STOR 151 SECTION 1 FINAL EXAM DECEMBER 11 2018 YOUR NAME PID SIGNATURE Honor pledge On my honor I have neither given nor received unauthorized aid in this exam Please write your answers in a blue book except for the graph in Question 7 which is at the end of this exam Please hand in this question paper as well as your blue book This is an open book exam Course text personal notes and calculator are permitted You have 3 hours to complete the test Personal computers and cellphones are not allowed If you have any queries about the meaning of a question ask the instructor for advice SHOW ALL WORKING even correct answers will not get full credit if it s not clear how Incorrect answers will gain substantial credit if the method of working is they were obtained substantially correct Answer six of the eight questions If you attempt more than six all the answers will be graded but only the best six complete questions will count Each question is worth a total of 20 points and the whole exam is worth 120 points which will be rescaled to a maximum of 40 for grading purposes Points for each individual part of a question are also given in square brackets Questions may be answered in any order 1 An irregular eight sided die has the following probabilities Number on face of the die Probability 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 7 8 6 0 1 x x a Find x 3 points b Let A be the event the die shows an odd number Let B be the event the die shows one of 1 2 3 4 Find P A P B P A and B and P A or B Are the events A and B independent 7 points c Suppose the die is tossed three times What is the probability that the total is greater than 22 5 points d If the same die is thrown a large number of times the mean score is 5 1 and the standard deviation is 2 39 you can assume these numbers without proof If the die is thrown 100 times what is the approximate probability that the average score is at least 5 5 5 points 1 2 A new test for type 2 diabetes classi es each patient on a 3 point scale where a score of 1 means the patient is highly likely to have type 2 diabetes a score of 3 means the patient is unlikely to have type 2 diabetes and a score of 2 is intermediate For a patient who has type 2 diabetes the probabilities of a score of 1 2 or 3 are respectively 0 6 0 25 0 15 For a patient who does not have type 2 diabetes the probabilities of a score of 1 2 or 3 are respectively 0 1 0 1 0 8 Within the population being tested 8 of all patients have type 2 diabetes a Represent this information in the form of a tree diagram What is the probability that a randomly chosen patient both has diabetes and gets a score of 1 5 points b What is the overall probability that a patient gets a score of 2 or 3 5 points c Given that a patient has a score of 1 what is the probability that he she has type 2 diabetes 5 points d A physician decides that it is too complicated for him to explain probabilities to his patients so he simply tells every patient with a score of 1 or 2 that they have type 2 diabetes and every patient with a score of 3 that they do not What proportion of all patients are given the wrong diagnosis by this physician 5 points 3 A large survey of 17 year old boys in the USA included the question Do you work out with weights at least twice a week 71 of the respondents said yes Assume that this proportion applies to the full population of 17 year old boys in the USA a Among a sample of 10 boys what is the probability that at least 8 of them work out with weights at least twice a week 5 points b Among a sample of 31 boys what is the probability that exactly 22 71 of 31 of them work out with weights at least twice a week 5 points c Among a sample of 121 boys what is the approximate probability that at least 91 of them work out with weights at least twice a week 5 points d Now suppose I tell you that the sample in part c actually came from Germany i e not part of the original survey and that the sample reported exactly 91 boys of the 121 who do work out with weights at least twice a week Does this sample provide evidence that 17 year old boys in Germany have di erent exercise habits from those in the USA State carefully what assumptions you are making and how you reached your conclusion 5 points 4 A sample of ten 15 year old girls from public schools in North Carolina has a mean height of 64 1 inches with a sample standard deviation of 2 7 inches a Calculate a 90 con dence interval for the mean height of all 15 year old girls from public schools in North Carolina 7 points b Test the null hypothesis that the mean height of all 15 year old girls from public schools in North Carolina is 65 8 inches Use a two sided test at signi cance level 0 05 and state any assumptions you make 8 points 2 c Suppose the true population standard deviation for the mean height of all 15 year old girls from public schools in North Carolina is 3 2 inches How large a sample size would you need for the total width of the con dence interval in a to fall below 1 inch 5 points 5 At the start of a new track season the coach has every member of his team run a 400 meter time trial Six weeks later he repeats the trial These are the recorded times to the nearest tenth of a second Runner Trial 1 Trial 2 Di erence A 53 3 52 6 0 7 B 54 3 53 4 0 9 C 51 5 51 3 0 2 D 51 9 51 5 0 4 E 52 6 52 1 0 5 F 51 3 51 1 0 2 G 50 2 50 3 0 1 H 54 9 53 8 1 1 I 51 0 50 9 0 1 Mean 52 333 51 889 0 444 SD 1 571 1 178 0 394 a Analyze the data as a two sample experiment Is there a statistically signi cant improve ment between the running times in Trials 1 and 2 Use a one sided test with signi cance level 0 05 7 points b Analyze the data as a paired comparison experiment Is there a statistically signi cant …
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