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Midterm exam sample problems 1. A fair coin is flipped 15 times. The probability of obtaining the expected number of heads is: (A) 15 (B) 0.15 (C) 0.1333 (D) 0.0667 (E) None of these 2. A population has μ = 83, σ = 3.25. Chebyshev’s theorem indicates that the proportion of the population between 70 and 96 must be: (A) At least 6.25% (B) At most 6.25% (C) At least 93.75% (D) At most 93.75% (E) None of these 3. The Poisson distribution differs from the binomial distribution in that: (A) The Poisson distribution is discrete (B) It cannot take negative values (C) It is always symmetric (D) All of these (E) None of these Problems 4 – 6 pertain to the following situation: The Nova Scotia Lottery Commission considers developing a new scratch-off lottery card to sell for $1. Each ticket will have probability 0.0001 of paying a $5000 prize, probability 0.0049 of paying a $5 prize, and probability 0.0555 of resulting in a free ticket. 4. If you were to buy one of these tickets, what is your expected loss? (A) $1 (B) $0.9395 (C) $0.4999 (D) $0.42 (E) None of these 5. What is the standard deviation of the outcome for this game? (A) $2500 (B) $0.42 (C) $50 (D) 0 (E) None of these 6. If the Commission were to sell 10,000 such tickets, their expected result would be: (A) To make $4,999 (B) To lose $4,999 (C) To make $4,200 (D) To lose $4,200 (E) None of these 7. Some positive constant number k is added to every element of a quantitative data set. As a result, (A) k is added to both the mean and the standard deviation (B) k is added to the mean, but not to the standard deviation (C) k is added to the standard deviation, but not the mean (D) Neither the mean nor the standard deviation are changed. (E) None of the above. 8. Obtaining a standard deviation of zero indicates that: (A) There are the same number of values above and below the mean (B) No values are equal to the mean (C) Both of the above (D) All values are equal (E) No two values are equal Problems 9 – 12 are based on the following situation: A manufacturing line fills containers with orange juice. The actual volume of juice (“X”) put into each container is normally distributed, with μ = 3780 mL, σ = 110 mL. 9. Find P(X ≤ 3700 mL). (A) 0.7273 (B) 0.2673 (C) 0.2327 (D) 0.7327 (E) None of these 10. Find P(3740 mL ≤ X ≤ 4000 mL). (A) 0.6178 (B) 0.3366 (C) 0.8366 (D) 0.9772 (E) None of these 11. Containers filled with less than 3600 mL of juice are rejected. If the manufacturing line processes 200,000 containers per day, how many are expected to be rejected for this reason? (A) 4950 (B) 8990 (C)16,400 (D) 10,100 (E) None of these 12. Suppose that the 3600 mL threshhold for rejection is producing too many rejected containers. What minimum volume should be chosen so that only 2.5% of containers are rejected for being not full enough? (A) 3996 mL (B) 3564 mL (C) 3706 mL (D) 3774 mL (E) None of these 13. If two events A and B are dependent, then: (A) P(A|B) = P(A) (B) P(B|A) = P(B) (C) P(AB) = P(A)P(B) (D) all of the above (E) none of the above 14. Data consisting of values that are neither numeric nor ordered are called: (A) Nominal (B) Ordinal (C) Continuous (D) Discrete (E) Numerate Problems 15 – 18 pertain to the following situation: Thirty percent of a very large population tests positive for a communicable disease. Twelve individuals are chosen at random and tested. 15. How many are expected to test positive? (A) 0.975 (B) 0.025 (C) 4 (D) 3.6 (E) None of these 16. What is the variance of the number who test positive? (A) 1.59 (B) 1.90 (C) 1.98 (D) 2.52 (E) None of these 17. What is the probability that all twelve people test positive? (A) 0.975 (B) 0.025 (C) 4 (D) 3.6 (E) None of these 18. What is the probability that the number of people who test positive is more than one, but less than five? (A) 1.0000 (B) 0.8683 (C) 0.6386 (D) 0.2397 (E) None of these For problems 19 – 22, Z is a standard normal random variable. 19. Find P(Z ≤ 1.44). (A) 0.4251 (B) 0.0749 (C) 0.4626 (D) 0.0000 (E) None of these 20. Find P(0 ≤ Z ≤ 1.44). (A) 0.4251 (B) 0.0749 (C) 0.4626 (D) 0.0000 (E) None of these 21. Find P(-1.25 ≤ Z) . (A) 0.8944 (B) 0.4472 (C) 0.3944 (D) 0.1056 (E) None of these 22. Find P(-2.46 ≤ Z ≤ -0.34). (A) 0.3600 (B) -0.3600 (C) 0.1835 (D) 0.6331 (E) None of these _________________________________________________________________ 23. Which of the following represents a probability distribution? (A) X 3 2 1 0 P(X) .18 .23 1.11 -.52 (B) X 9 8 7 P(X) 1.34 .12 .88 (C) X 1 0 -1 P(X) .11 .12 .77 (D) X 4 3 2 1 P(X) .17 .34 .77 -.28 (E) None of these are valid probability distributions. 24. Every element of a quantitative data set is multiplied by some positive constant number k. As a result, (A) The mean and standard deviation are both multiplied by k. (B) The mean is multiplied by k, but the standard deviation is unchanged. (C) The mean is unchanged, but the standard deviation is multiplied by k. (D) Neither the mean nor the standard deviation are changed. (E) None of the above. Problems 25 – 28 are based on the following frequency distribution of grades on an exam: score frequency 60 - 69 4 70 - 79 19 80 - 89 23 90 - 99 11 25. The mean as estimated from this table is: (A) 14.25 (B) 57 (C) 4656.50 (D) 81.69 (E) None of these 26. The relative frequency of the 80 – 89 class is: (A) 0.4035 (B) 23 (C) 46 (D) 0.8070 (E) None of these 27. The standard deviation as estimated from this table is: (A) 8.61 (B) 8.77 (C) 11.18 (D) 74.12 (E) None of these 28. The median score is in which class? (A) 70 – 79 (B) 80 – 89 (C) 90 – 99 Answers: 1. (E) 2. (C) 3. (E) 4. (D) 5. (C) 6. (C) 7. (B) 8. (D) 9. (C) 10. (A) 11. (D) 12. (B) 13. (E) 14. (A) 15. (D) 16. (D) 17. (E) 18. (C) 19. (E) 20. (A) 21. (A) 22. (A) 23. (C) 24. (A) 25. (D) 26. (A) 27. (A) 28. (B) (D) 74.5 (E) None of these