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İstanbul Kültür University Faculty of Engineering MCB1007 Introduction to Probability and Statistics Final Exam Fall 2015-2016 Number: Name: Department: – You have 90 minutes to complete the exam. Please do not leave the examination room in the first 30 minutes of the exam. There are six questions, of varying credit (100 points total). Indicate clearly your final answer to each question. You may use a calculator. During the exam, please turn off your cell phone(s). You can’t use any book or lecture note. You have one page “cheat-sheet”. The answer key to this exam will be posted on Department of Mathematics and Computer Science board after the exam. Good luck! Emel Yavuz Duman, PhD. M. Fatih Uçar, PhD. Arzu Yemişçi, PhD. Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. TOTAL Question 1. 7 + 7 points A fair die is tossed and its outcome is denoted by X. After that, X independent fair coins are tossed together and the number of heads obtained by Y . (a) P (Y = 4) =? Answer. (b) P (X = 5|Y = 4) =? Answer. Question 2. 6 + 6 + 6 points A study of the annual growth of certain cactus showed that, 64 of them, selected at random in a desert region, grew on the average 52.80 mm with a standart deviation of 4.5 mm. (a) Construct a 99% confidence interval for the true average annual growth of the cactus. Answer. (b) What should be the minimum sample size if the error does not exceed 1.2? Answer. (c) With what degree of confidence could we say that the mean of annual growth of all 64 cactus is (51.5, 54.1)? Answer. MCB1007 - Int. to Prob. and Statistics 2 Final Exam Question 3. 7 + 10 points An egg carton contains 10 eggs, 2 of which are cracked. A sample of 5 eggs is randomly selected from the carton. (a) Find the probability distribution for X, the number of cracked eggs in the sample. Answer. (b) What are the mean and variance of X? Answer. Question 4. 8 + 8 + 8 points A fair coin is tossed 5 times. Find the probability of getting 2, 3 or 4 heads by using (a) Binomial distribution. Answer. (b) The normal approximation to the binomial distribution. Answer. (c) The Poisson approximation to the binomial distribution. Answer. MCB1007 - Int. to Prob. and Statistics 3 Final Exam Question 5. 7 + 8 points From the set of numbers {3, 5, 7}, a random sample of size 2 will be selected with replacement. (a) Determine the sampling distribution of the mean. Answer. (b) Calculate the mean and standard deviation for the population distribution and for the distribution of X̄. Answer. Question 6. 12 points Estimate the mean, median and modal class of the following data: Number of Orders 10 − 12 13 − 15 16 − 18 19 − 21 Frequency 4 12 20 14 Answer. MCB1007 - Int. to Prob. and Statistics 4 Final Exam