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# Biostatistics 22S 101

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This 16 page Class Notes was uploaded by Cullen Conn on Friday October 23, 2015. The Class Notes belongs to 22S 101 at University of Iowa taught by Staff in Fall. Since its upload, it has received 174 views. For similar materials see /class/228086/22s-101-university-of-iowa in Natural Sciences and Mathematics at University of Iowa.

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Date Created: 10/23/15

8101 Announcements A note about the determination of course grades On the syllabus I stated that The Final will be a weighted sum of HW quiz midterm amp Final exam grades Final Vo 10HW 20Quiz 2 01 Midt 20239quot Midtoo 3 0FinalExam As arougl1 guide quotAquot 90 quotBquot 80 quotCquot 70 quotDquot 60 In fact I took the MAXIMUM of 3 calculations the one above amp Finaloo 10HW o 20Quiz 101 Midtoo 252 d Midtoo 35FinalExam o Finaloo 10HW o 20Quiz 251 Midtoo 10239quot Midtoo 35FinalExam o The latter two methods were to bene t students who had a midterm score that was low relative to their other midterm score amp their Final exam score So for example a student who had 100 on the HW 80 on the quizzes 40 on midterm l 70 on midterm 2 amp 60 on the Final exam would benefit from method 2 Final 10100 2080 1040 2570 3560 685 Note that perfect HW only added l 01 00 10 to the Final SEVERAL students have written to me expressing concern about the Final Exam length 25 questions The midterms were 15 questions over 75 minutes 5 minques while the Final will be 25 questions over 130 minutes 52 minques Please arrive 10 minutes early so that we can set up amp be ready to roll at the scheduled time I will allow a few extra minutes at the beginning amp end for a total time 130 min The last midterm went rather well with 30 students 13 of the class scoring 1315 11 students scoring a perfect 15 amp a median 1093 so I don t think my exams are unreasonably long or difficult In previous semesters the Final exam scores in my S 101 courses have been good overall A THEORY It may be that students who are running out of time are too dependent on their notes If you know the material most problems are straightforward amp can be done rather quickly If you need to look everything up amp are mechanically duplicating the steps on prototype problems you are wasting a huge amount of exam time It s far better to learn how to do the standard problems before the exam rather than during the exam v v n 11 all rules of probability conditional probability odds independent events mutually exclusive events expected value conditional expected value fair price histograms amp stemplots standard deviation Bayes rule tabular Bayes credible sets amp intervals density functions normal curve problems beta prior gt beta posterior normal appx to beta calculation of credible intervals amp posterior probabilities for rate difference between rates RR OR amp single means I will not ask about difference between means 3 reverse engineering of frequentist confidence intervals amp pvalues calculation of pvalues amp confidence intervals for p amp u using the test statistics amp confidence interval expressions on worksheet 9 frequentist test stats for p values amp con dence intervals credible intervals amp posterior probabilities for medians when a transformation is used Regression analysis including correlation residuals R2 sy x sums of squares logistic regression including estimation of odds probabilities amp relative odds and also credible intervalsposterior probabilities for relative odds construction of the Kaplan Meier survival function HERE ARE SOME WORKED OUT PROBABILITY PROBLEMS Hg Suppose 40 of adult Iowans has a college degree Find the probability that at least one of 5 randomly chosen Iowans has a college degree 1 l 7 Pall FAIL to have college degree 1 7 65 9222 2 Your company has made three independent bids to the cities of Davenport Iowa City and Sioux City The probabilities that your bids are accepted are l 3 and 6 respectively Find Pnone of the bids are accepted 974 252 3 In pblm 8 nd Pexactly two of the bids are accepted PSSF PSFS PFSS134 l76 936 216 Note the events SSF SFS amp FSS are mutually exclusive so we can add probabilities 4 Suppose 28 vehicles on US roads are trucks 70 are either a truck or are US made and 54 are US made What percentage are US made trucks Use the addition rule PT or U PT PU 7 PT and U 70 28 54 7PT and U PTand U 12 5 In 4 what is the probability that a randomly chosen vehicle