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 that 75% of the students at Clemson university wear orange color T-shirts on Friday. If 1000 students are selected at random from this university, what are the mean and standard deviation of the random variable X = number of selected students who wear the orange color t-shirts on Friday? X is a binomial random variable A) Mean = 750; standard deviation = 13.69 B) Mean = 75; standard deviation = 187.5 C) Mean = 75; standard deviation = 13.69 D) Mean = 750; standard deviation = 187.5 2. Which of the following will provide the smallest standard deviation of the sampling distribution of the sample mean, thus resulting in the sample mean differing the least from sample to sample? A) Random sample of size 100 from a population with μμ = 25 and σσ = 8 B) Random sample of size 20 from a population with μμ = 15 and σσ = 8 C) Random sample of size 1000 from a population with μμ = 25 and σσ = 15 D) Random sample of size 100 from a population with μμ = 15 and σσ = 12 3. A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the random variable X = actual capacity of a randomly selected tank has a distribution that is well approximated by a normal curve with mean 15.0 gallons and standard deviation 0.1 gallon. Which of the following probability statements represents the probability that a randomly selected tank will hold at most 14.8 gallons? A) PP(XX 14.8) B) PP(ZZ < 2.0) C) PP(XX < 14.8) D) All the above 1
4. Suppose the standard deviation of the sampling distribution of XX is A, which of the following would be the standard deviation if the sample size were only a quarter of the size? A) 2A B) 0.5A C) 4A D) 0.25A 5. The quality manager of a fortune cookie company believes that a larger than acceptable proportion of paper fortunes being used are blank. Suppose a sample of 290 fortune cookies was taken. If the true proportion of fortunes that were blank is 0.02, what is the probability that the sample proportion is at most 0.03? A) 0.1119 B) 5.8 C) 0.8888 D) 0.02 6. Let X denote the number of bars of service on your cell phone whenever you are at an intersection with the following probabilities: What is the expected value of X? A) 2.5 B) 1 C) 0.53 D) 0.0.5 x 0 1 2 3 4 5 P(X=x) 0.1 0.15 0.25 0.25 0.15 0.1 2
7. On average, the number of pumpkin spiced lattes ordered at a local coffee shop during the month of October is 17.2 per hour. Consider the random variable X that represents the number of pumpkin spiced lattes ordered per day in the month of October. The random variable X has which of the following probability distributions? A) A poisson distribution B) A binomial distribution C) A normal distribution D) None of the above 8. The thickness, x, of a protective coating applied to glass shower doors designed to ensure the longevity and clarity of the glass follows a uniform distribution over the interval from 10 to 40 microns. Find the probability that the coating is less than 20 microns thick. A) 0.3333 B) 0.6667 C) 0.5000 D) 0.3000 9. Which of the following situations describes a binomial random variable? A) You play two games against the same opponent. The probability that you win the first game is 0.4. If you win the first game, the probability you also win the second game is 0.2. You are interested in the probability that you win both games? B) A new restaurant opening in Greenwich Village has a 30% chance of survival during their first year. Assume restaurant openings are independent. You count the number of restaurants that survive out of the 16 new restaurants that open this year. C) You are training your dog, Sophie to catch a ball. You count the number of tosses before Sophie catches her first ball. Assume that Sophie s attempts to catch the ball are independent. D) You wish to obtain three out of the five prizes that are offered in Cocoa Krackers cereal boxes. You purchase Cocoa Krackers until all three prizes are obtained. 3
10. A roulette wheel has 38 slots, 18 are red, 18 are black, and 2 are green. Each slot is equally likely to be selected. You play five games and always bet on red. Approximately, what is the probability that you will win at least one game? A) 5 0 0.47370 (1 0.4737) 5 B) 5 1 0.47371 (1 0.4737) 1 C) 1-5 0 0.47370 (1 0.4737) 5 D) 5 0 0.47370 (1 0.4737) 5 + 5 1 0.47371 (1 0.4737) 1 11. Determine n and p for the following situation: A student is taking a 15 question multiple-choice Anthropology exam in which each question has four choices. Assuming that she has no knowledge of the correct answer to any of the questions, she has decided on a strategy in which she will place four balls (marked A, B, C, and D) into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question. This is a binomial situation with: A) n = 4, p = 0.5 B) n = 4, p=0.25 C) n = 15, p=0.5 D) n=15, p=0.25 4
12. In a batch of batteries 5% are defective. A random sample of 80 batteries is to be taken from a large production of batteries. Let X be the number of defective batteries out of 80. X is a binomial random variable. Which of the following is the best interpretation of the standard deviation of X? A) It is expected that there will be 4 defective batteries in a sample of 80 batteries. B) It is expected that the number of defective batteries in a sample of 80 batteries will be between 5 and 6. C) The number of batteries that are defective in repeated samples of 80 batteries in the long run will typically vary from the mean by approximately 1.95 batteries. D) The number of batteries that are defective in one single sample of 80 batteries once selected will typically vary from the mean by approximately 1.95 batteries. 13. On average there are 3 typing errors per page of text. What is the probability that on any given page of text there are 2 typing errors? A) 23 ee 2 3! B) 32 ee 3 2! C) 32 ee 3 2! D) 32 ee 3 2! + 32 ee 3 1! + 32 ee 3 1! + 32 ee 3 0! 5
