow for repairs is a radom sample of 100 items selected from the dail output of the machie
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationA 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
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationORF 245 Fundamentals of Engineering Statistics. Midterm Exam 2
Priceto Uiversit Departmet of Operatios Research ad Fiacial Egieerig ORF 45 Fudametals of Egieerig Statistics Midterm Eam April 17, 009 :00am-:50am PLEASE DO NOT TURN THIS PAGE AND START THE EXAM UNTIL
More informationTests of Hypotheses Based on a Single Sample (Devore Chapter Eight)
Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationBecause it tests for differences between multiple pairs of means in one test, it is called an omnibus test.
Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationGG313 GEOLOGICAL DATA ANALYSIS
GG313 GEOLOGICAL DATA ANALYSIS 1 Testig Hypothesis GG313 GEOLOGICAL DATA ANALYSIS LECTURE NOTES PAUL WESSEL SECTION TESTING OF HYPOTHESES Much of statistics is cocered with testig hypothesis agaist data
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationClass 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2
More informationChapter 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
More information[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:
PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationTMA4245 Statistics. Corrected 30 May and 4 June Norwegian University of Science and Technology Department of Mathematical Sciences.
Norwegia Uiversity of Sciece ad Techology Departmet of Mathematical Scieces Corrected 3 May ad 4 Jue Solutios TMA445 Statistics Saturday 6 May 9: 3: Problem Sow desity a The probability is.9.5 6x x dx
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationAsymptotic 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
More informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More informationIP 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
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationt distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference
EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More informationSequences 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 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More information1 Review of Probability & Statistics
1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5
More informationHYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018
HYPOTHESIS TESTS FOR ONE POPULATION MEAN WORKSHEET MTH 1210, FALL 2018 We are resposible for 2 types of hypothesis tests that produce ifereces about the ukow populatio mea, µ, each of which has 3 possible
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationOPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES
OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES Peter M. Maurer Why Hashig is θ(). As i biary search, hashig assumes that keys are stored i a array which is idexed by a iteger. However, hashig attempts to bypass
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationPower Comparison of Some Goodness-of-fit Tests
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 7-6-2016 Power Compariso of Some Goodess-of-fit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750
More informationCONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION
CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION Petros Maravelakis, Joh Paaretos ad Stelios Psarakis Departmet of Statistics Athes Uiversity of Ecoomics ad Busiess 76 Patisio St., 4 34, Athes, GREECE. Itroductio
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More information2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2
Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationINF Introduction to classifiction Anne Solberg Based on Chapter 2 ( ) in Duda and Hart: Pattern Classification
INF 4300 90 Itroductio to classifictio Ae Solberg ae@ifiuioo Based o Chapter -6 i Duda ad Hart: atter Classificatio 90 INF 4300 Madator proect Mai task: classificatio You must implemet a classificatio
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationMath 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency
Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the
More information- E < p. ˆ p q ˆ E = q ˆ = 1 - p ˆ = sample proportion of x failures in a sample size of n. where. x n sample proportion. population proportion
1 Chapter 7 ad 8 Review for Exam Chapter 7 Estimates ad Sample Sizes 2 Defiitio Cofidece Iterval (or Iterval Estimate) a rage (or a iterval) of values used to estimate the true value of the populatio parameter
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSampling Distributions, Z-Tests, Power
Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationSection 6.4: Series. Section 6.4 Series 413
ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationControl Charts for Mean for Non-Normally Correlated Data
Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies
More information( ) = p and P( i = b) = q.
MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationChapter 12 - Quality Cotrol Example: The process of llig 12 ouce cas of Dr. Pepper is beig moitored. The compay does ot wat to uderll the cas. Hece, a target llig rate of 12.1-12.5 ouces was established.
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationMA238 Assignment 4 Solutions (part a)
(i) Sigle sample tests. Questio. MA38 Assigmet 4 Solutios (part a) (a) (b) (c) H 0 : = 50 sq. ft H A : < 50 sq. ft H 0 : = 3 mpg H A : > 3 mpg H 0 : = 5 mm H A : 5mm Questio. (i) What are the ull ad alterative
More informationU8L1: Sec Equations of Lines in R 2
MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationSTAT431 Review. X = n. n )
STAT43 Review I. Results related to ormal distributio Expected value ad variace. (a) E(aXbY) = aex bey, Var(aXbY) = a VarX b VarY provided X ad Y are idepedet. Normal distributios: (a) Z N(, ) (b) X N(µ,
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationComparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading
Topic 15 - Two Sample Iferece I STAT 511 Professor Bruce Craig Comparig Two Populatios Research ofte ivolves the compariso of two or more samples from differet populatios Graphical summaries provide visual
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationChapter 13: Tests of Hypothesis Section 13.1 Introduction
Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed
More informationLecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS
Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationResponse Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable
Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationTopic 18: Composite Hypotheses
Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationLESSON 20: HYPOTHESIS TESTING
LESSN 20: YPTESIS TESTING utlie ypothesis testig Tests for the mea Tests for the proportio 1 YPTESIS TESTING TE CNTEXT Example 1: supervisor of a productio lie wats to determie if the productio time of
More informationChapter 11: Asking and Answering Questions About the Difference of Two Proportions
Chapter 11: Askig ad Aswerig Questios About the Differece of Two Proportios These otes reflect material from our text, Statistics, Learig from Data, First Editio, by Roxy Peck, published by CENGAGE Learig,
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More information1036: Probability & Statistics
036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso
More informationA Confidence Interval for μ
INFERENCES ABOUT μ Oe of the major objectives of statistics is to make ifereces about the distributio of the elemets i a populatio based o iformatio cotaied i a sample. Numerical summaries that characterize
More information