2009 solutions

2009 solutions - MATH 203 Final Examination April 2009...

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Unformatted text preview: MATH 203 Final Examination April 2009 Student Name: Student Number: McGill University Faculty of Science FINAL EXAMINATION MATH 203 Principles of Statistics I April 27th, 2009 9 am. « 12 Noon Answer directly on the test (use front and back if necessary). Calculators are allowed. One 8.5” X 11” two—sided sheet of notes is allowed. Language dictionaries are allowed. There are 17 pages to this exam and 2 pages of tables. The total number of marks for the exam is 100. Examiner: Professor Russell Steele Associate Examiner: Professor David Stephens MATH 203 Final Examination April 27, 2009 Question 1: (10 points) The Canadian government has decided that there was some suspicious7 potentially illegal activity regarding a company’s revenue stream. As part of their investigation, they wanted to establish that there was a significant increase in the mean transaction amount before and after a particular event. The government statistician took a random sample of transactions with 50 customers who were billed both before and after the date of interest (i.e. each customer was billed twice). The data are summarized in the table below: [Mean rStd Dev_| 25%ile [Median] 75%ile] Sample size . Before event 98.82 [— 20.49 83.99 101.2 109.3 50 . After event 108.3 . 17.94 96.25 107.8 120.1 50 | Difference [After — Before] 9.47 I 19.81 5.188 9.01 l 24.72 50 ] Test for a significant increase in the mean transaction amount before and after the event at a = 0.05. MATH 203 Final Examination April 27;, 2009 Question 2: (10 points) uiz one of his classes. The quiz had 3 questions. partial credit was given. Assume that the . 1 ity massgfunction) for the number of correctly answered Professor Steele gave out .. .. :- Each question was worth probability distribution (i.e. pro questions by a randomly selected student is: Number of correctly answered iq’uestions 0 1 2 3 Probability 0.05 0.20 0.40 0.35 (a) What is the distribution (i.e. the probability mass function) for the total score on the quiz for a randomly selected student? [2 points] (b) What is the expectation and variancef‘of the total quiz score for a randomly selected student? [2 points] E( Score} Z,‘:§_0(69.os‘)+ 2(01) 7" flex—10).; 4(3)») 7, / V~( Sm) = @mfo.or+ ‘ (24010.2 + (IA/Mm ,7 + ((0- y. I)? gr 12. W MATH 203 Final Examination April 27 , 2009 (c) What is the distribution (i.e. the probability mass function) for the average (or mean) of two randomly selected student quiz scores? [3 points] Wt I6PU$5"IDLL @mésbq£0~5' 0;.SCWS, Sana sz 1*» v (44mm) 0 O (0-Odo.os)=_w 00 j; 2(30) {my} 0 (0 2 265501»: my 2(.os)(.33)ro.o f 2 2 1 16mm» Z ‘7‘ L(,")v)(l‘{o)to K, 2 6 zcaoxas)‘ .H 1/ ‘I’ meow)», .Ic. _ , 9 2 20ml”): ‘2) "I 4 i (9 I t C “ibis-Mar): 4.1725" (d) What is the approxirn e distribution for the average (or mean) of 100 randomly selected student quiz scores? [3 points] ' A. 7. 2,? X” 9/07”) 37:72: i I MATH 203 Final Examination April 2 Question 3: (6 points) In a 1995 paper in Astrophysics and Sp1 e Science, Higgins and Henrikson discussed the distribution of gamma-ray bursts in halo neufifOn star—comet models. They based most of their conclusions on a simulation model where the elocities of comets ejected from globular cluster stellar systems were assumed to be Normally istributed with mean velocity (in km / 8) equal to 200 and standard deviation 110. (a) Find the proportion of comets ejected th velocities less than 150 km / s. [2 points] l§0m200>t ‘— 90. 