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MATH_203_december2006

# MATH_203_december2006 - MATH 203 Final Examination December...

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Unformatted text preview: MATH 203 Final Examination December 7, 2006 Student Name: Student Number: McGill University Faculty of Science FINAL EXAMINATION MATH 203 Principles of Statistics I December 7th, 2006 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 15 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 Keith Worsley MATH 203 Final Examination December 7, 2006 Question 1: (8 points) The Mississippi Department of Transportation collected data on the number of cracks (called crack intensity) in an undivided two—lane highway using van-mounted state—of—the-art Video technology (Journal of Infrastructure Systems, March 1995). The mean number of cracks found in a sample of eight 50—meter sections sections of the highway was (E = 2.10 with a variance of s2 = 0.011. Suppose the American Association of State Highway and Transportation Ofﬁcials (AASHTO) recommends a maximum mean crack intesity of 1.0 for safety purposes. Test the hypothesis that the true mean crack intensity of the Mississippi highway exceeds the AASHTO recommended maximum. Use a = 0.10. State your assumptions. MATH 203 Final Examination December 7, 2006 Question 2: (6 points) normally distributed random variable with a mean of 9.0 square meters and a standard deviation of 0.6 square meters. (a) What percentage of social workers’ ofﬁces are larger than 10.5 square meters? (2 points) (b) What percentage of social workers’ oﬁices are between 8.5 and 10.5 square meters? (2 points) (0) What percentage of social workers’ oﬁices are exactly 9.0 square meters? (2 points) MATH 203 Final Examination December 7, 2006 Question 3: (10 points) The ﬁgure below shows two side-by-side boxplots. The one on the left is a boxplot of 45 anxiety test scores for mothers Who have very low birthweight babies, Where a higher score implies more anxiety. The one of the right is a boxplot of 45 anxiety scores from the same test for fathers of very low birthweight babies. The second and third plots summarize the same data, only in histogram form, with the mothers at the top and the fathers at the bottom. Assume that the mothers and fathers are independently sampled, i.e. they are not paired. 80 T0 :Eg ..___ ID | _‘1_‘ Npuf: (mu Histogram of MSA :i DEM [— |"— F "—1 _|_ r—J —I 20 3D 40 54] 50 3'0 30 90 15 Fraqua ncy Mmhers‘ Amt!er Scares Histogram of FSA In é \— g CIZD: 5 ” 333:3: IL ° !— -:——| —I_““ r—‘I— —: 20 an 40 50 60 TD so 90 Falhers'AnxlalyScores MATH 203 Final Examination December 7, 2006 Here are the summary statistics for the two groups: II Mean Variance125%ile I Median I 75%ile I Min I Max I Mothers 57.42 137.16 48.00 55.00 64.00 37.00 84.00 Fathers 52.16 128.81 45.00 50.00 59.00 34.00 80.00 (a) Construct a 90% conﬁdence interval for a mother’s mean anxiety score for those mothers that have very low birthweight children. (5 points) (b) Would you conclude that there is a difference between mother’s and father’s mean anxiety scores from these data? Why or why not? Carry out a suitable test at signiﬁcance level a = 0.05. (5 points) 01 MATH 203 Final Examination December 7, 2006 Question 4: (8 points) separated or divorced. What is the smallest sample size needed to guarantee that the margin of error for a 95% conﬁdence interval for the proportion of students who have separated or divorced parents is no greater than 4% (i.e. the interval has total width 8%)? (8 points) MATH 203 Final Examination December 7, 2006 Question 5: (10 points) From the late 1980’s it began to become popular for young people to wear their baseball caps backwards. Trinkaus gathered data at two locations — the campus of a private university and the business school of a public university. He took a random sample of people wearing caps and recorded whether it was forwards or backwards. The data are compiled in the table below. I Private University I Business School I Total number observed 236 319 Cap backwards 29 107 (a) Calculate a 95% conﬁdence interval for the true proportion of private university students who wear their caps backwards. (5 points) (b) Could you conclude there difference between public business school students and private university students when it comes to wearing hats backwards? Explain your reasoning statistically by carrying out a suitable test. (5 points) ' MATH 203 Final Examination December 7, 2006 Question 6: (14 points) The owner of a local restaurant chain was interested in increasing the number of poutines that his employees could make. He talked to some Master Poutine Makers (MPM’s) and found two trainers who claimed that they could teach employees to make poutines more quickly. So