is US made given that it is a truck You can draw a Venn diagram with 100 marbles There are 4 categories in the diagram 12 marbles in T and U 16 in T but not U 42 in U but not T amp 30 in neither From the diagram we conclude that PU l T 1228 429 Or use the formula PU l T PU and TPT 1228 429 6 7 8 9 10 11 Suppose that 30 of U1 students smoke 60 of smokers drink coffee amp 45 of nonsmokers drink coffee Find Psmoke 1 drink coffee SOLUTION Psmokel drink co iee Pdrinkl smokePsmoke Pdrinkl smokePsmoke Pdrinkl don39t smokePdon39 I smoke 6030 6030 4570 3636 Continuing problem 6 suppose 10 UI students are selected randomly Find the probability that exactly 4 of these students smoke 10C4304 706 200 note 10C4 210 Continuing problem 6 find the probability that fewer than 3 students smoke P lt 3 P0 P1 P2 10C0300 7010 10C1301709 10C2302 708 383 In problem 6 find the proportion of students who drink coffee amp smoke Pdrink and smoke PsmokePdrink l smoke 3060 18 note this is the numerator of the Bayes expression in problem 16 Two independent observations are made from a standard normal distribution Find the probability that BOTH observations are greater than 1 a 003 b 012 c 018 d 025 e 0406 P obs gt 1 1587 from normal table Pboth gt 1 15872 Four of my 40 Biostat students are lefthanded Suppose we choose 2 of them at random for an experiment Find the probability that at least one is a lefty a 192 b 222 c 264 d 288 e 325 1 7 Pall righties 1 7 36403539 12 13 14 15 16 17 18 It was reported in Campus Review June 2003 that 32 of college students in the US work more than 20 hours per week Find the probability that exactly four often randomly chosen US college students work more than 20 hours per week a 084 b 125 c 164 d 200 e 218 10C4324686 Continuing 12 if 200 US college students are chosen to participate in a survey what is the approximate probability that fewer than 28 of them work more than 20 hours per week a 088 b 098 c 113 d 142 e 164 standardize 28 by subtracting 32 amp dividing by sqrt 3268200 033 Your accountant has informed you that one or more of your last three taX returns may be audited by the IRS The probabilities of audit are 2 3 and 5 for the years 2001 2002 amp 2003 Find Pn0ne of the three returns are audited a 12 b 18 c 24 d 28 e 30 Pno audit in 2001Pno audit in 2002Pno audit in 2003 875 28 In pblm 14 Pexactly two of the three returns are audited a 18 c 25 d 32 e 35 PSSF PSFS PFSS 235 275 835 Suppose 62 of Acme customers are women 28 of customers are college educated and 80 are either women or college educated What percentage of Acme customers are college educated women a 10 b 12 c 16 d 18 e 20 addition rule 80 62 28 7 X In 16 nd the prob that a customer is NOT college educated given that she is a woman a 64 b 75 c 84 d 90 e 92 draw a Venn diagram Coperto Insurers of Davenport offers 5 different insurance plans all of which are equally popular with customers If each of three different customers acting independently chooses a plan what is the probability that at least two of them match hint this is like the birthday problem 19 1 4 Pno match 1 4 554535 20 of Acme laptop computer screens have dead pixels 25 of Glaxco screens have dead pixels 80 of the laptops used by UI students have Acme screens while the rest have Glaxco screens If a U1 student s screen has a dead pixel what is the probability that it is an Acme screen Use Bayes theorem note I changed the numbers in the problem PAcme dead 20 21 22 23 Pdead AcmePAcme Pdead AcmePAcme Pdead GlaxC0PGlaxC0 20 80 2080 2520 The copy machine in 241 SH breaks down up to 4 times per month with the probabilities indicated number of break downs 0 l 2 3 4 probability 3 3 2 l 1 Find the probability that the machine breaks down at least twice in one month a 3 b 4 c 5 d 6 e 7 42 of Stat Professors are boring 28 are poor dressers and 60 are either boring or poor dressers What percentage of Stat Professors are boring and poor dressers a 10 b 12 c 16 d 18 e 20 addition rule 60 42 28 7 x In problem 21 nd Pboring l not a poor dresser a 328 b 363 c 388 d 444 e 475 use your answer in 17 to draw a Venn diagram A light bulb lasts more than 60 hours with probability 84 The same bulb lasts more than 100 hours with probability 52 Find the probability that the bulb lasts more than 100 hours given that it lasts more than 60 hours a 40 b 45 c 50 d 58 e 62 24 25 26 27 28 P gt 100 l gt 60 Pboth Pgiven P gt 100 and gt 60 P gt 60 Pgt100Pgt 60 52 84 Three batteries are chosen randomly from a box containing 4 dead and 6 live batteries Find the probability that all three