Download Math 111 Final Exam Spring 1997: Probability and Statistics and more Exams Probability and Statistics in PDF only on Docsity! Math 111 FINAL EXAM Spring 1997 There are eight answer pages. Put the answer to only one question on each page. Put the problem number and. your name on each answer page, as well as the name of your instructor. Write solutions completely showing all work for full credit. DNC means Do Not Compute. 1. In a class of 150 seventh graders, 60 have had chicken pox, 50 have had measles and 40 have had pertussis. Moreover, 30 have had both chicken pox and measles, 15 chicken pox and pertussis, 10 measles and pertussis, and 5 have had all three. a) (10 points) Draw a Venn diagram corresponding to the above data. Fill in the numbers corresponding to every cell. b) (10 points) Whci is the probability that a student in this class has not had any of these three diseases? c) (10 points) In this group of students, are the two events of 'had chicken pox' and 'had measles' independent events? Why or why not? 2. A self-administered (diabetes test has an accuracy rate of 95%, both for those who have and who do not have the disease. Assume that in the general population, 10% of people over 40 years old have diabetes. a) (15 points) If an individual who is over 40 is chosen at random and takes this test, what is the probability she will test positive for diabetes? b) (15 points) Given that she tests negative, what is the probability that she does not have diabetes? 3. A flower shop has 100 roses, 40 tulips and 12 orchids in stock. A customer in a hurry randomly picks out 6> of these for a gift. a) (10 points) What is the probability that the gift will consist of 3 roses, 2 tulips and 1 orchid? DNC b) (10 points) What is the probability that the gift will contain no orchids? DNC 4. Suppose that the daily high temperature in Washington, D.C. during the summer is a uniformly distributed random variable T that ranges from 90° to 108°. a) (10 points) What is the probability that on August 7, 1997, 92.5 < T < 98.5? b) (15 points) A compulsive gambler wants to make the following bet with you. If the high temperature on August 7, 1997 is within 3° of 95.5°, then he wins $10, otherwise you win $3. Is this a fair bet? If not, does it favor you or the gambler? 5. Let X be a random variable with mean 3 and standard deviation a = 2. a) (5 points) Use Chebychev's Inequality to estimate P( 3 < X < 9). b) (5 points) Let / 8. Use Chebychev's Inequality to estimate P(3 — l < X < 3 + /). a) (5 points) In order to conclude that P(3 I < X < 3 + 1) > .96, should we choose / larger than 8 or smaller than 8? 6. The life span of white mice is approximately normally distributed with a mean of 15 months and a standard deviation of 4 months. a) (15 points) What is the probability that a randomly chosen white mouse will live between 14 and 17 months? b) (15 points) Find an age w so that the probability that a randomly chosen mouse will live longer than w is 0.0668. 7. A vending machine malfunctions with a probability of .15. Suppose 10 people line up to use the machine. Let X' be the number of times it malfunctions and let X be the number of times it works properly. a) (10 points) Write a relation between X' and X. b) (15 points) Use Table 2 to compute P(X = 7). 8. (25 points) Let X be a binomial random variable with n — 200 and p = 0.7. Use the normal approximation to estimate P(X < 150).