32‘ 7 (b) Find the proportion of comets ejected velocities between 100 km / s and 240 km / s. [2 points] an —- 2o 0 2% ’2; “B T "1 4’ V( < 2< ~, R(~0g{371< z <036) = meme-6.1917 '=— 6.45'72 (c) Find the proportion of comets ejected with velocity equal to exactly (without rounding to whole numbers) 200 km/s. [2 points] [4 (X: o w x mm s MATH 203 Final Examination April 27, 2009 Question 4: (10 points) In Alberta7 away for organizations to raise funds for charitable or religious purposes is to obtain a licence for a casino gaming event. A variety of card and roulette—based games are permitted, with much of the revenue going to the good cause. A gaming official claims that about 10% of these casino gaming licences are granted to support the arts. (a) An auditor is testing the claim, at a = 005, and selects a random sample of 190 licenses granted in the past year. She finds that 15 of licenses were used to support the arts. What conclusion should be reached? [6 points] Ho? p20,“) 15+ '22:» 14: alo 1% ’ 3 Ito A p? ’“(dolZ/DJ‘I'S) Z: —: 470 6‘10 j” “’0‘ ok/ 24h 2 [776 ?/< LQé) 50 wacaxzqé'f lyeUL H3 0910/ aw/ Conc/Vét flip $0.10, (b) If the auditor wanted the margin of error for the sample proportion to be i1%, how large would her sample size need to be? [4 points] W @L z (/éafasros) 51152207me 9. 0'1 OZ 0"" f.‘ 3, few; ”‘ >, 2¥¢®Zh 2 2%??? MATH 203 Final Examination April 2 Question 5: (10 points) Animal behaviourists have discovered that th more domestic chickens peck at objects placed in their environment, the healthier the chick: s seem to be. White string has been found to be a particularly attractive pecking stimulu 11 one experiment, 72 chickens were exposed to a string stimulus. Instead of white string, colored string was used. The number of pecks each chicken took at the blue string over a sfimcified interval of time was recorded. Summary statistics for the 72 chickens were 5? = 1.13 and s = 2.21 pecks. (a) Construct a 99% confidence interval toiestimate the population mean number of pecks made by chickens pecking at blue stringi :Interpret the result. [7 points] 11,2 3 r 0.4 7 t 0,4,6, / Yo) [4.2L m ??% Cu. mean num$¢rrfitcés fi't 5/“ 5"“); )9 Chink-rim hemlv/flifi [454.) [.94 ‘ ' 1“. m: (b) Previous research has shown that n = 7.5 pecks if chickens are exposed to white string. Based on the results found in part (axis there evidence that chickens are more apt to peck at white string than blue string? EXplain. [3 points] yfig. gézquu Luc' av: .WGL‘SAD '.S 1% hmJ’W‘v/ (of/G) / MM fi-n'S/S céo/{a 921%.; (3%, .VL MATH 203 Final Examination April 27, 2009 Question 6: (12 points) A bike rental chain owner owns rental locations in both Westmount and the Plateau. The owner was interested in trying to detect a difference betWeen the two locations in the mean distance that the renters travelled on the bikes. The ownerirandomly selected a particular day and only rented bikes to customers that had odometers on thefubikes so that he could measure the distance (in miles) travelled by each renter. The table and figures below contain the data for all of the renters at the two bike shops on that particular day. __] Mean 1 Std Dev 25%ile Median | 75%ile ] Sample size LWestmount bike shop 13.15 I 10.56 8.478 T 11.47 14.97 13 Plateau bike shop 21.38 i 9.298 13.98 I 22.82 30.23 11 > stem(Westmount) The decimal point is,1 digit(s) to the right of the | O | 14589 1 | 011256 2 | 8 3 | 4 | O > stem(Plateau) The decimal point is 1 digit(s) to the right of the | O | 6 1 I 1179 2 | 349 3 I 113 MATH 203 Final Examination April 27";2009 (a) Test to see if there was a statistically si’: 3 cant difference in mean bike distances between the two offices. (Use 0; = 0.05). What : Inption(s) need(s) to be made in order for the inference to be valid? Are they met i 'V is particular problem? [9 points] W pooLOL, Z’SWQ {7‘} a ?” )(A "3‘; O .52, ,3;— :3 l/ 5 ~20) (‘2.oll < t2,)o'°?5.32.°> CZ) VZ’ ‘ MAL C‘uno (- WJ‘LJ #9 (w-‘L (MC/a» “(1; 7110+ 770% {s a 0/; 0; ‘ C3) A/O/m..