Explanation Given info: There were a total of 50 sites a patient might contain lesions. A biopsy selects 8 sites randomly. Calculation: Hypergeometric distribution: In a population with N objects, K objects classified as success and N − K are classified as failures. A sample of size n is randomly selected. Where K ≤ N and n ≤ N . The random variable X denotes number of success in the sample. P ( X = x ) = ( K x ) ( N − K n − x ) ( N n ) x = max { 0 , n + K − N } to min { K , n } The random variable X denotes that the number of sites with lesions in the sample. Here, N = 50 , K , and n = 8 . Substitute these values in the above formula. Therefore, f ( x ) = ( K x ) ( 50 − K 8 − x ) ( 50 8 ) x = max { 0 , 8 + K − 50 } to min { K , 8 } The probability of at least one selected lesions is greater than or equal to 0.95. P ( X ≥ 1 ) = 1 − P ( X = 0 ) ≥ 0.95 Therefore, P ( X = 0 ) ≤ 1 − 0.95 = 0.05 That is, P ( X = 0 ) = ( K 0 ) ( 50 − K 8 − 0 ) ( 50 8 ) 0.05 ≥ ( K ! 0 ! ( K − 0 ) ! ) ( ( 50 − K ) ! 8 ! ( 50 − K − 8 ) ! ) 50 ! 8 ! ( 42 ) ! 0.05 ≥ ( ( 50 − K ) ! 8 ! ( 50 − K − 8 ) ! ) 536 , 878 , 650 Taking K = 15 Then ( ( 50 − K ) ! 8 ! ( 50 − K − 10 ! ) ) 536 , 878 , 650 = ( ( 50 − 15 ) ! 8 ! ( 50 − 15 − 10 ! ) ) 536 , 878 , 650 = ( ( 35 ) ! 8 ! ( 35 − 10 ! ) ) 536 , 878 , 650 = 23 , 535 , 820 536 , 878 , 650 = 0.0438 If K = 15 , then ( ( 50 − K ) ! 8 ! ( 50 − K − 8 ! ) ) 536 , 878 , 650 < 0.05 . Taking K = 14 Then ( ( 50 − K ) ! 8 ! ( 50 − K − 10 ! ) ) 536 , 878 , 650 = ( ( 50 − 14 ) ! 8 ! ( 50 − 14 − 10 ! ) ) 536 , 878 , 650 = ( ( 36 ) ! 8 ! ( 36 − 10 ! ) ) 536 , 878 , 650 = 30 , 260 , 340 536 , 878 , 650 = 0

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118802250

Author: Montgomery

Publisher: WILEY

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Chapter 3, Problem 210SE

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Find the minimum number of sites with lesion so that the probability of at least one selected lesions is greater than or equal to 0.95.

Find the minimum number of sites with lesion so that the probability of at least one selected lesions is greater than or equal to 0.99.