the owner, who knows a little bit about statistics, took one random sample of 5 employees from his restaurant in the Plateau and one random sample of 5 employees from his restaurant in Old Montreal. In order to set up a baseline speed for each employee, the owner timed each of the employees’ speed in making a single poutine BEFORE they Went to the trainers. Then he sent each group of employees to a separate trainer (the Plateau group went to Trainer A and the Old Montreal group went to Trainer B) to learn how to speed up their poutine creation. When the employees came back, he timed them all again to see how much each had improved. Here are the results from his experiment (given in the number of seconds it takes to make 5 poutines). The tables gives, for each group, the sample mean time BEFORE going to the trainers, the sample mean time AFTER going to the trainers, and the sample mean change in time. Similarly, for each group, the table gives the sample standard deviation of the times BEFORE going to the trainers, the sample standard deviation of the times AFTER going to the trainers, and the sample standard deviation of the changes in times. Before n ifBefore SBefore {E Plateau / Trainer A group 5 100 5 Old Montreal/ Trainer B group 5 110 4 (a) Was there a statistically signiﬁcant difference in mean speeds of the employees in the two groups BEFORE they were sent to the two trainers (use a = 0.05)? (5 points) MATH 203 Final Examination December 7, 2006 (b) Test to see if there was a statistically signiﬁcant difference in the abilities 'of the trainers to speed up the employees’ mean abilities to make poutines. (Use a = 0.05) (5 points) (0) What assumptions about the population of employees and the data collected are necessary for your tests in parts (a) and (b) to be valid? ( 4 points) MATH 203 Final Examination December 7, 2006 Question 7: (10 points) Scientiﬁc journals publish many papers on various topics and typically the journals want their studies to Show statistically signiﬁcant scientiﬁc discoveries. Assume that a particular scientiﬁc journal (“Journal of Reality Television Research”) has a policy of accepting any study that has a scientiﬁcally interesting ﬁnding with a data analysis that yields a p—Value rejecting the null hypothesis that is less than 0.05. Also assume that there is only one scientiﬁc hypothesis tested in each paper. (a) If the Journal of Reality Television Research receives 2000 independent article submissions examining questions Where the null hypothesis is actually true, what is the distribution of the number of articles that will contain Type I errors if all papers reject H0 at a = 0.05? (2 points) (b) What is the approximate probability that at least 10 articles contain Type 1 errors? (8 points) 10 MATH 203 Final Examination December 7, 2006 Question 8: (10 points) Scleroderma is an auto-immune disease that causes inﬂammation of bodily tissue, most notably skin—related tissue. The activity of the disease is measured by a questionnaire that is scored between 0 and 10. The following plot shows a boxplot and histogram for 390 scleroderma patients in the Canadian Scleroderma Research Group registry. Assume that these patients are a random sample of the population of scleroderma patients. Activity Scorn 0246510 E Hlstogram of Activlly Score Activity Seoul Mean Sthev 2.5%ile 25%ile 50%ile 75%ile 97.5%ile 2.8 2.07 0.00 1.43 2.33 4.12 7.74 (a) Construct a 99% conﬁdence interval for the mean of the activity score for scleroderma patients. (5 points) Frequency 0 20 40 60 80 (b) Note that the histogram of activity scores is highly skewed. Will this affect the accuracy of the conﬁdence interval in (a)? Explain. (5 points) 11 MATH 203 Final Examination December 7, 2006 _ Question 9: (12 points) Suppose there are only two cashiers who work at your lucky cash register line at the Metro grocery store on Avenue du Parc / Robert Bourassa. One’s nickname is Speedy and the other’s nickname is Slowpoke. The amount of time that it takes Speedy to process a customer has a uniform distribution in the interval, 0 minutes to 6 minutes. The amount of time it takes Slowpoke to process a customer has a uniform distribution in the interval, 2 minutes to 10 minutes. (a) What is the probability that it takes Speedy fewer than 3 minutes to ﬁnish with your groceries? (4 points) (b) What is the probability that it takes Slowpoke fewer than 3 minutes to ﬁnishwith your groceries? (4 points) (c) Let’s say that you don’t know who will be working on the day that you will go to the store and assume that you always go to your favorite cash register line. Let there be a 50% chance that Slowpoke will be the cashier at your favorite line on a given day and a 50% chance that it will be Speedy, what is the probability that it will take