Test 2 VERSION A STAT 3090 Fall 2017

Transcription

1 Multiple Choice: (Questions 1 20) Answer the following questions on the scantron provided using a #2 pencil. Bubble the response that best answers the question. Each multiple choice correct response is worth 3 points. For your record, also circle your choice on your exam since the scantron will not be returned to you. Only the responses recorded on your scantron will be graded. 1. The Oconee Newspaper Company sometimes makes printing errors in its advertising in the next issue of the paper. The managing editor has done a study of this problem and found the following data: Number of Errors Probability What is the probability that the next issue of the paper will have at least 1 error? A B C D Refer to the previous question. What is the expected number of printing errors in the next issue of the paper? A. 0.8 errors B. 0 errors C. 0.5 errors D. 1 error 3. Compute the variance of random variable, Y. y P(y) A. 0.5 units 2 B units C units 2 D. 0.5 units 1

2 4. A special coin has the probability of 0.6 of landing heads. Assume tosses of this coin are independent. What is the expected number of heads tossed in 18 tosses of the coin? A. 11 heads B. 0.6 heads C heads D. 7 heads 5. Your company president has told you that the company experiences product returns at the rate of two per month with the number of returns distributed as a Poisson random variable. Determine the probability that next month there will be one return. A e 2 1! B e 2 C. 20 e 2 0! D. 21 e 2 1! 0! 6. Suppose that on average, there are 1.2 defects per 25 square foot roll of wallpaper and the number of defects follows a Poisson distribution. Determine the expected number of defects per 50 square foot roll. A. 1.2 defects B. 1.2 defects C. 2.4 defects D. none of these 7. Suppose the following is a valid probability distribution. Find the value of m. x P(x) m m A B C D. cannot be determined 8. Evaluate P( 1.1 Z 0.21) A B C D

3 9. A major airline conducted a survey and determined that 80% of its customers were in favor of a system in which seats were assigned on a first-come, first-served basis. Assume that customer views are independent. What is the probability that exactly 3 out of 7 randomly selected customers will be in favor of this seating system? Note: nc r = ( n r ) A e 0.8 3! B. ( 7 0 ) (0.8)0 (0.2) 7 + ( 7 1 ) (0.8)1 (0.2) 6 + ( 7 2 ) (0.8)2 (0.2) 3 + ( 7 3 ) (0.8)3 (0.2) 2 C. ( 7 3 ) (0.8)3 (0.2) 4 D e 0.8 0! e 0.8 1! e 0.8 2! e 0.8 3! 10. The following uniform distribution describes the wait time (in minutes) for passengers of the Catbus at the stop in front of Sikes Hall. What is the probability that a randomly selected passenger will wait between 5 and 10 minutes? P(time) A. 5/10 B. 5/9 C. 4/9 D. 4/ Find the Z-score in the standard normal distribution such that the area to the left of Z is A B C D X X 3

4 12. It has been determined that the mean amount of time that computer science majors spend on homework each week is approximately normally distributed with a mean of 15.2 hours and standard deviation 3.1 hours. What is the probability that a randomly selected computer science major will spend more than 14.5 hours on homework in a given week? A B C D It is known that the resistance of carbon resistors is approximately normally distributed with µ=1200 ohms and σ = 120 ohms. Determine the approximate resistance that separates the lower 15% from the rest of this distribution. A ohms B ohms C ohms D ohms 14. According to the Food Marketing Institute, U.S. consumers make an average of 1.5 trips to the grocery store in a typical week with a standard deviation of 0.65 trips. Which of the following expressions gives the correct z-score calculation needed to find the probability that in a random sample of 100 U.S. consumers, the average number of trips to the grocery store in a typical week exceeds 2 trips? A. B. C D. Without knowing the shape of the population it is not appropriate to use the normal distribution to determine this probability. 4

