CONCORDIA UNIVERSITY Department of Economics ECON 221/2 SECTIONS A, B, C and DD STATISTICAL METHODS I FALL 2016 - ASSIGNMENT 1 Due: Monday, October 17, before 3:00 pm Name: I.D: Section: Points Total: 100 points 1. (16 points) Of the employees that work for Hapolonians Incorporated, 35 percent have an MBA and 40 percent are over the age of 35. Of those employees that have an MBA, 30 percent are over the age of 35. a. (2 points) Calculate the probability that a randomly chosen employee has both an MBA and is over the age of 35. b. (2 points) Calculate the probability that a randomly chosen employee who is over the age of 35 has an MBA. c. (2 points) Calculate the probability that a randomly chosen employee has an MBA or is over the age of 35. 1 d. (2 points) Calculate the probability that a randomly chosen employee who is not over the age of 35 does not have an MBA. e. (2 points) Determine whether having an MBA is independent of being over the age of 35. f. (2 points) Determine whether having an MBA and being over the age of 35 are mutually exclusive. g. (2 points) Determine whether having an MBA and being over the age of 35 are collectively exhaustive. h. (2 points) Complete the following table assuming Hapolonians Incoporated employs 200 people. Number of employees A A Total M M Total 200 2 2. (4 points) Hapolonians Incorporated's 12-member board of directors is to be selected from a list of 8 men and 8 women. a. (2 points) Calculate how many different boards of directors are possible. b. (2 points) Calculate the probability that a majority of the board of directors will be men. 3. (12 points) Delays plague the airport in Flin Flon, Manitoba. The probability distribution for the number of delays on any given day is as follows: Number of delays 0 1 2 3 4 Probability 0.10 0.26 0.42 0.16 0.06 a. (2 points) Calculate the cumulative probability distribution. Number of delays 0 1 2 3 4 Cumulative probability b. (2 points) Calculate the probability that no more than three flights are delayed on any given day. 3 c. (2 points) Calculate the expected number of delayed flights on any given day. d. (2 points) Calculate the standard deviation of delayed flights on any given day. Each delay costs the airport $1500. e. (2 points) Calculate the expected cost of delayed flights on any given day. f. (2 points) Calculate the variance of the cost of delayed flights on any given day. 4 4. (8 points) In US football, a team that scores a touchdown must decide whether to try to get one or two bonus points. If it succeeds, it receives the number of bonus points that it attempted to get; if it fails, it gets nothing. One coach believes that the probability of succeeding on a two-point bonus play is 35 percent and that each attempt is an independent event. In one particular game, the coach's team attempted the two-point bonus play four times. a. (2 points) Calculate the probability that at least two of these attempts were successful. b. (2 points) Calculate the expected total number of points resulting from the four attempts. c. (2 points) Calculate the standard deviation of the total number of points resulting from the four attempts. d. (2 points) If the coach were to try the easier one-point bonus play all four times, calculate the required success rate to achieve the same expected outcome obtained in part (b). 5 5. (4 points) Sheep escape Michelle's farm according to a Poisson distribution at an average of three per week. a. (2 points) Calculate the probability that more than one sheep escapes in any given week. b. (2 points) Michelle can be 99-percent certain that no more than this many sheep escape in any given week. 6. (12 points) Corduroy and Tootie each run their own hedge fund, although it is thought that they share investment strategies because they are such close friends. It is found that they both make money 70 percent of the time and they both lose money 30 percent of the time, but Corduroy never makes money when Tootie loses money and Tootie never makes money when Corduroy loses money. Let X = 1 represent when Corduroy makes money and X = 0 when he loses money. Similarly, let Y = 1 represent when Tootie makes money and Y = 0 when he loses money. a. (2 points) Calculate the marginal probability distributions for X and Y. 6 b. (2 points) Calculate the expected values of X and Y. c. (2 points) Calculate the variances of X and Y. d. (2 points) Calculate the covariance of X and Y. e. (2 points) Calculate the correlation for X and Y. f. (2 points) Based on your answer to part (h), do you believe that Corduroy and Tootie share investment strategies? 7 7. (10 points) The random variable X follows a normal distribution with = 0.2 and 2 = 0.0025. ( ) a. (2 points) Calculate Pr X > 0.4 . ( ) b. (2 points) Calculate Pr 0.15 < X < 0.28 . ( ) c. (2 points) Calculate Pr X < 0.1 . ( ) d. (2 points) Calculate the value of x such that Pr X > x = 0.025 . e. (2 points) Calculate the values of x0 and x1 that are equidistant from the mean such that ( ) Pr x0 < X < x1 = 0.34 . 8 8. (4 points) The amount of time that Professor Hapoleon spends with students during office hours averages 10 minutes and follows an exponential distribution. a. (2 points) Calculate the probability that he spends more than 20 minutes with a given student. b. (2 points) Calculate the probability that he spends between 10 and 15 minutes with a given student. 9. (16 points) A random sample of 12 statistics students have the following ages (in years): 21 22 27 36 18 19 22 23 22 28 41 33 a. (10 points) Calculate the various elements of the five-number summary. b. (2 points) Calculate the sample mean. 9 c. (2 points) Calculate the sample standard deviation. d. (2 points) Use Chebyshev's theorem to state the range of ages, centered about the mean, that contains at least 75 percent of the observations. 10. (4 points) A random sample of 25 ECON221 students needed the following number of hours to complete this assignment: Completion time 0.00 - 3.99 4.00 - 7.99 8.00 - 11.99 12.00 - 15.99 16.00 - 20.00 Number of students 3 7 8 5 2 a. (2 points) Calculate the sample mean. b. (2 points) Calculate the sample variance. 10 11. (10 points) Excel Question. Consider two random variables: X represents the throw of a fair die and Y satisfies your birth month and Pr Y = 2 = Pr Y = 3 = Pr Y = 4 = Pr Y = 5 = Pr Y = 6 . Pr Y = 1 = 15 The sample space for each variable is {1, ..., 6}. ( ) ( ) ( ) ( ) ( ) ( ) a. (2 points) Fill in columns 2 and 3 in the following table where each entry is the probability of the random variable having the indicated outcome. b. (3 points) In Excel, simulate 100 draws of the random variable X; then 100 draws of the variable Y. Record the results as proportions in columns 4 and 7. Then repeat the simulations where the number of draws is 1000; record the results in columns 5 and 8. Finally, take 10000 draws of the two variables and record results in columns 6 and 9. X variable # of draws Y variable # of draws N P(X = n) P(Y = n) 100 1000 10000 100 1000 10000 1 2 3 4 5 6 c. (2 points) Briefly comment on the results. d. (3 points) Attach two pages. On one, place the 3 histograms for the X variable; on the other, the 3 histograms for the Y variable. 11