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Chapter 3, Problem 209SE

Chapter 3, Problem 211SE

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Chapter 3 Solutions

Applied Statistics and Probability for Engineers

Ch. 3.1 - For each of the following exercises, determine the...Ch. 3.1 - For each of the following exercises, determine the...Ch. 3.1 - For each of the following exercises, determine the...Ch. 3.1 - For each of the following exercises, determine the...Ch. 3.1 - For each of the following exercises, determine the...Ch. 3.2 - 3-16. The sample space of a random experiment is...Ch. 3.2 - For Exercises 3-17 to 3-21, verify that the...Ch. 3.2 - For Exercises 3-17 to 3-21, verify that the...Ch. 3.2 - For Exercises 3-17 to 3-21, verify that the...Ch. 3.2 - For Exercises 3-17 to 3-21, verify that the...Ch. 3.2 - For Exercises 3-17 to 3-21, verify that the...Ch. 3.2 - 3-22. Consider the hospital patients in Example...Ch. 3.2 - 3-23. An article in Knee Surgery, Sports...Ch. 3.2 - Prob. 24E Ch. 3.2 - Prob. 25E Ch. 3.2 - 3-26. The space shuttle flight control system...Ch. 3.2 - 3-27. A disk drive manufacturer sells storage...Ch. 3.2 - 3-28. Marketing estimates that a new instrument...Ch. 3.2 - 3-29. The distributor of a machine for cytogenics...Ch. 3.2 - 3-30. An assembly consists of two mechanical...Ch. 3.2 - 3-31. An assembly consists of three mechanical...Ch. 3.2 - 3-32. The data from 200 endothermic reactions...Ch. 3.2 - 3-33. Actual lengths of stay at a hospital’s...Ch. 3.2 - 3-34. The distribution of the time until a Web...Ch. 3.2 - 3-35 The following table shows the typical depth...Ch. 3.2 - 3-36. Consider the wafers with contamination...Ch. 3.2 - 3-37. Consider the circuit in Example 2-32. Assume...Ch. 3.3 - 3-38. Determine the cumulative distribution...Ch. 3.3 - 3-39. Determine the cumulative distribution...Ch. 3.3 - 3-40. Determine the cumulative distribution...Ch. 3.3 - 3-41. Determine the cumulative distribution...Ch. 3.3 - 3-42. Determine the cumulative distribution...Ch. 3.3 - Prob. 43E Ch. 3.3 - Prob. 44E Ch. 3.3 - 3-45. Determine the cumulative distribution...Ch. 3.3 - Prob. 46E Ch. 3.3 - 3-47. Determine the cumulative distribution...Ch. 3.3 - 3-48. Determine the cumulative distribution...Ch. 3.3 - Verify that the following functions are cumulative...Ch. 3.3 - Verify that the following functions are cumulative...Ch. 3.3 - Verify that the following functions are cumulative...Ch. 3.3 - Verify that the following functions are cumulative...Ch. 3.3 - 3-53. Determine the cumulative distribution...Ch. 3.3 - 3-54. Determine the cumulative distribution...Ch. 3.3 - 3-55. Determine the cumulative distribution...Ch. 3.3 - 3-56. Determine the cumulative distribution...Ch. 3.4 - 3-57. If the range of X is the set {0,1,2,3,4} and...Ch. 3.4 - 3-58. Determine the mean and variance of the...Ch. 3.4 - 3-59. Determine the mean and variance of the...Ch. 3.4 - 3-60. Determine the mean and variance of the...Ch. 3.4 - 3-61. Determine the mean and variance of the...Ch. 3.4 - Prob. 62E Ch. 3.4 - 3-63. Determine the mean and variance of the...Ch. 3.4 - Prob. 64E Ch. 3.4 - 3-65. The range of the random variable X is [0, 1,...Ch. 3.4 - 3-66. In a NiCd battery, a fully charged cell is...Ch. 3.4 - 3-67. The space shuttle flight control system...Ch. 3.4 - 3-68. Trees are subjected to different levels of...Ch. 3.4 - 3-69. An article in the Journal of Database...Ch. 3.4 - 3-70. Calculate the mean and variance for the...Ch. 3.4 - 3-71. Calculate the mean and variance for the...Ch. 3.4 - 