Suggested Exercises from M&M Chapter 4 [Homegrown exercises begin on next page] These pages were updated on September 16

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1 Suggested Exercises from M&M Chpter 4 [Homegrown exercises egin on next pge] These pges were updted on Septemer 16 To strt with, do some of the odd-numered exercises. nswers to ll odd-numered exercises re given on textook pges S-1 onwrds. Do some or ll of the following even-numered exercises. You re sked to hnd in nswers to designted ones.. see the list, nd the dedline, on the min course pge. Some of these will e discussed in tutorils or nswers to them posted on the course we pge * 4.10 * ecuse it strts the strem of rndom numers using seed which is determined (I suspect) y how long your computer hs een running since you lst strted up. Unlike if you use the Tool, there is no wy to control the strting vlue ("seed") if you use the function -- see the Help on the Rndom Numer Genertion Anlysis Tool]. If you press the F9 key, it will reclculte nd give you nother! [you cn control the re-clcultion using the Clcultion T under Preferences under the Tools menu] How to turn this rndom numer (uniform on 0 to 1) into the result of Bernoulli (0/1) tril where you expect 73% to e 1 (yes) nd 27% to e 0(no)? First, see the simpler exmple 3.21 on p 269 of text, simulting 60% 1's nd 40% 0's using single-digit 'pre-drwn' rndom numers from Tle B. To simulte 73% yes nd 27% no, you need to tke 2 digit rndom numers from tle B. [you cn mp the 73 possile 2 digit numers 00 to 72 into 'yes' nd the 27 remining ones (73 to 99) into 'no']. With Excel you do likewise: you tke dvntge of the fct tht proportion 0.73 or 73% of the "Uniform on 0 to 1" distriution is in the intervl 0 to 0.73, nd 27% etween 0.73 nd 1.0 (see "spinner" on p 317). If the rndom numer drwn flls in the intervl 0 to 0.73 (s it will in some 73% of drws!), you will cll it 'yes"; if it flls in the intervl 0.73 to 1.0 (s it will in some 27% of drws!) you will cll it 'no'. * Simulting rndom phenomen using Excel So you just get Excel to determine whether the drwn numer is ove or Excel hs two wys of generting rndom numers tht fit certin pttern elow i.e. use the or "distriution." IF(expression, result if true, result if flse) One is the Rndom Numer Genertion Anlysis Tool, which you my hve to sk Excel to lod ech time. It is more extensive, nd does more of the work for you if you wnt specific distriution, ut it is not s flexile if you wnt to repet the process with new smple [i.e. you cnnot sve the process y typing formul into cell] A second wy is to uild them yourself. The uilding lock is the RAND() function under the 'Mth nd Trig' ctegory of Functions. Just type =rnd() into cell [or sk the wizrd!] The formul will yield rndom numer etween nd (effectively etween 0 nd 1). You cnnot tell wht the result will e function =IF( RAND() <0.73, "yes", "no") [or do it in 2 steps -- using cell for the rndom numer nd nother for the result of the IF.. tht wy you cn check tht it is doing wht you wnt!] For more complex exmples, see some of the 'simulte' spredsheets on the course 323 we pge. A good one is the "Gmling 17th century: deméré 100 gmes", where I simulte the result of rolling die (ctully severl dice ll t once!). There, one needs n integer etween 1 nd 6. As you will see from the formul in sy the A1 cell, I tke rndom numer uniform on 0.0 to , multiply it y 6 so tht it is uniform on 0 to , dd 1 so it is uniform on 1.0 to , then tke the integer prt of it, so it is uniform on the integers 1 to 6 inclusive. pge 1