fewer than 3 minutes for whoever is working that day to ﬁnish with your groceries? (4 points) 12 MATH 203 Final Examination December 7, 2006 Question 10: (12 points) Information on C.diﬂicle from the Canadian Public Health Agency’s website: “Clostridium diﬁicile or C. difficile is a bacterium that causes diarrhea and more serious intestinal conditions such as colitis. It is the most common cause of infectious diarrhea in hospitalized patients in the industrialized world. The use of antibiotics increases the chances of developing C. difﬁcile diarrhea. Treatment with antibiotics alters the normal levels of good bacteria found in the intestines and colon. When there are fewer of these good bacteria, C. difﬁcile can thrive and produce toxins that can cause an infection. “ Let’s say that a hospital is trying to determine policies for placing patients in rooms. They are always worried that they will place a person with C difﬁcile bacteria in the room with a pa— tient whowould be extremely susceptible to the bacterium because of high levels of antibiotics. Make the following assumptions to answer the questions below: I If a patient shares a room with a person carrying the bacteria, they will deﬁnitely come in contact with the bacterium (ie. the probability of a patient acquiring the bacteria from sharing a room with someone who has it is 1). o The probability of any patient coming into the hospital carrying the C difﬁcile baterium is 0.01 a The probability of a susceptible patient (one on antibiotics) developing diarrhea without being exposed to the bacterium is 0.10 o The probability of a susceptible patient (one on antibiotics) developing diarrhea after being exposed to the bacterium is 0.70 CONTINUED ON NEXT PAGE 13 MATH 203 Final Examination December 7, 2006 (a) Assume that the hospital is not very crowded right now so there are only two patients per room. Patient Joe Schmo (who right now we know does not carry the C diﬁicile bacterium) is on heavy antibiotics and so is very susceptible to the bacterium and he has had a single roommate who was randomly assigned to room with Joe. Assuming that there are no other ways for Joe Schmo to be exposed to C—diﬁicile other than through his roommate, what is the probability that Joe will have symptoms of diarhrhea after having contact with the other patient in the room? (4 points) (b) The hospital obviously wants to quickly diagnose patients who they believe have been exposed to the Virus. If in a few days, Joe starts showing symptoms of diarrhea, what is the probability that the diarrhea is being caused by the C—difﬁcile bacterium? (4 points) CONTINUED ON NEXT PAGE 14 MATH 203 Final Examination December 7, 2006 (c) Now assume everything above is the same, except instead of having only 2 patients per room, there are 5 patients per room (that is, assume that Joe has four roommates instead of just one). With 5 patients per room, if Joe starts showing symptoms of diarrhea, what is the probability that the diarrhea is being caused by the C-difﬁcile bacterium? (4 points) 15 Appendix A Tables TABLE IV Normal Curve Areas I z .08 .09 3 .0199 .0239 .0279 .0319 .0359 g .0596 .0636 .0675 .0714 .0753 * .0987 1026 .1064 .1103 .1141 .1368 1406 .1443 .1480 .1517 .1736 1772 .1808 .1844 .1879 .2088 .2157 .2190 .2224 .2422 2454 .2486 .2517 .2549 .2734 .2764 .2794 .2823 .2852 .3023 .3051 .3078 .3106 .3133 .3289 .3315 .3340 .3365 .3389 .3531 .3577 .3599 .3621 .3749 3770 .3790 .3810 .3830 .3944 .3962 .3980 .3997 .4015 .4115 .4131 .4147 .4162 .4177 .4265 .4279 .4292 .4306 .4319 .4394 .4418 .4429 .4441 .4505 .4515 .4525 .4535 .4545 .4599 .4608 .4616 .4625 .4633 .4678 .4686 .4693 .4699 .4706 .4744 .4750 .4756 .4761 .4767 .4798 .4808 .4812 .4817 .4842 .4846 .4850 .4854 .4857 .4878 .4881 .4884 .4887 .4890 .4906 .4909 .4911 .4913 .4916 .4929 .4931 .4932 .4934 .4936 .4946 . .4949 .4951 .4952 .4960 .4961 .4962 .4963 .4964 .4970 .4971 .4972 .4973 .4974 .4978 .4979 .4979 .4980 .4981 .4984 .4985 .4986 .4986 _ _W Appendix A Tables 809 TABLE VI Critical Values oft f0) : 1 c1 / J r E i Q .5 1 2 3 4 5 6 7 8 9 . 1o . 4.587 11 1.363 4.437 12 1.356 4.318 13 1.350 4.221 14 1.345 4.140 15 1.341 4.073 16 1.337 4.015 '5 17 1.333 3.965 18 1.330 3.922 19 1.328 3.883 j- 20 1.325 3.850 21 1.323 3.819 22 1.321 3.792 23 1.319 3.767 24 1.318 3.745 E 25 1.316 3.725 g. 26 1.315 3.707 31' 27 1.314 3.690 g 28 1.313 3.674 29 1.311 3.659 30 3.646 3.551 3.460 3.373 E Source: This table is reproduced with Ihc kind permission of Ilw Trustees of Bicmclrika r ram 11'. S. [‘carsnn and HO. Hartley (eds). Thu 8101118”ka Tuba-1 for .S'rm‘r‘srim'mu. Vol. 1.311 cd.. Biumclrika, I966. ...
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MATH_203_december2006 - MATH 203 Final Examination December...

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