University of Notre Dame Math 114: Statistics Exam III Nov 21, 2003 Instructor: Bei Hu

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1 University of Notre Dame Math 114: Statistics Exam III Nov 21, 2003 Instructor: Bei Hu Name: Record your answers to the multiple choice problems by placing an through one letter for each problem on this answer sheet. There are 20 multiple choice questions for 5 points each. Everyone is under the The Academic Code of Honor. No books and notes are allowed on this exam and it is a violation of The Academic Code of Honor to use them. You may use your graphing calculator. 1. Fill in the blank. A negative standard Z score indicates. (a) the observed score is below the mean score. (b) a mistake has been made because the value must be between 0 and 1. (c) the observed score is above the mean score. (d) the area to the left of the observed score is negative. None of the other answers. 2. The variable Z is N(0, 1). Find the quartiles Q 1 and Q 3 for Z. (a) Q 1 = 0.67, Q 3 = (b) Q 1 = 1, Q 3 = 1 (c) Q 1 = 0, Q 3 = 1 (d) Q 1 = 0.67, Q 3 = 1.34 Q 1 = 0, Q 3 = 2 3. The distribution of a random variable X is symmetric, with a mean of 500 and a variance of 100. You are asked to provide two numbers such that about 68% of the values of X are between the two numbers. (a) Can t Tell. (b) ( , ), i.e., (400, 600). (c) (500 10, ), i.e., (490, 510). (d) (500 2(10), (10)), i.e., (480, 520). (500 3(10), (10)), i.e., (470, 530).

2 2 4. A dairy refuses to accept raw milk having more than 5000 bacteria per milliliter (ml). The bacteria count varies from shipment to shipment. Assume that the count of bacteria per milliliter for a shipment is normally distributed with a population standard deviation of 16. The dairy wants to test that the mean bacteria count for shipments is less than 5000 per ml. The null and alternative hypotheses for this problem in terms of µ, the mean bacteria count for shipments are: versus H 0 : µ = 5000 H 1 : µ < The next shipment gives a count of 4970 bacteria per ml. What is the p-value for testing the above null hypothesis? (a) 0.03 (b) 0.97 (c) 0.17 (d) In the population of students who took the Graduate Record Exam in a particular year, the analytical ability scores were normally distributed with a mean of 520 and a standard deviation of 10. What proportion of these students had scores between 500 and 540? (a) 95%. (b) 68%. (c) 99.7%. (d) 97.5%. 50%. 6. Fill in the blank. If two events A and B have no outcomes in common, A and B are said to be. (a) dependent. (b) independent. (c) conditional events. (d) None of the other answers. mutually exclusive or disjoint. 7. Fill in the blank. If knowing that event A has occurred does not changes the probability of event B occurring, then A and B are said to be. (a) mutually exclusive or disjoint. (b) dependent. (c) independent. (d) conditional events. None of the other answers. 8. Let X be the random variable representing the number of calls received in an hour by a 911 emergency service. A portion of the probability distribution of X is given below. Value X = x Probability P (X = x) Suppose the probability that X = 1 and the probability that X = 2 are the same. What are these probabilities (a) 0.3 (b) 0.6 (c) 0.15 (d)

3 3 9. In the neighborhood called Pine Springs, there are a lot of pine trees in people s yards. Let X be the random variable representing the number of pine trees per back yard in the neighborhood. A frequency table is shown below. Number of trees Frequency Suppose we randomly sample one back yard from this neighborhood. What is the probability that it has at least 1 pine tree knowing that this yard has fewer than 4 trees. (a) 32/42 (b) 35/42 (c) 18/42 (d) 35/45 18/ A typist knows from previous experience that even though she is an experienced typist, she does make mistakes. Let X be the random variable representing the number of mistakes made by the typist on a randomly selected page from all those she has typed. The probability mass function of X is given below. Number of mistakes x Probability P (X = x) What is the expected value of the random variable X? (a) 1.07 (b) 1.37 (c) 1.00 (d) 0.45 None of the other answers. 11. At a certain university there are 500 professors. Among them, 300 are assistant professors, 50 are associate professors, and 150 are full professors. The percentages of female professors for these three groups are 15%, 40%, and 30%, respectively. Suppose we randomly select one professor from this university. If we know that this professor is female, what is the probability that she is a full professor? (a) 0.30 (b) 0.41 (c) 0.28 (d) In California, 65% of teenagers have blond hair. Suppose we randomly select 5 California teenagers. What is the probability that exactly 4 of the teenagers have blond hair? ( ) (a) (0.35) 4 (0.65) 1 5, i.e., 0.01 (b) (0.65) 4 (0.35) 1, i.e., 0.31 (c) (0.65) 4 (0.35) 1, i.e., 0.06 (d) , i.e., ( 5 4 ) (0.35) 4 (0.65) 1, i.e., 0.05

4 4 13. Fill in the blank. A statistic is said to be if the mean of the sampling distribution of a statistic is equal to the corresponding population parameter value. (a) biased. (b) dependent. (c) unbiased. (d) independent. parametric. 14. You have applied for a summer job at Store A and also at Store B. You estimate that probabilities of getting the job at the stores A and B are 0.1 and 0.7 respectively. You also estimate that the probability of getting both jobs is only What is the probability that you get at least one of the jobs? (a) 0.72 (b) 0.04 (c) 0.76 (d) 0.8 none of the other answers. 15. Suppose that the percentage of drivers in Jefferson County who prefer to drive a car with manual transmission is 35%. Can you determine the approximate sampling distribution of ˆp for a sample size of n = 100? (a) Yes, it is N(0.35, ) (b) Yes, it is N(0.65, ) (c) Yes, it is N(0.35, ) (d) Yes, it is N(0.65, ) No, need more information. 16. Let X be the sum of the faces obtained by rolling a fair die twice. Find the probability of the event X = 4: (a) 1/9 (b) 1/18 (c) 1/12 (d) 1/6 1/ Twenty-nine percent of all members of the West Coast Bear Collectors Club prefers to collect teddy bears made with mohair fur instead of plush fur. In a random sample of 180 members of this Bear Club, only 20% were found to be collectors of mohair bears. In this story, what is the value of the parameter? How many of the sampled members are mohair collectors? (a) The value of the parameter is 0.20 (the population proportion) = 52 (b) The value of the parameter is 0.29 (the population proportion) = 52 (c) The value of the parameter is 0.20 (the population proportion) = 36 (d) The value of the parameter is 0.71 (the population proportion) = 36 The value of the parameter is 0.29 (the population proportion) = 36

5 5 18. The density for waiting times in minutes at an elevator is sketched below. Fill in the blank. The value of 1 is called the. (a) mean (b) median (c) Q 3 (d) Q 1 none of the other answers. 19. Fill in the blank. A is the imitation of random behavior using a random device such as a random number generator. (a) simulation (b) probability (c) phenomenon (d) random process event 20. The variable T is U(18, 28). Find the proportion of values below 18, the proportion of values above 21, the proportion of values between 21 and 24, and the proportion of values equal exactly to 21. They are: (a) 0, 0.5, 0.3, 0 (b) 0, 0.7, 0.3, 0.1 (c) 0.1, 0.5, 0.5, 0.1 (d) 0.1, 0.7, 0.3, 0 0, 0.7, 0.3, 0