Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 The Blood Testig Problem Suose that you have a large oulatio that you wish to test for a certai characteristic i their blood or urie (for examle, testig all CAA athletes for steroid use or all US military ersoel for a articular disease) Each test will be either ositive or egative Sice the umber of idividuals to be tested is quite large, we ca exect that the cost of testig will also be large How ca we reduce the umber of tests eeded ad thereby reduce the costs? If the urie could be ooled by uttig a ortio of, say, 0 samles together ad the testig the ooled samle, the umber of tests might be reduced If the ooled samle is egative, the all the idividuals i the ool are egative, ad we have checed 0 eole with oe test If, however, the ooled samle is ositive, we ow oly that at least oe of the idividuals i the samle will test ositive Each member of the samle must the be retested idividually ad a total of tests will be ecessary to do the job The larger the grou size, the more we ca elimiate with oe test, but the more liely the grou is to test ositive Would oolig 5 secime be better tha oolig 0? What is the relatioshi betwee the robability of a idividual testig ositive ad the grou size that miimizes the total umber of tests required? Certaily, we aticiate that the larger the robability of a idividual testig ositive the smaller the grou size, while the smaller the robability, the larger the grou size required Problem Statemet You have a large oulatio () that you wish to test for a certai characteristic i their blood Each test will be either ositive or egative Sice the umber of idividuals to be tested is quite large, you wish to reduce the umber of tests eeded to scree everyoe ad thereby reduce the costs If the blood could be ooled by uttig samles together ad the testig the ooled samle, the umber of tests required might be reduced What is the relatioshi betwee the robability of a idividual testig ositive () ad the grou size () that miimizes the total umber of tests required? Use your solutio to determie the umber of tests required to fid 00 idividuals who will test ositive i a oulatio of,000,000 The Basic Model A essetial asect to develoig ay model is to cosider the simlest case that embodies the essece of the roblem For the grou testig roblem, this is a solutio that uses oly oe grou test, ad the tests everyoe remaiig idividually If the studets caot solve this roblem, they will ot be able to solve a more ivolved model that is erhas more realistic Further, the solutio to the simlest situatio ofte is helful i arrivig at a more geeral solutio What follows is the aroach tae by most studet grous Sice there are eole to be tested i grous of size, the iitial umber of tests eeded to test these grou is The robability of testig ositive is, so we exect that eole will test ositive With the worst case assumtio that exactly oe erso i each grou will test ositive, this meas that of the grous will test ositive, ad eole remai to be tested The umber of tests eeded for this testig rotocol is give by Daiel J Teague C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 T( ) = + I the worst case, if ay grou tests ositive, all members of the grou will test ositive So the umber of tests eeded to test usig oe grou test ad the testig everyoe remaiig idividually is T = + = + Sice the factor roduces a vertical stretch i the grah of T, the value of that miimizes the umber of tests will ot be affected by, so we set = for coveiece For a give value of, we ca determie the grou size which miimizes the umber of tests, ad therefore the costs, by usig a grahig calculator to zoom ad trace For, examle, if = 00, we ca fid the miimum value o the grah of T = + ( 00) as show i Figure below Figure : rahs of T = + for = 05 ad = 00 Reeatig the rocess for differet values of, we geerate the table below: 05 00 05 00 005 00 00 0005 000 00005 00000 0 58 6 7 577 00 6 7 000 By usig techiques of data aalysis o the scatterlot for this data, we ca create a model relatig the best grou size to the robability Figure : Scatterlot of Best rou Size vs Probability Figure : Log-log Re-exressio to Liearize Data Daiel J Teague C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 By re-exressig the data with a log-log lot, we liearize the data The least-squares equatio is l ( ) = 000005 05 l ( ), so = If =, the total umber of tests eeded is T = + = + = I our examle, 00 idividuals testig ositive ca be foud i a oulatio of oe millio eole i aroximately 0,000 tests Further, if, the, ad it is couterroductive to test i grous Refiig ad Imrovig the Model Of course, there is o reaso to re-test everyoe idividually Sice is ideedet of, we could retest all of the remaiig after the first grou test i similar grous We already ow that = However, sice we have already elimiated a large umber of eole i the first hase of testig, the value of will be much larger for the secod grou test There are eole that we exect to test ositive ad eole remaiig to be retested The robability of testig ositive i the secod roud is = = So the ext test should be doe with = = Cotiuig i this fashio, we fid the grou sizes to be First rouig ew Probability Secod rouig ew Probability Third rouig 8 ew Probability 8 th rouig ew Probability Sto grouig whe the ew robability is greater tha 05 Whe do you sto grouig ad test everyoe idividually? We wat to ow for what l( ) is Solvig for, we fid that = l l() l() The iterated model just created wors quite well, reducig the umber of tests dramatically, eve though i ractice we violate the assumtio uo which the solutio was based I the examle of fidig 00 ositive idividuals i a oulatio of,000,000, testig i grous of 00, 0, ad, ad the idividually requires oly,6 tests However, this is clearly ot a otimal solutio I creatig the model, we assumed that we would grou oly Daiel J Teague