Ch07 Continuous Probability Distributions.docx

Chapter 7 Continuous Probability Distributions True/False 1. The Empirical Rule of probability can be applied to the uniform probability distribution. Answer: False 2. Areas within a continuous probability distribution represent probabilities. Answer: True 3. The total area within a continuous probability distribution is equal to 100. Answer: False 4. The total area within any continuous probability distribution is equal to 1.00 Answer: True 5. For any continuous probability distribution, the probability, P(x), of any value of the random variable, X, can be computed. Answer: False 6. For any discrete probability distribution, the probability, P(x), of any value of the random variable, X, can be computed. Answer: True 7. The uniform probability distribution's standard deviation is proportional to the distribution's range. Answer: True 8. For any uniform probability distribution, the mean and standard deviation can be computed by knowing the maximum and minimum values of the random variable. Answer: True 9. In a uniform probability distribution, P(x) is constant between the distribution's minimum and maximum values. Answer: True 10. For a uniform probability distribution, the probability of any event is equal to 1/(b-a). Answer: False 11. The uniform probability distribution is symmetric about the mode. Answer: False 12. The uniform probability distribution's shape is a rectangle. Answer: True 13. The uniform probability distribution is symmetric about the mean and median. Answer: True 14. Asymptotic means that the normal curve gets closer and closer to the X-axis but never actually touches it. Answer: True 15. A continuity correction factor compensates for estimating a discrete distribution with a continuous distribution. Answer: True 16. The normal curve falls off smoothly in either direction from the central value. Since it is asymptotic, the curve gets closer and closer to the X-axis, but never actually touches it. Answer: True 17. When referring to the normal probability distribution, there is not just one; there is a "family" of distributions. Answer: True 18. Some normal probability distributions have equal arithmetic means, but their standard deviations may be different. Answer: True 19. Some normal probability distributions have different arithmetic means and different standard deviations. Answer: True 20. Some normal probability distributions are positively skewed. Answer: False 21. For a normal probability distribution, about 95 percent of the area under normal curve is within plus and minus two standard deviations of the mean and practically all (99.73 percent) of the area under the normal curve is within three standard deviations of the mean. Answer: True 22. The area under the normal curve within plus and minus one standard deviation of the mean is about 68.26%. Answer: True 23. The total area under the normal curve is 100%. Answer: True 24. A z-score is the distance between a selected value (X) and the population mean () divided by the population standard deviation (). Answer: True 25. In terms of a formula the standardized value of z is found by z = (X – )/. Answer: True 26. The mean () divides the normal curve into two identical halves. Answer: True 27. The normal probability distribution is generally deemed a good approximation for the binomial probability distribution when n? and n(1 – ) are both greater than five. Answer: True 28. The number of different normal distributions is unlimited. Answer: True 29. A z-score is also referred to as the standard normal deviate or just the normal deviate. Answer: True 30. The mean of a normal distribution is represented by . Answer: False 31. The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. Answer: True 32. A computed z for X values to the right of the mean is negative. Answer: False 33. A computed z for X values to the left of the mean is positive. Answer: False 34. Non-stop Airlines determined that the mean number of passengers per flight is 152 with a standard deviation of ten passengers. Practically all flights have between 142 and 162 passengers. Answer: False 35. The binomial can be used to approximate the normal distribution. Answer: False 36. The number of different standard normal distributions is unlimited. Answer: False Multiple Choice 37. The shape of any uniform probability distribution is A) Negatively skewed B) Positively skewed C) Rectangular D) Bell shaped Answer: C 38. The mean of any uniform probability distribution is A) (b - a)/2 B) (a + b)/2 C) D) n Answer: B 39. The standard deviation of any uniform probability distribution is A) (b – a)/2 B) n ( 1 – ) C) D) Answer: C 40. The upper and lower limits of a uniform probability distribution are A) positive and negative infinity B) plus and minus three standard