Sample Questions for Mastery #5

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1 Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could the normal approximation not be applied? a. n = 0 and p = 0.5 c. n = 00 and p = 0.4 b. n = 0 and p = 0. d. n = 0 and p = 0.. 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. c. 0 b..5 d. 5. A coin is tossed five times. What is the probability of observing exactly three heads? a. 5 c. 6 b. d 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..5 c. 0 b. 0 d. 5. The z-score for a particular value of X is 0.8. What is the total probability of this or any smaller value of X occurring? a. 4.9% c. 96.4% b. 57.% d..6% 6. In order to approximate the probability that X = 5 or X = 6 for a binomial distribution, what boundary values should we choose to obtain our z-scores? a. 5 < X < 6 c. 4.5 < X < 6.5 b. 4 < X < 7 d. none of the above 7. The probability distribution for a binomial experiment with n = 0 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 = will be higher than the bar above X = 5.

2 Name: 8. The normal approximations to two binomial experiments are compared. Both have 0 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 = 0 and p = 0.45 d. n = 0 and p = The probability of a parent allowing a child to play in the sprinkler when the temperature is above 8 C is 0.8. What is the probability that exactly 5 of 0 children will be allowed to play in their sprinklers on a day above this temperature? a c b d 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. c. 6 b. 600 d What is the z-score associated with observing 5 or fewer heads for 8 tosses of a fair coin? a. c. b. 5 d A die is rolled times. Calculate the probability that a 5 appears exactly twice. a c b d 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. 9.4% c..6% b. 80.6% d. 97.4% 5. An antibiotic is effective against a particular strain of streptococcus 70% of the time. What is the probability that at least 70 of 00 cases will respond when treated with this antibiotic? a. 54.4% c. 70% b. 45.6% d. 49% Short Answer 6. What is the mean value for a binomial experiment with 45 repetitions and a probability of success of 0.5?

3 Name: 7. Calculate the standard deviation for the normal approximation to the binomial distribution for an experiment with 00 trials and a probability of success of Express the z-score for a particular value of X in a binomial distribution in terms of X, n, p, and q where q = p. 9. 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 out of 7 putts or better on a given day, what boundary value should be chosen for X in order to find the appropriate z-score? 0. What is the z-score needed to approximate the probability of observing 5 or fewer heads when tossing a fair coin 8 times?. The probability of success for a binomial experiment is 0.5. What is the smallest number of trials for which we may apply the normal approximation?. The probability that a customer will actually buy a pair of shoes if she tries them on is 0.. What is the probability that Sapna will sell exactly pairs of shoes if she waits on 5 customers who try on shoes?. A bag contains 0 marbles, of which 5 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 of 6 marbles drawn will be white? 4. What is the mean value for the normal approximation to a binomial experiment with 75 repetitions and a probability of success of %? 5. A student guesses at all 0 questions on a true/false test. What is the probability that he will get exactly 5 answers correct? 6. It is known that 0% of the population is left-handed. In a random sample of 00 people, what is the probability that exactly 8 are left-handed? 7. Compare the standard deviation for the normal approximation to a binomial experiment with p = 0.4 for 00 trials and 400 trials. 8. What is the range of probabilities for which the normal approximation may be applied if 40 Bernoulli trials are performed?

4 Sample Questions for Mastery #5 Answer Section MULTIPLE CHOICE. B. C. A 4. D 5. B 6. C 7. C 8. D 9. D 0. C. A. C. B 4. C 5. A SHORT ANSWER From z = x x x np, we replace to obtain z = σ npq. 9. A value of in the discrete distribution corresponds to the range from.5 to.5 in the continuous approximation. We need to calculate P(X.5). We will also need to recognize that the z-score will give the cumulative probability for all values less than.5, so we will need to use minus the probability from the z-score table The smallest number of trials allowed would be.. The probability is or less than %.. The probability is 0.98 or almost 0% The probability is 0.46 or almost 5% For 00 trials, σ = 00(0.4)(0.57) = For 400 trials, σ = 400(0.4)(0.57) = When the number of trials is increased by a factor k, the standard deviation of the normal approximation increases by k.

5 8. We are allowed to use the normal approximation when the values of np and n( p) are both greater than 5. If we consider the smallest value, we must solve 40p > 5, which gives a solution of p > 0.5. Switching the values for p and ( p ), we get a maximum possible value for p of The range of allowable probabilities for applying the normal approximation would be 0.5 < p <

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