C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 oce ad the retest idividually The grou size = was determied o the basis of that assumtio Is it ossible to determie the umber of tests eeded by taig the additioal grou tests ito accout? A secod model exteds this iitial solutio The model just created wors well, reducig the umber of tests dramatically I the examle of fidig 00 ositive idividuals i a oulatio of,000,000, testig i grous of 00, 0, ad, ad the idividually requires oly,6 tests Roud of Testig umber to Test,000,000 0,000,000 00 rou Size 00 0 umber of Tests 0,000,000 00 umber to Retest 00(00) 00(0) 00() 0 Calculus Solutio: This is a straight-forward calculus roblem for first semester studets We have the same model T = + To determie the best grou size, we differetiate with resect to dt = +, d ad fid the value of that maes this derivative zero If dt = 0, the = The rest of the roblem roceeds as before, ad we fid that,6 d tests are eeded to fid 00 out of,000,000 This is a large reductio, but we ca do eve better! The Multile rou, Iterated Model As with the iitial solutio, the startig oit is with the simlest model that cotais the essece of the roblem I this case, it is a model that allows for two grou tests ad the testig everyoe remaiig idividually So, T = + + P It seems that T is a fuctio of two variables, ad, which is beyod the scoe of a itroductory course i calculus Is ossible to rewrite this as a sigle variable roblem? A few hits are geerally i order at this oit With some ecouragemet, studets will realize that they ow the solutio to the last art of the roblem, the roblem of miimizig the umber of tests with oe test ad the testig everyoe remaiig idividually Recall that this grou size was ideedet of the umber beig tested So, i fact, Daiel J Teague C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 we ow =, where is the robability of testig ositive after the first test But is just the exected umber testig ositive divided by the total umber i the reset oulatio So = = Substitutig, we ow that = The total umber of tests ca ow be writte as a fuctio of the sigle variable T( ) = + The dt = + / d ad elemetary calculus shows that the otimum value for is / If two grouigs are used, the sizes of the grous should be / ad / Extedig the grouig to three cotiues the atter If we ow that = ( ) / ad ( ) / Rewritig, we fid that T = + + + P, / =, with = So = / ( ) = + T ad / = Agai, elemetary calculus shows that the otimum grou sizes are /, /, ad / Reeatig the aalysis with four grous geerates the otimum grou sizes 5 / 5, / 5, / 5, ad / I geeral, if a total of grouigs are used, the grou sizes are give by = ( ) + + +, + + with the th grou of size o grou of studets has yet verified this geeralized grouig, but they lace their faith i the atter geerated from usig,,,, ad 5 grous Provig the geeral case seems beyod their reset abilities Based o the clear atter develoed by iitial secific cases, studets argue the total umber of tests required with grouigs is Daiel J Teague 5 C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 T / = + What umber of grouigs is otimum for a give iitial robability? If we cosider T as a fuctio of, we fid that Differetiatig, we fid that + T ( ) = ( + ) dt l( ) + = + d + The otimal umber of grouigs is = l( ) Recall that we are igorig ay o-iteger asects of the roblem The total umber of tests required is give by ( l( )) T = e If = 0 000, this is a reductio by a factor of 00 If 00 out of,000,000 had the sought for characteristic, they could be foud i aroud,500 tests Fidig the Otimal rou Size With this value of, we ca also determie the otimum size of the th grou We ow that ( ) +, with = l( ), so the th grou should size should be l( ) l( ) While most grous leave their exressio for the th grou size i this form, be simlified l( ) l( ) l ( ) l( ) l( ) l If, the l ( ) = l ( ) ad l = l( ) l = l or So ( ) ( ) l( ) l( ) ca = e Studets are always surrised to see e show u i the solutio Of course, while this theoretical result is leasig, it may ot be realizable, for = may be too may secime to hadle e i a sigle grou This roblem offers may imortat teachig oits both about mathematical modelig ad about calculus The roblem of imrovig the iitial solutio by violatig the assumtios Daiel J Teague 6 C School of Sciece & Mathematics
Teachig Cotemorary Mathematics Coferece Feb 6-7, 00 creates a iterestig discussio about mathematical theory ad ractice, ad the imortace of a good aroximate solutio over a ideal urealizable oe The imortace of iteratig the model ad refiig the solutio based o rior wor is clear ad covicig i this settig o studets have foud the more sohisticated multile grou solutio without first worig through the sigle grou solutio ad beig dissatisfied with it Also, the imortace of cosiderig "what questio does this ew solutio as?" is see i several laces We obtai a solutio, ad immediately use it to further the roblem First, we cosider T as a fuctio of, the as a fuctio of The coversatios surroudig the solutio ad i the rocess of solvig this roblem ecourage essetial asects of modelig Fial ote The Precalculus modelig techique will also wor o the average case sceario I this situatio, the total umber of tests is the sum of the iitial umber of tests ad all the retests ( ( ) ), so the model is ( ) ( ( ) ) ( ) T = + = + You may wat to verify that calculus is absolutely o hel here! However, we ca create a table for ad best as before ad fit a model to the data The result is = + REFERECES Dilwy Edwards ad Mie Hamso, uide of Mathematical Modelig, CRC Mathematical uides, CRC Press, Boca Rato, Florida, 989, 99-08 William Feller, A Itroductio to Probability Theory ad Its Alicatios, Vol, rd Ed Joh Wiley ad Sos, Ic, ew Yor, 968, 5 Paul L Meyer, Itroductory Probability ad Statistical Alicatios, d Ed, Addiso-Wesley Publishig Co, Readig, Massachusetts, 970, - Daiel J Teague 7 C School of Sciece & Mathematics