deviations. C) 0 and 1 D) the maximum and minimum values of the random variable. Answer: D 41. When using the binomial distribution to approximate a normal distribution, what is the value of the continuity correction factor? A) 1.00 B) 0.50 C) 100 D) 1.96 Answer: B 42. A new drug relieves nasal congestion in 90 percent of those with the condition. The new drug is administered to 300 patients with the condition. What is the probability that more than 265 will be relieved of nasal congestion? A) 0.0916 B) 0.1922 C) 0.8078 D) 0.3078 Answer: C 43. What is an important similarity between the uniform and normal probability distributions? A) The mean, median and mode are all equal. B) The mean and median are equal C) They are negatively skewed D) About 68% of all observations are within one standard deviation of the mean. Answer: B 44. Which of the following is NOT true regarding the normal distribution? A) Mean, median and mode are all equal B) It has a single peak C) It is symmetrical D) The points of the curve meet the X-axis at z = –3 and z = 3 Answer: D 45. For the normal distribution, the mean plus and minus 1.96 standard deviations will include about what percent of the observations? A) 50% B) 99.7% C) 95% D) 68% Answer: C 46. For a standard normal distribution, what is the probability that z is greater than 1.75? A) 0.0401 B) 0.0459 C) 0.4599 D) 0.9599 Answer: A 47. What is the area under the normal curve between z = 0.0 and z = 1.79? A) 0.4633 B) 0.0367 C) 0.9599 D) 0.0401 Answer: A 48. What is the area under the normal curve between z = –1.0 and z = –2.0? A) 0.0228 B) 0.3413 C) 0.1359 D) 0.4772 Answer: C 49. What is the area under the normal curve between z = 0.0 and z = 2.0? A) 1.0000 B) 0.7408 C) 0.1359 D) 0.4772 Answer: D 50. The mean amount spent by a family of four on food per month is $500 with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month? A) 0.2158 B) 0.8750 C) 0.0362 D) 0.1151 Answer: D 51. Which of the following is NOT a characteristic of the normal probability distribution? A) Positively-skewed B) Bell-shaped C) Symmetrical D) Asymptotic Answer: A 52. What is the proportion of the total area under the normal curve within plus and minus two standard deviations of the mean? A) 68% B) 99.7% C) 34% D) 95% Answer: D 53. The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed. What percent of the students scored below 320? A) About 50.82% B) About 34.13% C) About 7.86% D) About 0.82% Answer: D 54. The mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100. What is the z-score for an income of $1,100? A) 1.00 B) 2.00 C) 1.683 D) -0.90 Answer: A 55. A new extended-life light bulb has an average service life of 750 hours, with a standard deviation of 50 hours. If the service life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours? A) 95% B) 68% C) 34% D) 99.7% Answer: D 56. A study of a company's practice regarding the payment of invoices revealed that an invoice was paid an average of 20 days after it was received. The standard deviation equaled five days. Assuming that the distribution is normal, what percent of the invoices were paid within 15 days of receipt? A) 15.87% B) 37.91% C) 34.13% D) 86.74% Answer: A 57. An accelerated life test on a large number of type-D alkaline batteries revealed that the mean life for a particular use before they failed is 19.0 hours. The distribution of the lives approximated a normal distribution. The standard deviation of the distribution was 1.2 hours. About 95.44 percent of the batteries failed between what two values? A) 8.9 and 18.9 B) 12.2 and 14.2 C) 14.1 and 22.1 D) 16.6 and 21.4 Answer: D 58. The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. What is the area between 415 pounds and the mean of 400 pounds? A) 0.5000 B) 0.1932 C) 0.4332 D) 0.3413 Answer: C 59. The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes? A) $30,000 and $40,000 B) $36,400 and $38,000 C) $34,800 and $39,600 D) $35,600 and $38,800 Answer: B 60. Which of the following is true in a normal distribution? A) Mean equals the mode and the median B) Mode equals the median C) Mean divides the distribution into two equal parts D) All of the above are correct Answer: D 61. Tables of normal distribution probabilities are found in many statistics books. These probabilities are calculated from a normal distribution with A) a mean of 1 and a standard deviation of 1 B) a mean of 100 and a standard deviation of 15 C) a mean of 0 and a standard deviation of 15 D) a mean of 0 and a standard deviation of 1 Answer: D 62. Two normal distributions are compared. One has a mean of 10 and a standard deviation of 10. The second normal distribution has a mean of 10 and a standard deviation of 2. Which of the following it true? A) the locations of the distributions are different B) the distributions are from two different