Test 2 VERSION B STAT 3090 Spring 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. Suppose we have a loaded die that gives the outcomes 1 to 6 according to the probability distribution X _ P(X) Note that for this die all outcomes are not equally likely, as it would be if this die were fair. If this die is rolled many times, then the mean of the number of spots on the many rolls, should be about A B C D Find the z-score in the standard normal distribution such that the area to the right of z is A B C D Suppose that the amount of time that it takes a clerk to process an employment application is uniformly distributed between 5 minutes and 12 minutes. What is the probability that the clerk will take more than 7 minutes to process a randomly selected application? A. 3/7 B. 4/7 C. 5/7 D. cannot be determined 1

2 4. At Ingles Market is has been determined that customers arrive at the checkout section according to a Poisson distribution at an average rate of 12 customers per hour. What is the probability that at least 1 customer will arrive at the checkout section in the next hour? A. 121 e 12 1! B e 12 0! C. 120 e 12 0! D e 12 1! 5. Which of the following criteria ensures that the sampling distribution for the sample proportion is approximately normally distributed? A. The population distribution is approximately normally distributed. B. The mean must be greater than 5. C. np 5 and n(1 p) 5 D. n Suppose that individuals applying for a driver s license in South Carolina are given 4 attempts to pass the driver s test. The following probability distribution shows the number of attempts, X, that were required by individuals who passed their exam in X P(X) What is the probability that a randomly selected individual who attained a driver s license required more than one attempt? A B C D Evaluate P( 1.25 Z 2.21) A B C D

3 8. A special coin has the probability of 0.65 of landing heads. What is the probability that it will land heads exactly 3 times in 7 tosses? A. C 3 (0.65) 3 (0.35) 4 7 B. 7 C 0 (0.65) 0 (0.35) C 1 (0.65) 1 (0.35) C 2 (0.65) 2 (0.35) 3 + 7C 3 (0.65) 3 (0.35) 2 C e ! D e ! e ! e ! e ! 9. It is known that the resistance of carbon resistors is approximately normally distributed with µ=1200 ohms and σ = 120 ohms. If 10 resistors are randomly selected from a shipment, what is the probability that the average resistance will be less than 1250 ohms? A B C D A department store reports that 84% of their customers pay their bills on time. We are interested in the probability that at least 12 out of 15 randomly selected customers will pay their bill on time. Assume that customer payment habits are independent. What type of distribution appropriately describes this situation? A. binomial distribution B. Poisson distribution C. normal distribution D. none of these 11. X is a binomial random variable with n=10 and p=0.9. Which of the following statements is false? A. P(1 < X < 5) = P(2 X 4) B. P(X 5) = 1 P(X 5) C. μ X = 9 D. σ X =

4 12. For the uniform distribution, illustrated below, which of the following is/are true? I. The mean is smaller than the median. II. P(0.1 < X < 0.7) = 0.6. III. P(X > 0.5) = 0.75 A. I only B. I and II C. II and III D. III only 13. The time that it takes a Clemson student to find parking once they have arrived on campus is approximately normally distributed with a mean 25 minutes and standard deviation 5 minutes. If a student arrives on campus at 8:15 AM, what is the probability that the student will find parking before 8:45 AM? A B C D Random samples of size n were selected from a population with a known standard deviation. How is the standard deviation of the sampling distribution of the sample mean affected if the sample size is increased from 50 to 200? A. It remains the same. B. It is multiplied by four. C. It is divided by four. D. It is divided by two. 15. The height of an adult male is known to be normally distributed with mean of 175 cm and standard deviation 6 cm. What is the value of Q3 in this distribution of heights? A cm B cm C cm D cm 16. Let X be a random variable that has a bimodal distribution with mean 12 and standard deviation 1.5. Based on random samples of size 400, the sampling distribution of x is A. highly skewed with mean 12 and standard deviation 1.5 B. slightly bimodal with mean 12 and standard deviation 1.5 C. approximately normal with mean 12 and standard deviation D. approximately normal with mean 12 and standard deviation X 4