( ,myala Alas ’9 [Vailv W drsi‘q‘éu [95,. fly“ (I 0702 é¢ 47(0me . (b) Calculate the p—value (or an approximate p-Value) related to the hypothesis test in part (a). [3 points] 2.0! 51(3 tovsjzz 4‘4 corp/>— so it [(5 2(r025)‘:'05’an6/ 26m éjf’joJo, MATH 203 Final Examination April 27, 2099 Question 7: (10 points) The Canadian Food Inspection Agency has become worried about synthetic hormone levels in Canadian beef products. They would like to devise a control process that would allow them to detect when the mean synthetic hormone level of aparticular supply of beef is above a particular level. Obviously, they cannot test every package of beef, so they randomly sample 200 packages from each supplier in order to decide whether the mean hormone level for that supplier is too high (anything over 200 ppm of the synthetic hormonégwould be considered unsafe). Assume that the distribution of the hormone levels from a supplier are normally distributed with unknown mean and with known standard deviation 10. ,§(a) Describe what a Type I error and what a Type II error would be in the context of their ix 7, hormone testing. [4 points] is”? _ Ms above 20 0 Win #ng m4. (b) What is the approximate distribution of thesample mean hormone level of the 200 ran— domly sampled packages from a particular supplier if the true mean hormone level is 203 ppm? [2 points] CONTINUED ON NEXT PAGE MATH 203 Final Examination Ap r 27, 2009 lves to a Type I error probability of 0.0017 what the true mean hormone level were actually (0) If we assume that they want to limit t would their approximate Type II err 203 ppm? [4 points] 2:? Our (l/afW“ '1’: ‘34:)“, 751/\ W 24. value: maid lac. Lug—o RJCJ (/21 or? Z w/M— 5M“ % 3.07. A2 M202 to 3 J m A??? wow 1‘» ML m 5°C elm; 3?; moi/avg majw" fl/‘im ' Krrkiflj‘ec‘lwn (AVA-L A; A7 but“ [at 3'07 : Xar" [0/ zaiééjji 200+ 2J2: 76 “3263 r bat/f 5 ) Pr( M MATH 203 Final Examination April 27, 2009 Question 8: (10 points) In a recent newspaper article on health costs, the author reported prevalence and incidence statistics for diabetes in Canada. The annual preValence of a disease is the total number of cases of the disease in the population divided by the number of people in the population for a given year. The annual incidence of a disease is the total number of new cases in the population during the year. The newspaper article reported that the prevalence of diabetes in Canada is 350 cases per 10000 people in the population the incidence of diabetes in Canada is 26 cases per 10000 people. (a) Assuming the prevalence stated in the newspaper is the true prevalence, what is the probability of observing more than 40 peoplefwith diabetes in a random sample of 1000 Canadians? ; A Iowa 7 0.07») M . X/V film/‘4‘! n’lOOo/ F 1 M x > yo) (wean we mm M - 5‘ ’70.)“ 5’5“ >31: > tit/(C13 > 0.?5‘) 7 'O. /7// (“Cing (b) Assuming the incidence stated in the newspaper is the true prevalence: what is the prob— ability of observing more than 5 people withgdiabetes in a random sample of 1000 Cana— dians? X“) Kimmie ((524900) F: VFCX > $34: if Able“ MJu/Io *- ‘ifufif 'Tfk‘ig‘ Mai 43 ‘I 7 < ?> [‘80) (gage —- 3 A - 3f, 0‘33. MATH 203 Final Examination April 2%32‘009 Question, 9: (12 points) A student who was interested in sociology a of mothers to look for the predictive ability _ particular, she was interested in various cult j and the gender of the baby. There were two 'jublic health carried out an interesting survey ome old wives’ tales concerning pregnancy. In heories of attributes of the mother’s pregnancy Sicular ”tales” that she was interested in. Torning sickness (beyond just a general queasi— ' probability of having a girl 0 Whether the mother had serious bouts: ness) which supposedly indicates a hig o What the mother thought she was havin fore finding out officially (either via ultrasound or at the birth) She sent out a survey to a very large number (firecent mothers (restricting her sample to only those who had non-multiple births, i.e. no tw ‘ ' triplets7 etc.) asking three questions: 1. Did you have very serious morning during your pregnancy? 