families C) the dispersions of the distributions are different D) the dispersions of the distributions are the same Answer: C 63. A random variable from an experiment where outcomes are normally distributed A) can have any value between -? and +?? B) can have only a few discrete values C) can have a mean of 0 and a standard deviation of 1 D) can have no values Answer: A 64. The total area of a normal probability distribution is A) between –3.0 and 3.0 B) 1.00 C) dependent on a value of 'z'. D) approximated by the binomial distribution. Answer: B 65. An area of a normal probability distribution represents A) a permutation B) a combination C) a likelihood D) a shaded area Answer: C 66. The standard normal probability distribution is one which has: A) A mean of 1 and any standard deviation B) Any mean and a standard deviation of 1 C) A mean of 0 and any standard deviation D) A mean of 0 and a standard deviation of 1 Answer: D 67. The weekly mean income of a group of executives is $1000 and the standard deviation of this group is $100. The distribution is normal. What percent of the executives have an income of $925 or less? A) About 15% B) About 85% C) About 50% D) About 23% Answer: D 68. The weight of cans of fruit is normally distributed with a mean of 1,000 grams and a standard deviation of 50 grams. What percent of the cans weigh 860 grams or less? A) 0.0100 B) 0.8400 C) 0.0026 D) 0.0001 Answer: C 69. What is the distribution with a mean of 0 and a standard deviation of 1 called? A) Frequency distribution B) z-score C) Standard normal distribution D) Binomial probability distribution Answer: C 70. The seasonal output of a new experimental strain of pepper plants was carefully weighed. The mean weight per plant is 15.0 pounds, and the standard deviation of the normally distributed weights is 1.75 pounds. Of the 200 plants in the experiment, how many produced peppers weighing between 13 and 16 pounds? A) 100 B) 118 C) 197 D) 53 Answer: B 71. Ball-Bearings, Inc. produces ball bearings automatically on a Kronar BBX machine. For one of the ball bearings, the mean diameter is set at 20.00 mm (millimeters). The standard deviation of the production over a long period of time was computed to be 0.150 mm. What percent of the ball bearings will have diameters 20.27 mm or more? A) 41.00% B) 12.62% C) 3.59% D) 85.00% Answer: C 72. A national manufacturer of unattached garages discovered that the distribution of the lengths of time it takes two construction workers to erect the Red Barn model is approximately normally distributed with a mean of 32 hours and a standard deviation of 2 hours. What percent of the garages take between 30 and 34 hours to erect? A) 16.29% B) 76.71% C) 3.14% D) 34.13% Answer: D 73. A large manufacturing firm tests job applicants who recently graduated from college. The test scores are normally distributed with a mean of 500 and a standard deviation of 50. Management is considering placing a new hire in an upper level management position if the person scores in the upper 6 percent of the distribution. What is the lowest score a college graduate must earn to qualify for a responsible position? A) 50 B) 625 C) 460 D) 578 Answer: D 74. An analysis of the grades on the first test in History 101 revealed that they approximate a normal curve with a mean of 75 and a standard deviation of 8. The instructor wants to award the grade of A to the upper 10 percent of the test grades. What is the dividing point between an A and a B grade? A) 80 B) 85 C) 90 D) 95 Answer: B 75. The annual commissions per salesperson employed by a manufacturer of light machinery averaged $40,000 with a standard deviation of $5,000. What percent of the sales persons earn between $32,000 and $42,000? A) 60.06% B) 39.94% C) 34.13% D) 81.66% Answer: A 76. The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95 percent of the observations lie between what two values? A) 475 and 525 B) 480 and 520 C) 400 and 600 D) 350 and 650 Answer: B 77. A cola-dispensing machine is set to dispense a mean of 2.02 liters into a container labeled 2 liters. Actual quantities dispensed vary and the amounts are normally distributed with a standard deviation of 0.015 liters. What is the probability a container will have less than 2 liters? A) 0.0918 B) 0.3413 C) 0.1926 D) 0.8741 Answer: A 78. The employees of Cartwright Manufacturing are awarded efficiency ratings. The distribution of the ratings approximates a normal distribution. The mean is 400, the standard deviation 50. What is the area under the normal curve between 400 and 482? A) 0.5000 B) 0.4495 C) 0.3413 D) 0.4750 Answer: B 79. Suppose a tire manufacturer wants to set a mileage guarantee on its new XB 70 tire. Tests revealed that the tire's mileage is normally distributed with a mean of 47,900 miles and a standard deviation of 2,050 