5 17. A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean 3 cm and standard deviation 0.1 cm. The specifications call for corks with diameters between 2.9 and 3.1 cm. A cork not meeting specifications is considered defective. What proportion of corks will be considered defective? A B C D Suppose that on average, 5 students who are enrolled in a state university in Atlanta, GA have their cars stolen during the semester and the number of cars has a Poisson distribution. Determine the expected number of cars stolen per semester. A. 5 cars B. 5 cars C. 25 cars D. none of these 19. Plywood contains minor imperfections that can be repaired with small plugs. One customer will accept plywood with a maximum of 3.5 plugs per sheet on average. Suppose that a shipment was sent out to this customer, and when the customer inspected the two sheets at random, 10 plug defects were counted. What is the standard deviation of this Poisson distribution? A plugs B plugs C. 3.5 plugs D. 3.5 plugs For which of the following will the sample proportion tend to differ least from sample to sample? A. Random samples of size 50 from a population with p = 0.1 B. Random samples of size 60 from a population with p = 0.1 C. Random samples of size 40 from a population with p = 0.5 D. Random samples of size 50 from a population with p = 0.6 5

6 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. 1. Let X describe the number of defective tires on a randomly selected sport utility vehicle (SUV) at a certain inspection center. Assume that the following is a valid probability distribution. X P(X) ?? A. Find P(X = 4). (1 point) P(X=4) = pt for correct answer no probability statement is required B. Find the expected number of defective tires on a randomly selected SUV. Show work and round your answer to two places. (3 points) μ X or E(X) = 0(0.52)+1(.23)+2(0.04)+ 3( (0.17) = 1.11 tires 1 point - correct symbol for mean 1 point - correct justification, only first and last term required ---minus ½ if 0(0.42) is omitted 1 point - correct answer --- minus ½ if no unit C. What is the probability that the number of defective tires on a randomly selected SUV exceeds the mean? Include a probability statement and justify clearly. (3 points) P(X>1.11) or P(X>=2) = = point - correct probability statement 1 point correct justification 1 point correct answer 6

7 2. The Keowee Corporation buys parts from suppliers all over the country. One part is currently being purchased from a supplier in California under a contract that calls for at most 5% of the parts to be defective. When a shipment arrives, the Keowee Corporation randomly samples 10 parts. If it finds 2 or fewer defective parts in the sample, it keeps the shipment; otherwise, it returns the entire shipment to the supplier. For part A and part B include a probability statement. Also, you may justify your work with calculator syntax as long as the parameters are clearly defined. Round your response to four decimal places. A. Assuming that the conditions for a binomial distribution are satisfied, what is the probability that the sample will lead the Keowee Corporation to reject the shipment if the defect rate is actually 5%? (3 points) P(X>2) = 1 binomialcdf(10,0.05,2) = OR P(X > 2) = 1 (( 10 0 ) (0. 05)10 (0. 95) 0 + ( 10 1 ) (0. 05)9 (0. 95) 1 + ( 10 2 ) (0. 05)8 (0. 95) 2 ) = OR may calculate the probability without using the complement 1 point correct probability statement 1 point correct justification with formula or calculator syntax - no credit for justification if calculator syntax is used and parameters are not defined 1 point correct answer - deduct ½ for improper rounding B. Suppose that actually 10% of the shipment from the supplier is defective. What is the probability that the sample will lead Keowee to keep the shipment anyway? (3 points) P(X<=2) = binomialcdf(10,0.10,2) = OR P(X=0) + P(X=1) + P(X=2) = 10C0 (.10) 10 (0.9) 0 +10C1(.10) 9 (0.9) 1 +10C2(.10) 8 (0.9) 2 ) = point correct probability statement 1 point correct work with formula or calc syntax no credit for work if calculator syntax is used and parameters are not defined 1 point correct answer 7