batteries are live a 167 b 196 c 225 d 256 e 288 6 1 0 5 94 8 Three boxes contain red and white Box 1 Box 2 Box 3 marbles as shown A marble is chosen from each Find the probability that a 4 white 7 white 8 white red marble is chosen from at least one 6 red 3 red 2 red a 565 b 600 c 656 d 688 e 776 1 7 Pall white 1 7 478 In problem 25 nd the probability that different colors are chosen from Boxes 1 amp 2 a 38 b 42 c 48 d 50 e 54 Pwht red Pred wht 43 67 Suppose P smoke 1 drink coffee 40 Pdrink coffee 50 amp Psmoke 30 Find P drink coffee 1 smoke answer 667 by multiplication rule Pdrink and smoke Psmoke l drinkPdrink 4050 20 Now draw a Venn diagram In problem 21 if 6 stat profs are chosen randomly what is the probability that exactly 2 of them are boring a 20 b25 c 30 d 36 e 42 binomial 6C2422584 29 Consider the following probability histogram for T the number of times the spinner shown lands in the red zone when spun 5 times Find PT gt 1 red zone rectangle areas 237 396 264 088 015 001 0 1 2 3 4 5 a 237 b 367 c 396 d 428 e 556 add up the red probabilities 30 At SW state college 25 of students are management majors 54 of the management majors have completed Stat 101 while only 32 of the other majors have completed Stat 101 Find the probability that a randomly selected student is a management major given that she has completed Stat 101 a 28 b 32 c 36 d 42 e 48 Bayes formula P1101 mgtPmgt P101 l mgtPmgt P101 l otherPother 22S101Quiz 4 due 30 April 2007 Print name see NOTE you can use minitab to check your work but you must SHOW the work to receive full credit The age of 10 patients and their recovery times after a surgical procedure are as follows lpatientll 2 3 4 5 6 7 I8 I9 I10 I age I 38 28 I30 I 62 76 25 45 30 70 56 ldays 7 6 6 10 12 4 I8 5 15 7 1 Find the equation of the regression line that predicts the patient s time to recovery from hisher age avr age 46 SD 188 avr days 8 SD 34 r 90 2 Complete the table lpatientll 2 3 4 5 I6 7 I8 9 I10 I tted res1dua1 3 Calculate Total SS Residual SS Regression SS 4 Calculate R2 s 22S101Quiz 2 due 28 March 2007 Print name see problem 1 Suppose that 34 of 80 drinkers of caffeinated coffee report high anxiety whereas only 28 of 100 drinkers of decaffeinated coffee report high anxiety a Find a 90 credible interval for A uA425 280l45 039 V 0713 95 credible l45il6507l b Find PAgt10 data 63 2 Prob 736 071 problem 2 In a retrospective study of the relation between smoking and lung cancer the observed odds ratio was 267 and the reported 95 con dence interval was 240 290 a Find POR gt 260 I data u log 267 982 10 290 g 240 2096 POR gt 260 I data PlogOR gt 956 I data 956 982 54 2 POR gt 260 I data 705 048 b In the lung cancer group there were 2 smokers for every 3 nonsmokers Thus in the nonlung cancer group there were 2 smokers for every nonsmokers Hint use the value 267 ofthe observed odds ratio 267 a so thatX 8 3x problem 3 889 hospital administrators participated in a prospective study on the role of diet in heart disease The following table was constructed heart disease no heart disease total lownormal fat diet 64 lt ll in 469 533 high fat diet risk 57 299 356 lt ll in factor a Find the increased risk of heart disease associated with a high fat diet 57356 133 Increased risk 33 64533 12 b Find a 95 credible interval for RR u 10g133 285 84 88 77 169 35616 53312 95 credible 285 i 196 169 MIDTERM 2 answers FORM A 1D 2B 9C 10D FORM B 1D 2E 9E 10C MIDTERM 1 answers FORM A 1C 2E 9D 10A FORM B 1B 20 9D 10C MIDTERM 1 Topics 3D 11E 3B 110 3A 11D 3B 11E 12C 4D 12C 4D 12C 4D 12B 130 SC 13B 5A 13B 5B 13A 14B 6D 14A 6E 14D 6C 14B 15A 7A 1 5E 7C 15B 70 15D 8E 8B 8E 8E Descriptive statistics Histogram boxplot mean median quartiles standard deviation Probability Rules of probability addition complementation multiplication Pat least one success calculation of probability by partitioning Conditional probability de nition Bayes theorem tabular Bayes credible sets Independent events mutually exclusive events expected value conditional expected value odds of an event vs probability of an event probability density functions ONE 85 x 11 PAGE ofnotes are allowed An interesting Link httpexploringdatacgueduau70 drafthtm Here are some solutions to the worksheet probability problems 1 The probability is 45 that George is hired by ACME P amp V the probability that Martha is hired is 62 the probability that at least one is hired is 81 Pboth are hired PG PM PG and M Pexactly one is hired Draw a Venn Diagram Are the events quotGeorge hiredquot amp quotMartha hiredquot mutually