3-72. Calculate the mean and variance for the...Ch. 3.4 - 3-73. Calculate the mean and variance for the...Ch. 3.4 - 3-74. Calculate the mean and variance for the...Ch. 3.4 - 3-75. Calculate the mean for the random variable...Ch. 3.5 - 3-76. Let the random variable X have a discrete...Ch. 3.5 - 3-77. Let the random variable X have a discrete...Ch. 3.5 - 3-78. Thickness measurements of a coating process...Ch. 3.5 - Prob. 79E Ch. 3.5 - 3-80. The lengths of plate glass parts are...Ch. 3.5 - 3-81. Assume that the wavelengths of...Ch. 3.5 - 3-82. The probability of an operator entering...Ch. 3.5 - 3-83. Suppose that X has a discrete uniform...Ch. 3.5 - 3-84. Show that for a discrete uniform random...Ch. 3.5 - 3-85. The number of pages in a PDF document you...Ch. 3.5 - 3-86. Suppose that nine-digit Social Security...Ch. 3.5 - 3-87. Suppose that 1000 seven-digit telephone...Ch. 3.5 - 3-88. The probability that data are entered...Ch. 3.5 - 3-89. Each multiple-choice question on an exam has...Ch. 3.5 - 3-90. Consider the hospital data in Example 2-8....Ch. 3.6 - 3-91. For each scenario (a)–(j), state whether or...Ch. 3.6 - 3-92. Let X be a binomial random variable with p =...Ch. 3.6 - 3-93. Let X be a binomial random variable with p =...Ch. 3.6 - 3-94. The random variable X has a binomial...Ch. 3.6 - 3-95. The random variable X has a binomial...Ch. 3.6 - 3-96. The random variable X has a binomial...Ch. 3.6 - 3-97. Sketch the probability mass function of a...Ch. 3.6 - 3-98. Determine the cumulative distribution...Ch. 3.6 - Prob. 99E Ch. 3.6 - Prob. 100E Ch. 3.6 - 3-101. The phone lines to an airline reservation...Ch. 3.6 - 3-102. A multiple-choice test contains 25...Ch. 3.6 - 3-103. A particularly long traffic light on your...Ch. 3.6 - 3-104. Samples of rejuvenated mitochondria are...Ch. 3.6 - Prob. 105E Ch. 3.6 - 3-106. Heart failure is due to either natural...Ch. 3.6 - 3-107. A computer system uses passwords that are...Ch. 3.6 - 3-108. Samples of 20 parts from a metal punching...Ch. 3.6 - 3-109. Because all airline passengers do not show...Ch. 3.6 - 3-110. This exercise illustrates that poor quality...Ch. 3.6 - 3-111. Consider the lengths of stay at a...Ch. 3.6 - 3-112. Consider the patient data in Example 2-8....Ch. 3.6 - 3-113. Assume that a Web site changes its content...Ch. 3.6 - 3-114. Consider the endothermic reactions in...Ch. 3.6 - 3-115. The probability that a visitor to a Web...Ch. 3.6 - 3-116. Consider the circuit in Example 2-34....Ch. 3.6 - 3-117. Consider the time to recharge the flash in...Ch. 3.6 - 3-118. Consider the patient data in Example 2-8....Ch. 3.7 - 3-119. Suppose that the random variable X has a...Ch. 3.7 - Prob. 120E Ch. 3.7 - 3-121. Consider a sequence of independent...Ch. 3.7 - Prob. 122E Ch. 3.7 - 3-123. The probability of a successful optical...Ch. 3.7 - 3-124. In a clinical study, volunteers are tested...Ch. 3.7 - 3-125. Assume that each of your calls to a popular...Ch. 3.7 - 3-126. A player of a video game is confronted with...Ch. 3.7 - 3-127. Heart failure is due to either natural...Ch. 3.7 - 3-128. A computer system uses passwords...Ch. 3.7 - 3-129. A trading company uses eight computers to...Ch. 3.7 - 3-130. Assume that 20 parts are checked each hour...Ch. 3.7 - 3-131. A fault-tolerant system that processes...Ch. 3.7 - Prob. 132E Ch. 3.7 - Prob. 133E Ch. 3.7 - 3-134. Consider the endothermic reactions in...Ch. 3.7 - 3-135. A Web site randomly selects among 10...Ch. 3.7 - Prob. 136E Ch. 3.7 - 3-137. Consider the time to recharge the flash in...Ch. 3.7 - 3-138. An