14. Suppose a sample of 400 people is used to perform a taste test. If the true proportion of the population that prefer Pepsi is 0.5. What is the standard deviation of the sampling distribution of proportion of people that prefer Pepsi in a sample of 400? A) 100 B) 10 C) 0.025 D) 0.000625 15. The average GPA at a particular school is 2.89 with a standard deviation 0.63. A random sample of 55 students is collected. Let XX denote average GPA from a sample of 55 students. What can be said about XX? A) The sampling distribution of XX may not be normally distributed because the sample size is too small. B) The sampling distribution of XX is normally distributed because it comes from a normally distributed population. C) The sampling distribution of XX is approximately normally distributed because the sample size is small enough. D) The sampling distribution of XX is approximately normally distributed because the sample size is large enough. 16. Which of the following statement(s) is/are true for standard normal distribution? I. The total area under a probability distribution is equal to 1 II. The probability associated with one particular value P(X=x) = 0 III. The distribution has a median of 0. A) Only I B) Only I and II C) Only I and III D) Only II and III E) All above are true 6
17. Let Z be a standard normal random variable. What is the probability that Z will greater than 1.23? A) 0.8907 B) 0.1093 C) 0.3907 D) 0.6093 18. Suppose Y = the number of broken eggs in a randomly selected carton of one dozen eggs. The probability distribution of Y is as follows: Y 0 1 2 3 4 P(y) 0.65 0.20 0.10 0.04? What is the probability that the number of broken eggs is at least 2? A) 0.85 B) 0.95 C) 0.15 D) 0.05 19. Let Z be a standard normal random variable. Find the value of ZZ such that 0.67 of the area under the curve lies to the right of ZZ. A) zz = 0.44 B) zz = 0.44 C) zz = 0.7486 D) zz = 0.6293 7
20. Coliform bacteria are randomly distributed in a certain Arizona river at an average concentration of 0.5 per 10 cc of water. A test tube contains 10 cc of liquid. Let X be the number of coliform bacteria per test tube of water. X is a Poisson random variable. What is the standard deviation of the number of coliform bacteria per test tube of water? A) 0.5 B) 0.71 C) 3.16 D) 10 8
Free Response: The Free Response questions will count 40% of your total grade. Read each question carefully. In order to receive full credit you must show legible and logical (relevant) justification which supports your final answer. You MUST show your work. Answers with no justification will receive no credit. 1. (5 pts) Explain which of the conditions for a binomial experiment is NOT met for the following random variable. A football team plays 12 games in its regular season. X = number of games the team wins. 2. (4 pts) Suppose that the time students wait for a bus can be described by a uniform random variable X, where X is between 0 and 20 minutes. On the axes below draw the probability density function for the random variable X. Make sure to use proper labels. 9
3. Suppose the pulse rates of women is normally distributed with a mean of 75 and a standard deviation of 8. A) (5 pts) What is the probability that a randomly selected woman has a pulse rate greater than 92? Provide the probability statement (ie, P( )), show work, and provide value to 4 decimal places. Work shown may be in calculator syntax as long as appropriate parameters are properly labeled. B) (5 pts) What is the probability that 15 randomly selected woman has an average pulse rate less than 80? Provide the probability statement (ie, P( )), show work, and provide value to 4 decimal places. Work shown may be in calculator syntax as long as appropriate parameters are properly labeled. 10
4. (5 pts) The Graduate Record Examination (GRE) is a standardized test that students usually take before entering graduate school. According to a publication by the Educational Testing Service, the scores on the verbal portion of the GRE are approximately normally distributed with mean 462 points and standard deviation 119 points. What score would a student need to achieve in order to be at the 90 th percentile? Work shown may be in calculator syntax as long as appropriate parameters are properly labeled. 5. (5 pts) Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. Let X be the number of passengers that arrive within one-minute. X then follows a Poisson distribution. What is the probability that exactly 40 passengers arrive within 4 minutes. Provide the probability statement (ie, P( )), show work, and provide value to 4 decimal places. Work shown may be in calculator syntax as long as appropriate parameters are properly labeled. 11
6. (5 pts) A study conducted by the Pew Research Center showed that 75% of 18 to 34 years olds living with their parents say they contributed to household expenses. Suppose that a random sample of fifteen 18 to 34 year olds living with their parents is selected and asked if they contribute to household expenses. Let X be the number that say they contribute to household expenses out of the 15. X is a binomial random variable. What is the probability that between 12 and 14 (inclusive) of those selected say they contribute to household expenses? Provide the probability statement (ie, P( )), show work, and provide value to 4 decimal places. Work shown may be in calculator syntax as long as appropriate parameters are properly labeled. 12
7. (5 pts) Suppose that fund-raisers at a university call recent graduates to request donations for campus outreach programs. They report the following information for last year s graduates: Size of donation $0 $10 $25 $50 Proportion of calls 0.45 0.30 0.20 0.05 Consider the random variable X = amount donation for a person selected at random from the population of last year s graduates of this university. What is the standard deviation of X? Show work, provide answer to 2 decimal places and provide correct units. Correct SCANTRON: If your scantron is correctly bubbled with a #2 pencil, with your correct XID, your correct test version, AND the front of your test is completed with your signature on the academic integrity statement, you earn 1 point. END OF TEST 13