exclusive 2 independent 2 Each day Larry sells up to 5 loaves of seedless rye bread at his bakery sold 0 1 2 3 4 5 prob 1 1 1 2 4 1 PLarry sells at least 2 loaves of seedless rye bread 8 PLarry sells an odd number of loaves of seedless rye bread 4 3 PLarry sells 5 loaves of seedless rye in 2 days 3 white 5 black Two marbles are drawn without replacement from the box shown that the 1st is white and the 2nd is black 3857 ln 3 suppose three marbles are drawn without replacement P all 3 are black 584736 If you spin the spinner toss the coin amp draw a card what is the probability that you spin a toss heads and draw a quot2quot 2 Spinner has prob 1A of landing on 1 P2 from box 25 Pexactly 2 ofthese events occur PSSF PSFS PFSS ln problem 5 what is the probability that exactly one ofthe 3 events occur PSFF PFSF PFFS What is the probability that at least one ofthe 3 events occur 1 Pfail fail fail A coin having Pheads 4 is tossed 5 times Pat least one head 1 65 A medical test is designed to detect the presence of a particular drug When applied to a drug free person there is a 2 chance that the result is quotpositivequot and when applied to a person in whom the drug is present there is a 95 chance that the test is positive Suppose 1 ofthe population is using this drug If 100 drug free people are tested Pat least one positive test result 1 98100 10 If 200 drug free people are tested Pat least one positive test result 1 98200 If 500 drug free people are tested Pat least one positive test result 1 98500 In problem 9 suppose a person tests positive What is the probability that heshe is using the drug 2 Use Bayes theorem Suppose PE 48 PF 75 and PE and F 32 PE or F 48 75 32 Pneither E nor F draw the Venn diagram with 100 marbles Put 32 in the overlap intersection of E amp F 16 in E but outside of F 43 in F but not E amp 9 ouside of both E amp F 11 PF but not E 43 PE F 3275 PF not E 4352 PE or F but not both 59 A captain is chosen randomly from each of two basketball teams A Johnson R Cooper T Jones C Williams Team A H Smith Team B M Smith R Smith B Cooper J Barnes S Jones Pboth captains are named Smith 2515 Pboth captains have the same last name Pboth Smith Pboth Jones 2515 1515 325 12 13 14 Five men and eight women occupy an office Three people are chosen at random from this group Pall 3 are men 513412311 Pall 3 are the same gender 513412311 813712611 Suppose PL gt 70 56 amp PL gt 80 24 Find PL gt 80 l L gt 70 Use the de nition of conditional probability Pboth Pgiven PL gt 80 AND L gt 70 PL gt 70 PLgt 80PLgt 70 24 56 GlaXCo produces 20 of the defribulators currently in use Acme produces the rest One in ten GlaXCo defribulators are defective while only one in twenty Acme s are We encounter a defective defribulator Find the probability that it is a GlaXCo PG i def Pdef l GPGPdef l GPG Pdef lAPA 15 Amarble is transferred from 2 red 3 wht gt 1 red 4 wht Then a marble is chosen from the 2quotd box It is red Find Pthe transferred marble was red Ptrans R l obs R Pobs R l trans RPtrans R Pobs R l trans RPtrans R Pobs R l trans WPtrans W 16 Suppose that you purchase 20 of your steel rods from company A 50 from B and 10 from C Suppose also that 5 of the rods from A are defective 10 from B are defective and 2 from C are defective Given that a particular rod is defective what is the probability that it was made by company A B C PB i def Pdef lBPBPdef l APA Pdef lBPB Pdef lCPC rules of probability conditional probability odds independent events mutually exclusive events eXpected value fair price histograms amp stemplots standard deviation Bayes rule tabular Bayes credible sets amp intervals density functions normal curve problems beta prior gt beta posterior normal apr to beta calculation of credible intervals amp posterior probabilities for rate difference between rates RR OR amp single means I will not ask about difference between means reverse engineering of frequentist confidence intervals amp pvalues calculation of pvalues amp confidence intervals for p amp u using the following test statistics amp confidence interval eXpressions p usenormal amp faiz M p1p 4 n Y 7 ts usetdfnl 7 Xii quot w J credible intervals amp posterior probabilities for medians when a transformation is used Regression analysis including correlation residuals R2 sy x sums of squares logistic regression including estimation of odds probabilities amp relative odds and also credible intervalsposterior probabilities for relative odds construction of the KaplanMeier survival function

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