array of 30 LED bulbs is used in an...Ch. 3.7 - Prob. 139E Ch. 3.7 - 3-140. Customers visit a Web site, and the...Ch. 3.8 - 3-141. Suppose that X has a hypergeometric...Ch. 3.8 - Prob. 142E Ch. 3.8 - Prob. 143E Ch. 3.8 - Prob. 144E Ch. 3.8 - 3-145. A research study uses 800 men under the age...Ch. 3.8 - Prob. 146E Ch. 3.8 - Prob. 147E Ch. 3.8 - 3-148. A state runs a lottery in which six numbers...Ch. 3.8 - Prob. 149E Ch. 3.8 - 3-150. Calculate the finite population...Ch. 3.8 - 3-151. Consider the visits that result in leave...Ch. 3.8 - 3-152. Consider the nonfailed wells in Exercises...Ch. 3.8 - 3-153. Consider the semiconductor wafer data in...Ch. 3.8 - 3-154. Suppose that a healthcare provider selects...Ch. 3.8 - Prob. 155E Ch. 3.8 - 3-156. A utility company might offer electrical...Ch. 3.9 - 3-157. Suppose that X has a Poisson distribution...Ch. 3.9 - Prob. 158E Ch. 3.9 - Prob. 159E Ch. 3.9 - 3-160. The number of telephone calls that arrive...Ch. 3.9 - 3-161. Astronomers treat the number of stars in a...Ch. 3.9 - 3-162. Data from www.centralhudsonlabs.com...Ch. 3.9 - Prob. 163E Ch. 3.9 - 3-164. The number of flaws in bolts of cloth in...Ch. 3.9 - Prob. 165E Ch. 3.9 - 3-166. The number of cracks in a section of...Ch. 3.9 - Prob. 167E Ch. 3.9 - 3-168. The number of failures of a testing...Ch. 3.9 - 3-169. The number of content changes to a Web site...Ch. 3.9 - 3-170. The number of views of a page on a Web site...Ch. 3.9 - Prob. 171E Ch. 3.9 - 3-172. Orders arrive at a Web site according to a...Ch. 3.9 - 3-173. The article “An Association Between Fine...Ch. 3.9 - 3-174. Inclusions are defects in poured metal...Ch. 3 - Prob. 175SE Ch. 3 - 3-176. Let X denote the number of bits received in...Ch. 3 - 3-177. Batches that consist of 50 coil springs...Ch. 3 - Prob. 178SE Ch. 3 - 3-179. A total of 12 cells are replicated. Freshly...Ch. 3 - Prob. 180SE Ch. 3 - 3-181. A particularly long traffic light on your...Ch. 3 - 3-182. The probability is 0.6 that a calibration...Ch. 3 - Prob. 183SE Ch. 3 - 3-184. The probability that an eagle kills a...Ch. 3 - 3-185. Traffic flow is traditionally modeled as a...Ch. 3 - 3-186. A shipment of chemicals arrives in 15...Ch. 3 - Prob. 187SE Ch. 3 - 3-188. The probability that your call to a service...Ch. 3 - 3-189. The probability that your call to a service...Ch. 3 - 3-190. The number of messages that arrive at a Web...Ch. 3 - Prob. 191SE Ch. 3 - 3-192. The number of errors in a textbook follows...Ch. 3 - Prob. 193SE Ch. 3 - Prob. 194SE Ch. 3 - 3-195. In a manufacturing process that laminates...Ch. 3 - Prob. 196SE Ch. 3 - Prob. 197SE Ch. 3 - 3-198. A manufacturer of a consumer electronics...Ch. 3 - Prob. 199SE Ch. 3 - 3-200. The random variable X has the following...Ch. 3 - Prob. 201SE Ch. 3 - Prob. 202SE Ch. 3 - Prob. 203SE Ch. 3 - 3-204. An installation technician for a...Ch. 3 - Prob. 205SE Ch. 3 - Prob. 206SE Ch. 3 - 3-207. Messages arrive to a computer server...Ch. 3 - 3-208. Flaws occur in the interior of plastic used...Ch. 3 - Prob. 209SE Ch. 3 - Prob. 210SE Ch. 3 - Prob. 211SE Ch. 3 - Prob. 212SE Ch. 3 - Prob. 213SE Ch. 3 - Prob. 214SE Ch. 3 - Prob. 215SE Ch. 3 - Prob. 216SE Ch. 3 - Prob. 217SE Ch. 3 - Prob. 218SE Ch. 3 - Prob. 219SE

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Find the minimum number of sites with lesion so that the probability of at least one selected lesions is greater than or equal to 0.95. Find the minimum number of sites with lesion so that the probability of at least one selected lesions is greater than or equal to 0.99.