miles. The manufacturer wants to set the guaranteed mileage so that no more than 5 percent of the tires will have to be replaced. What guaranteed mileage should the manufacturer announce? A) 44,528 B) 32,960 C) 49,621 D) 40,922 Answer: A 80. The mean amount of gasoline and services charged by Key Refining Company credit customers is $70 per month. The distribution of amounts spent is approximately normal with a standard deviation of $10. What is the probability of selecting a credit card customer at random and finding the customer charged between $70 and $83? A) 0.1962 B) 0.4032 C) 0.3413 D) 0.4750 Answer: B 81. Management is considering adopting a bonus system to increase production. One suggestion is to pay a bonus on the highest 5 percent of production based on past experience. Past records indicate that, on the average, 4,000 units of a small assembly are produced during a week. The distribution of the weekly production is approximately normally distributed with a standard deviation of 60 units. If the bonus is paid on the upper 5 percent of production, the bonus will be paid on how many units or more? A) 6255 B) 5120 C) 3196 D) 4099 Answer: D Fill-in-the-Blank 82. About what percent of the area under the normal curve is within plus two and minus two standard deviation of the mean? _______ Answer: 95% (95.5%) 83. What is a graph of a normal probability distribution called? _________ Answer: normal curve 84. In a standard normal distribution, = ______ and = ______. Answer: zero and one 85. What type of probability distribution is the normal distribution? ______________ Answer: Continuous 86. What is the formula to convert any normal distribution to the standard normal distribution? _______________________ Answer: 87. In what units does the standardized z value measure distance from the mean? ______________________ Answer: standard deviation 88. What proportion of the area under a normal curve is to the right of a z-score of zero? _________ Answer: 50% or 0.50 89. The mean of a normal probability distribution is 60 and the standard deviation is 5. What percent of observations are between 50 and 70? _______ % Answer: 95.44 90. What does a z value of –2.00 indicate about the corresponding X value? _______________________ Answer: less than or to the left of the mean, or the X value is 2 standard deviations less than the mean 91. One of the properties of the normal curve is that it gets closer to the horizontal axis, but never touches it. What is this property of the normal curve called? _________________ Answer: asymptotic 92. What proportion of the area under a normal curve is to the right of z = –1.21? ________ Answer: 0.8461 93. What proportion of the area under a normal curve is to the left of z = 0.50? ________ Answer: 0.6914 94. What proportion of the area under a normal curve is to the left of z = –2.10? _________ Answer: 0.0179 95. A statistics student receives a grade of 85 on a statistics midterm. If the corresponding z-score equals +1.5 and the standard deviation equals 7, what is the average grade on this exam? _______ Answer: 74.5 Use the following to answer questions 96-105: A major credit card company has determined that customers charge between $100 and $1100 per month. Given that the average monthly amount charged is uniformly distributed, answer the following questions. 96. What is the average monthly amount charged? _______ Answer: $600 97. What is the standard deviation of the monthly amount charged? _________ Answer: 288.67 98. What percent of monthly charges are equal to $500? __________ Answer: Cannot calculate with a continuous distribution 99. What percent of monthly charges are between $600 and $889? ____________ Answer: 28.9% 100. What percent of monthly changes are between $311 and $889? ________ Answer: 57.8% 101. What percent of monthly charges is less then $100? _____________ Answer: None 102. What percent of monthly charges are between $100 and $1100? __________ Answer: 100% or all. 103. What is the probability that a person charges less than $200 per month? _____ Answer: 0.1 or 10% 104. What is the 3rd quartile of the distribution? _____________ Answer: $850 105. 75% of all monthly charges are greater than ______________? Answer: $350 Use the following to answer questions 106-114: A financial advising company has determined that the price-to-earnings ratios for 20 randomly selected publicly traded companies range between 0.9 and 2.9. Given that the price-to-earnings ratios are uniformly distributed, answer the following questions. 