2 "Homegrown" Exercises round M&M Chpter 4-1- Pooled Blood [from Colton Ch 3] *Hint: Let Y = which drw produces 1st duplicte (so Y = 2,..., 11). Ech time n individul receives pooled lood products, there is 2% chnce of his developing serum heptitis. An individul receives pooled lood products on 45 occsions. Wht is his chnce of developing serum heptitis? (Note tht the chnce is not 45x0.02=0.9 ) To keep it simple, ssume tht there is 2% chnce tht unit is contminted nd clculte the chnce tht t lest one of the 45 units is contminted. The 2% shows how old Colton's ook is! -2- "Clustering" of Crdiovsculr Risk Fctors? A Snté Queec survey found the prevlence of 4 hert disese risk fctors in certin ge-sex group to e: smoking: 32%; fmily history: 32%; SBP>155mmHg: 12%; dietes: 5%. If risk fctors re distriuted independently of ech other, wht is the proportion of the ge-sex group with () 4 risk fctors () 0 risk fctors (c) 1 or more risk fctors?. A tree digrm my help. -3- "Duplicte Numers" [mini-version of irthdy prolem] To pprecite the high proility of duplicte irthdys, tke simpler cse of drwing single digit numers t rndom from Rndom Numer Tle or spredsheet until one gets duplicte. (lso, try ctully doing it to see how mny drws it tkes) Clculte the proility tht in 5 drws one will not otin duplicte, i.e., the proility of sequence 1st# ; 2nd# [ 1st#] 3rd# [ 2nd# 1st#] 4th# [ 3rd# 2nd# 1st#] 5th# [ 4th# 3rd# 2nd# 1st#] Clculte, y successive sutrctions* or otherwise, the proility tht the first duplicte will show up on the [Y =] 2nd, 3rd,...11th drw. Plot the frequency distriution of the # drws, Y, until duplicte. pge 2 e.g. Pr[Y>6] = Pr[Y = 7 or 8 or 9 or 10 or 11] Pr{Y>7] = Pr[Y = 8 or 9 or 10 or 11] diff. = Pr[Y = 7] -4- Errors cused y rounding Suppose one hs to nlyze lrge numer of 3 digit numers. To mke the jo esier, one rounds ech numer to the nerest 10, e.g., 460 < > 470. If the ending numers of the unrounded dt were uniformly distriuted (ech ending digit hs proility of 1/10), clculte c the verge error per (rounded) numer the verge solute error per (rounded) numer the squre root of the verge squred error per (rounded) numer ['root men squred error', or 'RMSE' for short] -5- Sving on Binry Tests y Pooling (More Advnced) When inry lood test [one tht yields positive ("+ve") or negtive ("-ve") result] gives +ve results in only smll proportion π of lood smples, it my e possile to economize on the costs of testing y pooling m lood smples, ccording to the following procedure: ech lood smple is divided into two portions; one portion is kept in reserve while the other is pooled with the corresponding portions from m-1 other lood smples if the result of single test on the pooled loods is -ve, the m individul lood smples re considered -ve; if the result is +ve, then the m reserve loods re individully tested. With m = 20 nd π = 0.1, clculte the expected numer of tests required to determine the sttus of m lood smples. (Drwing tree digrm my help to trck the scenrios)

3 "Homegrown" Exercises round M&M Chpter 4-6- Life Tles The following [conditionl] proilities re tken from the ridged life tles for mles (M) nd femles (F) computed from mortlity dt for Quéec for the yer 1990, nd pulished y the Bureu de l sttistique du Quéec: [the proilities for 90-yer olds hve een modified slightly!]. In (current) life tle, one tkes current (in this cse 1990) i.e. cross-sectionl deth rtes nd pplies them to fictitious cohort to clculte wht % of the cohort would survive pst vrious irthdys -- if these rtes persisted -- nd to clculte the verge ge t deth (lso known s life expectncy t irth). pro tht person who lives to his / her xth irthdy will die during next 10 yers x M F Birthdy Proportion Surviving x = x = x = Clculte, y successive sutrctions* or otherwise, the [unconditionl] proportions [i.e. proportions of the entire cohort] who will die etween their xth nd x+10th irthdys (x = 0, 10, 20, 30,... 90). Plot them s histogrms. We will use these proportions to clculte life expectncy in susequent exercise. * e.g. Pr[Die fter 70th irthdy] = 0.wwww Pr{Die fter 80th irthdy] = 0.zzzz Pr[die etween 70 nd 80] = difference Complete the following tree digrm nd clculte the proportions of mles who survive pst their xth irthdy (x = 0, 10, 20, 30, ). Do likewise for femles. Plot the proportions vs. x (these plots re clled survivl curves). Mke sure to lel the xes correctly. pge 3

