STAT 2507 Assignment # 2 (Chapters 4,5) Fall 2016 Last Name -, First Name Student # Lab Section (IMPORTANT) - Due Tuesday November 1st (to be handed at the end of the class) N as ar i Total mark 100 Part I. Lab questions. Do not include ANY Minitab code to your assignment. Write your answers for the Minitab questions on the provided lines. on ta z er i 1. (Mean and variance of random variables) Consider the random variable X whose pmf is c M x 1 2 3 4 5 6 7 8 P (X = x) 0.004 0.021 0.074 0.171 0.267 0.267 0.156 0.04 Use the following Minitab commands to enter the pmf of X into Minitab so that the possible values of X are stored in column c1 and the probabilities are stored in column c2 by: (First click Editor Enable commands) and type the below commands MTB> set c1 DATA> 1 2 3 4 5 6 7 8 DATA> end MTB> set c2 DATA> 0.004 0.021 0.074 0.171 0.267 0.267 0.156 0.04 DATA> end a. Obtain the mean of X, i.e. = E(X) by typing the following command MTB>let c8= sum(c1*c2) (the mean will appear in the first cell of column c8) What is ? [2] . 1 b. By typing the following command obtain the variance V ar(X) = 2 of X. MTB>let c9= sum((c1**2)*c2) - (sum(c1*c2))**2 (the variance will appear in the first cell of column c9) What is 2 ? [2]. c. What is the the standard deviation ? [2] -. N as ar i 2. (Binomial Distribution) Suppose that X has a binomial distribution with n = 40 and p = 0.1. Use Minitab to simulate 50 values of X. MTB>random 50 c1; (don't forget to enter the semicolon at the end) SUBC>binomial 40 0.1. (don't forget to enter the dot at the end) on ta z er i a. What proportion of your values are strictly less than 5? [2] . Hint: to answer part (a) you can sort the simulated values of column c1 in another column, say c7 by typing the command: MTB> let c7=sort(c1) c M b. What is the exact probability that X will be strictly less than 5? [2] -. c. Using this cdf also obtain the following probabilities P (X 6) [2] P (X = 8) [2] - Hint: To answer the preceding parts (b) and (c) you need to find P (X k) for any k = 0, . . . , 40, use \"cdf\" command; this works by typing MTB>cdf; SUBC>binomial 40 0.1. d. Find the value a such that P (X a) = 0.98450. [2] To answer this question you need to use the \"inverse cdf\" command by typing MTB>invcdf 0.98450; SUBC>binomial 40 0.1. 3. (Poisson Distribution) 2 Suppose that Y has a Poisson distribution with mean = 4. Use the cdf command MTB>cdf; SUBC>poisson 4. Find [2] P (Y 6) = - and [2] P (Y = 8) = - . er i N as ar i 4. (Poisson approximation to Binomial) Refer the Questions 2 and 3. a. Compare P (X 6) you obtained in part (b) of Question 2 for the binomial random variable X to P (Y 6) you obtained in Question 3 for Poisson with mean = 4. Are their values relatively close to each other? [1] -. b. Compare P (X = 8)) you obtained in part (b) of Question 2 for the binomial random variable X to P (y = 8) you obtained in Question 3 for Poisson with mean = 4. Are their values relatively close to each other? [1] -. c. Should these probabilities in each of part (a) and part (b) of this question be close to each other or not? [1] on ta z d. Explain why they should be close or why they should not be close? [3] c M 5. (Hypergeometric Distribution) In a party there is an ice bin that contains 20 cans of soft drink of which 8 are Pepsi and the rest are Cokes. All cans are covered by ice so we cannot see their brand. If we randomly select 5 cans without replacement from the ice bin and let H be the number of Pepsi among the the selected 5. We know that H has hypergeometric distribution with N = 20, M = 8 and n = 5. Derive the cdf of H by: Type 0, 1, 2, 3, 4, 5 in column c10, use the command MTB>cdf c10; SUBC>hypergeometric 20 8 5. Using the cdf of H answer the following parts(a) and (b). a. What is the probability that there will be at most 2 (inclusive) Pepsi among the chosen drinks. [2] b. What is the probability that there will exactly 2 cokes among the chosen drinks. [2] - 3 Part II. Long-answer questions; Give the solutions for the following questions in details 1. An aerospace company has submitted bids on two separate federal government defense contracts, A and B. The company feels that it has a 60% chance of winning contract A and a 30% chance of winning contract B. If it wins contract B, it believes that it has an 80% chance of winning contract A. c M on ta z er i N as ar i a) [2] Are the events of winning contract A and winning contract B independent? Explain. b) [2] What is the probability that the company will win both contracts? c) [2] Are the events of winning contract A and winning contract B mutually exclusive? Explain. d) [2] What is the probability that the company will win at least one of the contracts? e) [2] What is the probability that the company will win neither contract? f) [2] What is the probability that the company will win contract A but not contract B? g) [2] If the company wins contract A, what is the probability that it will win contract B? h) [2] If the company wins contract B, what is the probability that it will not win contract A ? 2. Let P (A|B) = 0.5, P (B) = 0.25, P (A B) = 0.75. a) [6] Find (i) P (A B) , (ii) P (A) , (iii) P (B 0 |A0 ) (Hint: (A0 B 0 ) = (A B)0 \"De morgan Law\") b) [1] Are A and B independent events? 3. The results of three flips of a biased coin are observed. The probability of a head occurring is 0.8. Consider the following events: A: At least two tails are observed B: exactly one head is observed C: exactly two heads are observed a) [2] List an appropriate sample space S for this experiment. b) [1] Are the events of getting a head on flip independent for the 3 flips? c) [3] Assign probabilities to the outcomes in S. d) [3] Find (i) P (A) (ii) P (A B) (iii) P (B|A) 4 e) [2] If the random variable X= number of heads in 3 flips, set up the probability distribution in table form. 4. Each week a retail outlet accepts delivery of a certain item from 3 different suppliers, A, B and C. All the items received are put into an empty bin. A supplies 50% of these items, while B and C each supply 25%. From past experience, it is known that 2% of the items supplied by A are defective, 2% of the items supplied by B are defective and 4% of the items supplied by C are defective. Suppose an item is chosen at random from the bin. N as ar i a)[3] What is the probability that it is defective? b) [3] What is the probability that it came from supplier C? on ta z er i 5. When circuit boards used in the manufacture of compact disc players are tested, the long run percentage of defective is 5%. Let X= the number of defective boards in random sample of size n = 25, so X Bin(25, 0.05). a) [6] Find the probability that the number of defective boards is (i) At most 2 (ii) At least 5 (iii) Between 1 and 4, inclusive. b) [1] What is the probability that none of the 25 boards is defective? c) [2] Calculate the expected value and standard deviation of X. c M 6. The number of flaws on a magnetic tape produced continuously at a factory follows a Poisson distribution with an average of 0.02 flaws per meter. A standard tape contains 250 meters of magnetic tape. a) [3] What is the probability that there are at least two flaws in a single tape? b) [3] What is the probability that there are no flaws in a single tape; that is, a tape is flawless? c) [5] In a random sample of 25 tapes, what is the probability that at least one of them are flawless? 7. A box of candy contains 30 pieces. Twenty-six are made of chocolate and four are made of vanilla. a) [5] Five pieces are selected at random without replacement. What is the probability that four of them are chocolates? b) [5] What would be the answer to part (a) if the five pieces are selected at random with replacement (i.e., a selected piece is put back in the box before the next selection is made)? 5