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Name: ________________________ Class: ___________________ Date: __________ ID: A Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. For which of the following binomial experiments could the normal approximation not be applied? a. n = 10 and p = 0.5 c. n = 100 and p = 0.4 b. n = 20 and p = 0.2 d. n = 20 and p = 0.3 ____ 2. What is the value of x for the normal approximation to a binomial experiment with 50 trials and a 40% likelihood of success for any given trial? a. b. 12 22.5 c. d. 20 25 ____ 3. A coin is tossed five times. What is the probability of observing exactly three heads? 5 3 a. c. 16 32 1 3 b. d. 32 16 ____ 4. A Bernoulli trial has a probability of success of 0.4. What is the smallest number of trials for which a normal distribution can be used to approximate its probability distribution? a. 12.5 c. 10 b. 20 d. 13 ____ 5. The z-score for a particular value of X is 0.18. What is the total probability of this or any smaller value of X occurring? a. 42.9% c. 96.4% b. 57.1% d. 3.6% ____ 6. In order to approximate the probability that X = 15 or X = 16 for a binomial distribution, what boundary values should we choose to obtain our z-scores? a. 15 < X < 16 c. 14.5 < X < 16.5 b. 14 < X < 17 d. none of the above ____ 7. The probability distribution for a binomial experiment with n = 10 and p = 0.4 is graphed. Which of the following statements is least likely to be true? a. The highest bar should be above X = 4. b. The graph will be higher on the left than on the right. c. The graph will be highly symmetrical. d. The bar above X = 3 will be higher than the bar above X = 5. 1 Name: ________________________ ID: A ____ 8. The normal approximations to two binomial experiments are compared. Both have 20 trials, but the first has p = 0.4 and the second has p = 0.7. Which statement is not true? a. Both binomial distributions may be approximated by a suitable normal distribution. b. The first distribution is more symmetrical than the second. c. The highest point for the first distribution is found above X = 8. d. The normal approximation is a closer fit to the actual binomial distribution for the second experiment than for the first. ____ 9. Which of the following binomial experiments has the normal approximation with the smallest standard deviation? a. n = 50 and p = 0.45 c. n = 40 and p = 0.55 b. n = 20 and p = 0.45 d. n = 10 and p = 0.55 ____ 10. The probability of a parent allowing a child to play in the sprinkler when the temperature is above 28°C is 0.8. What is the probability that exactly 15 of 20 children will be allowed to play in their sprinklers on a day above this temperature? a. 0.75 c. 0.1746 b. 0.714 d. 0.315 ____ 11. What is the value of the standard deviation for the normal approximation to a binomial distribution with 600 trials and a probability of success of 0.6? a. 12 c. 6 600 b. d. 0.24 ____ 12. What is the z-score associated with observing 5 or fewer heads for 8 tosses of a fair coin? 1 3 a. c. 2 2 2 5 11 b. d. 8 16 ____ 13. A die is rolled 12 times. Calculate the probability that a 5 appears exactly twice. a. 0.1667 c. 0.0139 b. 0.2961 d. 0.0004 ____ 14. A novice competitor in biathlon hits 80% of her targets. What is the probability that she will hit more than 45 of 50 targets attempted? a. 19.4% c. 2.6% b. 80.6% d. 97.4% ____ 15. An antibiotic is effective against a particular strain of streptococcus 70% of the time. What is the probability that at least 70 of 100 cases will respond when treated with this antibiotic? a. 54.4% c. 70% b. 45.6% d. 49% Short Answer 16. What is the mean value for a binomial experiment with 45 repetitions and a probability of success of 0.25? 2 Name: ________________________ ID: A 17. Calculate the standard deviation for the normal approximation to the binomial distribution for an experiment with 300 trials and a probability of success of 0.27. 18. Express the z-score for a particular value of X in a binomial distribution in terms of X, n, p, and q where q = 1 – p. 19. A golfer makes 80% of her putts from a distance of 5 metres or less from the hole. In order to calculate the probability that she would make 23 out of 27 putts or better on a given day, what boundary value should be chosen for X in order to find the appropriate z-score? 20. What is the z-score needed to approximate the probability of observing 5 or fewer heads when tossing a fair coin 8 times? 21. The probability of success for a binomial experiment is 0.25. What is the smallest number of trials for which we may apply the normal approximation? 22. The probability that a customer will actually buy a pair of shoes if she tries them on is 0.3. What is the probability that Sapna will sell exactly 12 pairs of shoes if she waits on 25 customers who try on shoes? 23. A bag contains 120 marbles, of which 15 are known to be white. Marbles are drawn one at a time from the bag. The colour is recorded and the marble is returned to the bag. What is the probability that exactly 3 of 16 marbles drawn will be white? 24. What is the mean value for the normal approximation to a binomial experiment with 75 repetitions and a probability of success of 32%? 25. A student guesses at all 10 questions on a true/false test. What is the probability that he will get exactly 5 answers correct? 26. It is known that 10% of the population is left-handed. In a random sample of 200 people, what is the probability that exactly 18 are left-handed? 27. Compare the standard deviation for the normal approximation to a binomial experiment with p = 0.43 for 100 trials and 400 trials. 28. What is the range of probabilities for which the normal approximation may be applied if 40 Bernoulli trials are performed? 3 ID: A Sample Questions for Mastery #5 Answer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. B C A D B C C D D C A C B C A SHORT ANSWER 16. 11.25 17. 7.69 18. From z = x−x σ , we replace to obtain z = x − np npq . 19. A value of 23 in the discrete distribution corresponds to the range from 22.5 to 23.5 in the continuous approximation. We need to calculate P(X ≥ 22.5). We will also need to recognize that the z-score will give the cumulative probability for all values less than 22.5, so we will need to use 1 minus the probability from the z-score table. 20. 1.06 21. The smallest number of trials allowed would be 21. 22. The probability is 0.0268 or less than 3%. 23. The probability is 0.1928 or almost 20%. 24. 24 25. The probability is 0.2461 or almost 25%. 26. 0.0875 27. For 100 trials, σ = 100(0.43)(0.57) = 4.95. For 400 trials, σ = 400(0.43)(0.57) = 9.90. When the number of trials is increased by a factor k, the standard deviation of the normal approximation increases by 1 k. ID: A 28. We are allowed to use the normal approximation when the values of np and n(1 – p) are both greater than 5. If we consider the smallest value, we must solve 40p > 5, which gives a solution of p > 0.125. Switching the values for p and (1 – p ), we get a maximum possible value for p of 0.875. The range of allowable probabilities for applying the normal approximation would be 0.125 < p < 0.875. 2