106. What is the average price-to-earnings ratio? _______ Answer: 1.9 107. What is the standard deviation of the price-to-earnings ratio? _________ Answer: 0.58 108. What percent of price-to-earnings ratios are equal to 2.58? __________ Answer: Cannot calculate with a continuous distribution 109. What percent of price-to-earnings ratios are between 1.90 and 2.48? ____________ Answer: 29% 110. What percent of price-to-earnings ratios are between 1.32 and 2.48? ________ Answer: 58% 111. What percent of price-to-earnings ratios are less than 0.9? _____________ Answer: None 112. What percent of price-to-earnings ratios are between .9 and 2.9? __________ Answer: 100% or all. 113. What is the probability that a price-to-earnings ratio is less than 2.1? _____ Answer: 0.1 or 10% 114. What is the 3rd quartile of the distribution? _____________ Answer: 2.4 115. 75% of all price-to-earnings ratios are greater than ______________? Answer: 1.4 Use the following to answer questions 116-121: A sample of 500 evening students revealed that their annual incomes from employment in industry during the day were normally distributed with a mean income of $30,000 and a standard deviation of $3,000. 116. How many students earned more than $30,000? _______ Answer: 250 117. How many students earned between $27,000 and $33,000? ______ Answer: 341 118. How many students earned between $24,000 and $30,000? ______ Answer: 239 119. How many students earned between $20,000 and $40,000? ______ Answer: 500 120. How many students earned less than $22,500? _______ Answer: 3 121. How many students earned more than $36,000? _______ Answer: 11 Use the following to answer questions 122-128: The weight of a bag of corn chips is normally distributed with a mean of 22 ounces and a standard deviation of ½ ounces. 122. What is the probability that a bag of corn chips is less than 20 ounces? _____ Answer: 0.0 123. What is the probability that a bag of corn chips weighs more than 21 ounces? _____ Answer: 0.9772 124. What is the probability that a bag of corn chips is weighs more than 23 ounces? _____ Answer: 0.0228 125. What is the probability that a bag of corn chips weighs less than 24 ounces? _____ Answer: 1.0 126. What is the probability that a bag of corn chips weighs between 20.75 and 23.25 ounces? _____ Answer: 0.9876 127. What is the probability that a bag of corn chips weighs 22.25 ounces? _____ Answer: 0.0 128. What is the probability that a bag of corn chips weighs between 21.75 and 22.25 ounces? _____ Answer: 0.3830 Use the following to answer questions 129-131: Two business major students, in two different sections of economics, were comparing test scores. The following gives the students scores, class mean, and standard deviation for each section. 129. Which student scored better compared to the rest of the section? _______________ Answer: student from section 2 130. What is the z-score of the student from section 1? _____________ Answer: 1.28 131. What is the z-score of the student from section 2? _____________ Answer: 1.87 Multiple Choices Use the following to answer questions 132-134: The average score of 100 students taking a statistics final was 70 with a standard deviation of 7. 132. Assuming a normal distribution, approximately how many scored 90 or higher? A) 0.4979 B) 0.0021 C) 0.9979 D) 2.86 Answer: B 133. Assuming a normal distribution, approximately how many scored less than 60? A) 0.2271 B) 0.3729 C) 0.8929 D) – 1.14 E) None of the above Answer: E 134. Assuming a normal distribution, approximately how many scored greater than 65? A) 0.2611 B) 0.2389 C) 0.7611 D) –0.714 Answer: C Use the following to answer questions 135-138: Bottomline Ink, a forms management company, fills 100 orders a day with a 2% error rate in the completed orders. Assume this to be a binomial distribution. 135. What is the mean for this distribution? A) 0.02 B) 1.4 C) 2 D) There is no mean for this type of distribution. Answer: C 136. What is the standard deviation for this distribution? A) 0.02 B) 4 C) 2 D) There is no standard deviation for this type of distribution. Answer: B 137. What is the probability that there will be more than 5 order errors in a given day? A) 0.1894 B) 0.4838 C) 0.9838 D) 2.1428 Answer: A 138. The probability of less than 1 order error in a given day is A) 0.7143. B) 0.3520 C) 0.2611. D) 2.7611. Answer: B Fill-in-the-Blank 139. How is the expected value of a uniform distribution computed? __________ Answer: 1/(b-a) 140. How is the standard deviation of a uniform distribution computed? Answer: 141. How is the expected value of a normal distribution computed? Answer: ? X/ N 142. How is the standard deviation of a normal distribution computed? Answer: Essay 143. Identify two similarities of the uniform and normal distributions? Answer: Both distributions are continuous, areas for both distributions are 1.00 , 4 144. For a binomial distribution with both n and n ( 1 – ) greater than 5, what can we conclude about the characteristics of the distribution? Answer: the distribution is approximately normal with a mean of n and a standard deviation of n ( 1 – )