STAT:2020 Probability and Statistics for Engineers Exam 2 Mock-up. 100 possible points

Transcription

1 STAT:2020 Probability and Statistics for Engineers Exam 2 Mock-up 100 possible points Student Name Section [letter/#] Section [day/time] Instructions: 1) Make sure you have the correct number of pages. There are 9 pages and 16 questions. 2) Please use a pencil. 3) Questions can be clarified, but no hints will be provided. 4) You have 2 hours to complete the exam. 5) Show your student ID to your TA when handing the exam in. 6) Please hand-in your formula sheet with your exam. 7) If not stated, round answers to four significant digits. 8) You may begin, good luck. 1

2 Questions 1-3. Suppose random variables X and Y have the joint probability distribution below: 1. Find P (Y = 2) X Y a) 0.08 b) 0.22 c) 0.68 d) 0.05 e) None of the answers is correct to the second decimal place. 2. Find E[Y X = 0] a) 1.52 b) 0.15 c) 1.30 d) 1.60 e) None of the answers is correct to the second decimal place. 3. Are X and Y independent? a) Yes b) No 2

3 Question Which of the following is/are true? (a) The mode represents the balance point for a distribution. (b) An outlier has a large impact on the mean. (c) The mean is higher than the median in a left-skewed distribution (i.e. median<mean). (d) Both (b) and (c) (e) All of the above Question The following is a joint pmf for X and Y. x y Find the covariance of X and Y. a) 750 b) 0 c) 2850 d) 12,750 e) None of the answers are correct. 3

4 Questions 6-7. Suppose X and Y have the joint probability density function f XY (x, y) = 1 18 x2 y for 0 < x < 3 and 0 < y < Find P (X < 1.5) a) b) c) d) e) None of the answers is correct to four decimal places. 7. Find P (Y > 1 X < 1.5) a) 0.15 b) 0.64 c) 0.75 d) 0.56 e) None of the answers is correct to the second decimal place. 4

5 Question To build a component, workers must fit a rod into the hole of a metal ring. The dimensions of the parts can vary a bit due to manufacturing variability. The diameter of the hole follows a normal distribution with a mean of 12 inches and a variance of inches 2. The diameter of the rod follows a normal distribution with a mean of 11.8 inches and a variance of inches 2. What is the probability that a randomly chosen rod will fit into a randomly chosen ring? a) b) c) d) e) None of the answers is correct to four decimal place. Question Let X and Y have a bivariate normal distribution with µ X = 100, µ Y = 50, σ X = 7, σ Y = 9, ρ = 0.8. (You do not need to show work). Circle all correct answers below: (a) P (Y > 50) is less than P (Y > 50 X = 120) (b) P (Y > 50) is the same as P (Y > 50 X = 120) (c) P (Y > 50) is greater than P (Y > 50 X = 120) (d) P (Y > 50) is less than P (X > 120) (e) P (Y > 50) is the same as P (X > 120) (f) P (Y > 50) is greater than P (X > 120) 5

6 Questions Cracks in an oil pipeline follow a Poisson process at a rate of 0.8 cracks per mile. 10. What is the probability of three cracks occurring in a 5 mile stretch? a) b) c) d) e) None of the answers is correct to the second decimal place. 11. If you re inspecting the pipeline and start at a particular position, what is the probability that the first crack occurs between miles 2 and 3? a) b) c) d) e) None of the answers is correct to the four decimal places. 6

7 Question Let X be the quantity of cell phone minutes my family uses each month. From past history, I know that X has the probability density function below f(x) = ( ) , x2 + x for 0 x 600 In months where my family uses more minutes, my cell phone company charges me less per minute. Specifically, the monthly cost function, C(x), in dollars with respect to the number of minutes, x, used each month is: C(x) = 2x 1/2 What is the expected cost of my family using the cell phone in a month? (round to nearest dollar). a) $35 b) $34 c) $30 d) $28 e) None of the answers is correct to the nearest dollar. Question A soft-drink machine is regulated to discharge an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation of 15 milliliters, what is the probability that a cup contains between 191 and 209 milliliters? a) b) c) d) e) None of the answers is correct to the fourth decimal place. 7

8 Question The population mean outstanding bill for delinquent customer accounts for a national department store chain is $ with a standard deviation of $ In a simple random sample of 50 delinquent accounts, what is the probability that the sample mean outstanding bill is over $200? a) b) c) d) e) None of the answers is correct to the fourth decimal place. Question A facility for bottling soft drinks uses two fill and seal machines (1 and 2). It is assumed that the two machines have the same overall average fill amount (so, we will take this as truth). As part of quality control, a random sample of size 30 is taken from each machine and the sample means are compared. If the standard deviations of the machines are the same and are known to be σ = 0.3 oz, what is the probability that sample mean for machine 1 is more than 0.25 oz above the sample mean for machine 2? a) b) c) d) e) None of the answers is correct to the fourth decimal place. 8

9 Question Suppose the continuous random variable X has pdf f X (x) = { x 2 0 < x < 2 0 otherwise Let Y = X 1/4. Find the pdf of Y. Be sure to state the support of Y. 9