BIT Assignment - 3

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College of Charleston *

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342

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Statistics

Date

Apr 3, 2024

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BIT ASSIGNMENT - 3 Problem 04-12 (Conditional Probability) Question 1 of 5 » Hint(s) Check My Work According to a 2018 article in Esquire magazine, approximately 70% of males over age 70 will develop cancerous cells in their prostate. Prostate cancer is second only to skin cancer as the most common form of cancer for males in the United States. One of the most common tests for the detection of prostate cancer is the prostate-specific antigen (PSA) test. However, this test is known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present). Suppose there is a 0.02 probability that a male patient has prostate cancer before testing. The probability of a false-positive test is 0.75, and the probability of a false-negative (no indication of cancer when cancer is actually present) is 0.20. Let C = event male patient has prostate cancer + = positive PSA test for prostate cancer — = negative PSA test for prostate cancer (@) What is the probability that the male patient has prostate cancer if the PSA test comes back positive? Round your answer to four decimal places. 0.0213 @ (b) What is the probability that the male patient has prostate cancer if the PSA test comes back negative? Round your answer to four decimal places. 0.0161 @ (c) For older men, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is 0.3 rather than 0.02. What is the probability that the male patient has prostate cancer if the PSA test comes back positive? Round your answer to four decimal places. 0.3137 @ What is the probability that the male patient has prostate cancer if the PSA test comes back negative? Round your answer to four decimal places. 0.2553 @ (d) What can you infer about the PSA test from the results of parts (a), (b), and (c)? The difference between P(C|+) and P(C|-) in parts (a) and (b) is[ lower v ’ ) than the difference between P(C|+) and P(C|-) in part (c). Problem 04-15 Algo (Discrete Probability Distributions) 4 Question 2 of 5 » Hint(s) Check My Work The percent frequency distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers are as follows. The scores range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied). Job Satisfaction IS Senior IS Middle Score Executives (%) Managers (%) 1 5 4 2 9 10 3 40 19 - 42 46 5 4 21 If required, round your answers to two decimal places. (a) Develop a probability distribution for the job satisfaction score of a randomly selected senior executive. x f(x) 1 0.0s & 2 0.00 & 3 0.20 & 4 0.02 & > 0.0a & 1.00 &

(b) Develop a probability distribution for the job satisfaction score of a randomly selected middle manager. x f(x) Ceereeee@ 1.00 (c) What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5? 0.46 @ (d) What is the probability that a randomly selected middle manager is very satisfied? 0.21 @ (e) Compare the overall job satisfaction of senior executives and middle managers. d v @ than middle managers. Senior executives appear to be | less sat Problem 04-22 Algo (Discrete Probability Distributions) 4 Question3 of 5 » Hint(s) Check My Work Consider a Poisson distribution with 1 = 6. If needed, round your answer to four decimal digits. (@) Choose the appropriate Poisson probability mass function. - z 6 e z 6 O g =T @ -T2 i) 6%’ (iv) _ afet f@) = —— f@) == | option (i) 7;\@ (b) Compute £(2). 0.0446 @ (c) Compute f(1). 0.0149 @ (d) Compute P(x = 2). 0.9826 &

Problem 04-33 Algo (Continuous Probability Distributions) 4 Question4of 5 » Hint(s) Check My Work Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations. (@) The process standard deviation is 0.24, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.76 or greater than 10.24 ounces will be classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number. Probability of a defect: 0.3174 @ Number of defects: 317 @ (b) Through process design improvements, the process standard deviation can be reduced to 0.08. Assume that the process control remains the same, with weights less than 9.76 or greater than 10.24 ounces being classified as defects. If required, round your answer for the probability of a defect to four decimal places and for the number of defects to the nearest whole number. Probability of a defect: | 0.0026 | & Number of defects: 3@ (c) What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean? Reducing the process standard deviation causes a | substantial decrease @ in the number of defects. Problem 04-34 Algo (Continuous Probability Distributions) 4 Question 5 of 5 Hint(s) Check My Work Consider the following exponential probability density function. 1 fla) =% e /5 forx=0 If needed, round your answer to four decimal digits. (a) Choose the correct formula for P(x < xo). (i) P(z<m)= Pz < @) = 0/5 i) P(z<az)=1+e e /% (iv) P(z <zo)=1+e/° [option i) +|© (b) Find P(x < 2). 0.3297 @ (c) Find P(x 2 3). o.5a88 & (d) Find P(x < 5). 0.6321 & (e) Find P(2 < x < 5). 03024 &

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BIT Assignment - 3

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