5 15. Let X be a random variable that has a skewed right distribution with mean μ = 10 and standard deviation σ = 10. Which of the following histograms could display the distribution of the sample mean x from many random samples of size 400 from this population? A. B. C. D. 16. The distribution of cholesterol levels for patients in a cardiology practice follows a normal distribution with a mean of 210 and a standard deviation of 40. In this practice, the probability that a patient has a cholesterol level reading more than 290 is the same as the probability that a patient has a cholesterol level reading less than: A. 250 B. 170 C. 130 D

6 17. When proofreading a statistics textbook, one can expect to find an average of 11 errors per 200 pages of the book. Assume that the number of errors in a statistics book follows a Poisson distribution. What is the probability that when proofreading a statistics text, one finds more than 2 errors in the next 200 pages? A B C D Suppose the it is known that 6% of the light bulbs produced at a certain manufacturer are defective. Ten light bulbs are randomly selected to be tested. What is the probability that at most 2 bulbs of the ten selected will be defective? A B C D Turkeys found in a particular county have an average weight of 15.6 pounds with a standard deviation of 4.00 pounds. Thirty-five turkeys are randomly selected for a county fair. What is the probability that the average weight of the turkeys will be more than 16.2 pounds? A B C D Assume that the time required to receive confirmation that an electronic transfer has occurred is uniformly distributed between 30 and 90 seconds. What is the probability that a randomly selected transfer will take more than 75 seconds? A. 15/60 B. 15/90 C. 30/90 D. 45/60 6

7 Free Response: The free response questions will count as 40% of your total grade. Read each question carefully. In order to receive full credit, you must show logical (relevant) justification which supports your final answer. You MUST show your work. Answers with no justification will receive no credit. Use appropriate symbols for the values that you compute. 1. The Clemson Computer Store stocks four 10.5-inch versions of Apple s ipad Pro. If it has fewer than four ipad Pros available at the end of a week, the store restocks the item to bring the in-stock level up to four. If weekly demand is greater than the four units in stock, the store loses sales (so, there are weeks that the demand for this item is not met). The ipad Pro sells for $679 and costs the store $649. The store manager estimates that the probability distribution of weekly demand for the ipad is as follows: Weekly Demand (D) Probability A. Let D = the weekly demand for ipads What is the probability that the weekly demand will be less than or equal to four (that is, what is the probability that the store will be able to meet the demand)? Include a probability statement with your answer. (3 pts) P(D 4) = = pt Probability statement deduct ½ for not using D 1 pt Correct justification first term + + last term is ok 1 pt Correct answer B. What is the expected weekly demand? (3 pts) E(D) or μ D = 0(0. 05)+.. +7(0. 05) = 3. 6 ipads ½ pt Correct symbol or word for mean or expected value 1 pt Correct justification first term +..+ last term is ok 1 pt Correct answer ½ pt Correct units 7

8 D. What is the expected weekly profit from the ipad? Remember: There are only four ipads available in any week to meet demand. So, if D = 5, the store will still only profit from 4 ipads. (4 pts) VERSION A: μ profit or E(weekly profit) = $0(0.05)+ $30(0.05)+$60(0.10)+$90(0.20)+$120(0.60) = $97.5 Note: 30(μ D ) is incorrect because the number of ipads in stock in any given week is less than or equal to 4. VERSION B: μ profit or E(weekly profit) = $0(0.05)+ $50(0.05)+$100(0.10)+$150(0.20)+$200(0.60) = $ pt Correct symbol for mean or expected value of profit 2 pts Correct work first term + + last term is ok 1 pt Correct answer with unit deduct 1/2 for incorrect or no unit 8