2. What did you think your baby’s gendergwas before you had your 20—week ultrasound? 3. What was the eventual gender of your and found the following: had very serious morning sickness during their en who gave birth to boys had very serious 0 60% of the women who gave birth to a ' pregnancies whereas only 25% of the " morning sickness o 70% of the women who gave birth to gir hought they were going to have a girl and 80% of the women who gave birth to boys tihpiight that they were going to have a boy o 51% of the women surveyed gave birth :2 r oys 0 Whether or not one has morning sicknessflfand whether one believes that they are going to have a girl (or a boy) are independent the gender of the baby that the woman gives birth to ‘ CONTINUED ON NEXT PAGE MATH 203 Final Examination April 27, 2009 (a) If a woman has serious morning sickness during her pregnancy, What does the study say that her probability of having a boy should be? [4 points] Le‘f A h—Cma/nrvfldt/uzm. 7 18+ ’g & 7’21lL5AL4134 . M {5 1A) 2 MA 1&3 pr (/3) fiWrsx/Mgh- flax/33%ch) :’ ©i25‘)(0.50 r(o.5‘l)+ 0.96 (Ont?) (b) If a woman thinks that she is going to have a boy, What is the probability that she will have a boy? [4 points] ” leg“ C wflhhsr‘fkqéfj‘ La If; bum 52¢ 4MB. fir (8 J (A :‘A/«cza Mg) [Ham Maw /r(ctfl°)fl(flc) CONTINUED ON NEXT PAGE 0 (go (55/ . Oi E0(015)'>+(a§o)(5.m 14 a 0.735— MATH 203 Final Examination April 27, 2009 (c) If a woman has serious morning sicknes 77‘; to haVe a 1303’; What is the probability ring her pregnancy and thinks that she is going he eventually ends up having a boy? [4 points] “(e/Mi?! PM I flat/"'1 aflg}! flréAflci/ch/(g: v: e? W: 1&9 (7/ (c lab/4(8) 4m 16),? (c 05% cg) + fl (Mac/flat]? i a Z .3“ ( (9 . 3’0) [0.31) @2310, so)(a.rz) +— O.é0(6.?o) 6.4; a 3 3 Q MATH 203 Final Examination April 27, Question 10: (10 points) The table below contains the results for a study conducted on two treatments for kidney stones, Treatment A and Treatment B. The table containsthe number of successes and failures for each treatment for two kinds of stones: small stones and large stones. ___.__ Small Stenes Successes, Failures Total 162 12 174 y Treatment A Treatment B 468 72 540 i 630 84 714 Large Stones 1 Successes. Failures Total 142 526 50 Treatment A 384 , Treatment B (a) Estimate the difference in the pooled (i.e. stratified by type of stone) probability of success between the two treatments using a7'190% confidence interval. Which of the two treatments do you believe is better to use based on the pooled probability of success with the two treatments? [3 points] PA: SS‘IHCQ 2 0.73 [7971.526 r—OcS 1” O, O 35- CONTINUED ON NEXT PAGE 1‘ C~ opoéfj “(5.043) MATH 203 Final Examination Ap‘iii’l'27, 2009 (b) Estimate the difference in the probability: of sucess between the two treatments using a 90% confidence interval for each type ofkidney stone [small and large] separately. Which of the two treatments do you believe is better to use when looking at the two types > of stone separately? [4 points] 3W: Kl ,, b 43 c ‘12an » 3,8,1 73 [[29 pk 75L, ' ' I7@ 570 '" t/K' to" 237-“ 30.6573" 676 166 /\ 4 « parry; i/fi ($191+ may hf '4 43 ©.7$*O.(;82~ . a. f .é‘l fl? 5? / .flafiv 0m”) ’99 5‘79 .0"! 75“ ,fi’f /(;f 507%.09137 : ‘oet‘marr(b-ozl,00”) :97 2 (00,1, .0328) 2415 [29% g; A [S (9W9 > (c) Did your conclusions differ in parts (aband (b)? If so, state what phenomenon you’ve observed. [3 points] yfls/ ngmm (Sin/son’s %JAV.(CM 93 926 a ~ 17 ...
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2009 solutions - MATH 203 Final Examination April 2009...

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