4 "Homegrown" Exercises round M&M Chpter 4-7- Life Expectncy t irth vs. t ttined ges we serch for Redelmeier Oscr -- on longevity of screen ctors nd ctresses who win n Oscr hs trick nlysis From lifetle, one cn lso restrict ttention to (or "condition issue revolving round this point! It is the sme issue tht on"] those who hve lredy survived to certin ge nd mkes ishops live longer thn priests -- nd crdinl clculte the verge (or expected) longevity from tht point longer thn ishops -- nd full professors longer thn onwrds. Full lifetles from the Vitl Sttistics sections of ssocite professors -- nd jzz musicins longer thn ntionl sttistics gencies (Sttistics Cnd,... ) usully hve persons just orn!] column showing these s function of ttined ge. We-sed helth promotion groups hve recently strted using on-line life-expectncy clcultors, some generl (only input is ge nd sex), nd some very specific (user must input dditionl informtion on life-style, fmily history, etc.). During the My 2001 version of Course 323, I put some links to some of these on the 323 we pge, ut some of the links hve ecome out of dte rther quickly! You cn lwys find new ones y serching on key words such s 'life expectncy'. Of course, I cnnot vouch for how good they re. Here is one such clcultor tht I found [ the link worked in mid Septemer]: c In his "More informtion on the life expectncy clcultion" the uthor of the ove-cited link sttes tht "Any given individul will hve 50% chnce of living longer thn the life expectncy nd 50% chnce of dying erlier thn the life expectncy." Is the first of these sentences correct? Or is it the cse tht good del more thn hlf of us will live longer thn verge? You might find it helpful to use the rough distriution you derived from prt of -6- to strighten him out! He lso sttes tht I m struck y the morid file nme -- even though how long one lives nd the ge t which one dies re the sme quntity! "On verge, however, the tle will produce the correct vlue." Unlike others I hve seen, the youngest ttined ge it ccepts is 5 -- ut (unlike others) it does ccept ttined ges higher thn Is he correct in this? 100. d [Advnced, optionl, should e of interest to Epidemiology Use it (or ny other one you find to clculte life expectncy for persons of your ge nd sex [Is it OK to cll it your life expectncy?] s well s for persons who re older thn you y 10, 20, 30,... yers Explin to n older reltive why -- in these clcultions -- he/she hs higher life expectncy thn you [some vrint of the word 'conditionl' might e helpful] [the Spring 2001 rticle y D Redelmeier, U of Toronto -- pge 4 nd Biosttistics students] From the life expectncy t ech ttined ge, could you reconstruct the life tle itself? Hint: Think of the re under the lifetle curve s the totl (or verge, if the curve egins t Proportion Alive =1.0 t Age =0)

5 "Homegrown" Exercises round M&M Chpter 4-8- Correcting for guessing on multiple choice exms Suppose one wishes to estimte vi multiple choice exmintion [with k nswers to choose from for ech question], wht proportion π of questions student knows the nswer to (excuse the dngling preposition!). Show tht the simple proportion p of correctly nswered questions gives ised (over)estimte of π if the student simply rndomly guesses mong the k nswers on questions where (s)he doesn't know the nswer. Do this y clculting the expected vlue of p (i.e. the verge mrk per question) when ech nswer is mrked 1 if correct nd 0 if not. One cn "de-is" the estimte y mrking ech correct nswer s 1 nd ech incorrect one nswer s m (where m is presumly negtive quntity). Wht vlue of m will provide n unised estimte of π? Begin y finding the expected mrk per question, then set it to π nd solve for m Glton's wy of showing tht the heights of the mrried couples in his dtset were virtully uncorrelted See Q2 of "Exercises round Chpter 5" opposite F 18 in the Course 323 we pge Other Exercises from Course 323 "Exercises round Chpter 2" opposite J 3 in the Course 323 we pge Testing for HIV Opposite M 7 in the Course 323 we pge. pge 5