8 3. A manager of an online shopping website finds that on average 10 customers per minute make a purchase on Mondays and the number of customers is well approximated by a Poisson distribution. A. Let X = the number of customers who make a purchase in a one-minute interval on Monday. What is the probability that during a one-minute interval on Monday, exactly 3 purchases will be made? Be sure to give a probability statement, show the calculation (including formula) and round your answer to 4 decimal places. (3 points) P(X=3)= 103 e 10 = ! 1 point correct probability statement 1 point correct work must show formula 1 point - correct answer B. Let W = the number of customers who make a purchase in a 30-second interval on Monday. What is the probability that during a 30-second interval on Monday, at least 3 purchases will be made? Include a probability statement. You may use calculator syntax as long as the parameters are clearly defined. Round to 4 decimal places. (4 points) new lambda = 5 customers P(W>=3) = 1 P(W<3) = P(W 3) = 1 P(W < 3) = 1 ( 50 e e 5 1! 0! 1 point - new lambda must be clear 1 point correct probability statement using W 1 point - correct justification with formula or calculator syntax - calculator syntax is ok as long as parameters are defined 1 point - correct answer - deduct ½ for improper rounding + 52 e 5 ) = ! 8

9 4. The scores for Test 1 in STAT 3090 are approximately normally distributed with mean 80.3 points and standard deviation 6.8 points. A. What score would a student need to score in the top 22% of students? You may justify your response with calculator syntax as long as the parameters are clearly defined. Round to 2 decimal places. (3 points) Score = invnorm(0.78, 80.3, 6.8) = points OR 0.77 = (X 80.3) /6.8 - solve for X 1 point - correct answer 2 points - correct justification if calculator syntax is used, deduct 1 point for work if parameters are not defined - students may earn 1 point for correct substitution into the z-score formula. B. What is the probability that a randomly selected group of 33 students will have a mean score of more than 83 points? Include a probability statement. You use calculator syntax as long as the parameters are clearly defined. Round to four decimal places (4 points). P(X > 83) = point correct probability statement no credit if no x-bar 1 point - correct standard deviation 6.8/sqrt(33) = point - correct work - calculator notation is ok as long as parameters are defined - z-score with probability ok 1 point - correct answer 9

10 5. According to a report published by U.S. News and World Report, 41% of Clemson students live on campus. A. Is the proportion of Clemson students who live on campus in a sample of 60 students normally distributed? Support your response with the appropriate calculation(s). (2 points) Yes, 60(.41)>=5 and 60(1 0.41) >=5 1 point correct decision 1 point - check conditions must plug in values to receive credit. B. Calculate the mean and standard error of the sampling distribution of the proportion of Clemson students who live on campus in a sample of 60 students. Label your answers with the appropriate symbol. Round your answer for standard deviation to four decimal places. (3 points) μ p hat = σ p hat = 0. 41(0. 59)/60 = point for correct mean minus ½ if correct symbol is not used 1 point for correct standard error minus ½ for incorrect symbol 1 point for correct work for standard error C. What is the probability that 43.5% or more of the students sampled live on campus? Show all your calculations and label your answer with appropriate probability notation. You may use calculator syntax. (3 points) 0. 41(0. 59) P(p > ) = normalcdf(0. 435, 1E99,. 41, = Z-score = 0.39, probability = point - correct probability statement no credit if p-hat is not used 1 point - for correct work - calculator syntax is ok if the parameters are defined - or correct z-score and corresponding probability 1 point - correct answer 10

11 6. The following graph shows the uniform distribution of wait times, in minutes, for the Catbus at the bus stop in front of Sikes Hall time (min) A. Find the area of the shaded region. Justify your response. (1 point) Area = 3/11 1 point - for correct area B. Interpret the area that you found in part A using words. (3 points) The probability that a randomly selected student will have to wait between 3 and 6 minutes is 3/11. OR The proportion of all bus riders who will wait between 3 and 6 minutes is 3/11. 1 point links area to probability 1 point proper boundaries between 3 and 6 minutes 1 point states proper context wait time in minutes 11