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2003 - Student Name Student Number Faculty of Science FINAL...

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Student Name: Student Number: Faculty of Science FINAL EXAMINATION Mathematics 203 Principles of Statistics I Friday, December 19th 9 a.m. - 12 Noon Answer directly on the test (use front and back if necessary). Calculators are allowed. One 8.5” × 11” sheet of notes is allowed. Language dictionaries are allowed. There are 16 pages to this exam and 3 pages of tables. The total number of marks for the exam is 100. Examiner: Professor Russell Steele Associate Examiner: Professor Masoud Asgharian 1

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Question 1: (6 points) The GRE is a standardized test required for admission to many graduate school programs in the United States. The test has three sections, one of which is a Verbal section. Assume that the population Verbal scores have a normal distribution with mean 500 and standard deviation 100. (a) What is the probability that a randomly selected student from the population scores above a 550 on the Verbal section? (3 points) (b) One graduate school program in English will only admit students who score in the top 25% of the students. What is the lowest GRE score they will accept? (3 points) 2
Question 2: (10 points) Farmer MacGyver has a very reliable and special chicken that produces double-yolked eggs 10% of the time and produces lots of eggs in one sitting. Let’s assume that yesterday morning Farmer MacGyver went to the henhouse and found that his special hen laid 20 eggs. (a) What is the probability that there are exactly 4 eggs with double yolks in the batch of 20 eggs? (3 points) (b) What is the probability that at least 5% of the eggs in the batch of 20 eggs have double yolks? (3 points) Now assume that he goes to the henhouse today and finds that his chicken laid 75 (!!!) eggs. (c) What is the probability that less than 10% of the eggs in the batch of 75 eggs have double yolks? (4 points) 3

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Question 3: (6 points) In class, we learned about the Bernoulli random variable. A Bernoulli random variable, X , takes on two possible values, 1 and 0. Assume that X takes on the value 1 with probability p (i.e. Pr ( X = 1) = p ) and X takes on the value 0 with probability 1 - p (i.e. Pr ( X = 0) = 1 - p ). (a) (3 points) Show that E ( X ) = p . (b) (3 points) Show that V ar ( X ) = p * (1 - p ). 4
Question 4: (10 points) A clinical trial examined the effectiveness of aspirin in the treatment of cerebral ischemia (stroke). Patients were randomized into treatment and control groups. The study was double-blind in the sense that neither the patients nor the physicians who evaluated the patients knew which patients recieved aspirin and which the placebo tablet. After 6 months

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2003 - Student Name Student Number Faculty of Science FINAL...

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