9 2. A manufacturing firm produces a metal product that has a powder coating. The powder coating is applied and then the product cured under heat. This process is known to produce 12% defective items (for example, the surface will have a crack or blemish). Every hour, 20 products from thousands of products that are independently cured are sampled and the powder-coating process is inspected. A. What is the probability that exactly 6 defective items will be found in the next sample of 20? Include a probability statement, show the proper values substituted into the appropriate formula and give an answer rounded to four places. (3 pts) Let X = the number of defective items in the next sample of 20. P(X = 6) = ( 20 6 ) (0. 12)6 (0. 88) 14 = pt Correct probability statement (should define a variable deduct ½ if students does not define the variable) 1 pt Correct work 1 pt Correct answer B. On average, how many defective items would be found in each sample of 20? (3pts) μ X = 20(0. 12) = 2. 4 items 1 pt Correct symbol or words 1 pt Correct work 1 pt Correct answer no credit for answer if it is rounded Deduct ½ for no unit C. How likely is it that between 6 and 9 defective items (inclusive) will be found in a sample of 20? Include a probability statement, show the proper values substituted into the appropriate formula and give an answer rounded to four places. (4 pts) P(6 X 9) = ( 20 6 ) (0. 12)6 (0. 88) 14 + ( 20 7 ) (0. 12)7 (0. 88) 13 + ( 20 8 ) (0. 12)8 (0. 88) 12 + ( 20 9 ) (0. 12)9 (0. 88) 11 = pt Correct probability statement if credit was lost for no variable definition in part A, do not deduct them again 2 pts Correct work correct first term + + correct last term is ok 1 pt Correct answer 9

10 3. Emergency room crowding has become a widespread problem across the United States. A local hospital recently changed how it allocated staff to increase the number of patients that are able to be treated in its emergency room during the peak hours between 9 PM and 1 AM. After these changes, the hospital is now able to treat up to 8 patients per hour during the peak hours. Prior to the staffing changes, the hospital administrators determined that the number of patients requiring emergency room treatment during peak hours was well approximated by a Poisson distribution with a mean of 3.9 patients per hour. A. Let X = the number of patients who arrive at the hospital during peak hours who require emergency room treatment. What is the probability that in a one-hour interval during peak hours, more patients arrive than are able to be treated in the emergency room? Include a probability statement, show the proper values substituted into the appropriate formula and give an answer rounded to four places. (5 pts) P(X > 8) = 1 P(X 8) = 1 ( 3.90 e e 3.9 ) = pt Correct probability statement 1 pt Correct use of complement 2 pt Correct justification 1 minus (first term +..+ last term) is ok 1 pt Correct answer credit may be given if answer is consistent with justification 0! 8! B. Use your answer for part A to comment on the efforts of those at this hospital to address emergency room crowding. (4 pts) The probability of the emergency room having to treat more patients than they are able to handle is very low (0.0185). The administrators seem to have adequately addressed the issue of emergency room crowding. 2 pts Must mention the low probability that was computed in part A deduct 1 pt. for not explicitly stating the probability 2 pts State the issue seems to have been adequately addressed. 10

11 4. The weights of eggs produced by the hens at one local farm follow an approximately normal distribution with a mean of 2.15 ounces and a standard deviation of 0.44 ounces. Let x represent the mean weight of 12 (one dozen) randomly selected eggs from this local farm. A. Explain why x follows an approximately normal distribution. (3 pts) The population of eggs have weights that are approximately normally distributed. Deduct 2 points if the statement does not clearly state that the population of weights or weights of all eggs. The response needs have a clear reference to the population B. Find the mean and standard deviation of x. Label each value with the appropriate symbol. (3 pts) μ X = ounces σ X = ounces 1 pt Correct symbols deduct ½ for each incorrect symbol 1 pt Correct mean 1 pt Correct standard deviation C. The USDA sizing of eggs is based on weight per dozen eggs. To be classified as "jumbo" a dozen eggs must have an average weight of 2.5 ounces or more. Find the probability that one dozen randomly selected eggs from this local farm will be classified as jumbo. Include a probability statement that contains a z-score. (4 pts) NOTE: z = 2.76 P(X > 2. 5) = P (Z > ) = pt Correct z-score 1 pt Correct probability statement 1 pt may be earned for P(Z> wrong value) 2 pt Correct probability Have you bubbled your ID and test form? 11