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Chapter 6: Q. 78 (page 404)
78. Rhubarb Refer to Exercise 76. Would you be surprised if or more of the plants in the bundle die before producing any rhubarb? Calculate an appropriate probability to support your answer.
The probability is less than ,Thereforeand the result is surprising.
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Ms. Hall gave her class a -question multiple-choice quiz. Let the number of questions that a randomly selected student in the class answered correctly. The computer output below gives information about the probability distribution of X. To determine each student鈥檚 grade on the quiz (out of localid="1649489099543" ), Ms. Hall will multiply his or her number of correct answers by 10. Let localid="1649489106434" the grade of a randomly chosen student in the class.
localid="1649489113566"
(a) Find the mean of localid="1649489121120" . Show your method.
(b) Find the standard deviation of localid="1649489127059" . Show your method.
(c) How do the variance of localid="1649489132289" and the variance oflocalid="1649489138146" compare? Justify your answer.
A machine fastens plastic screwon caps onto containers of motor oil. If the machine applies more torque than the cap can withstand, the cap will break. Both the torque applied and the strength of the caps vary. The capping-machine torquefollows a Normal distribution with mean inch-pounds and standard deviation inch-pounds. The cap strength C (the torque that would break the cap) follows a Normal distribution with mean 10 inch-pounds and standard deviation inch-pounds.
(a) Explain why it is reasonable to assume that the cap strength and the torque applied by the machine are independent.
(b) Let the random variable. Find its mean and standard deviation.
(c) What is the probability that a cap will break while being fastened by the machine? Show your work.
Benford鈥檚 law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from to . In that case, the first digit of a randomly selected expense amount would have the probability distribution shown in the histogram.
(a). Explain why the mean of the random variable Y is located at the solid red line in the figure.
(b) The first digits of randomly selected expense amounts actually follow Benford鈥檚 law (Exercise 5). What鈥檚 the expected value of the first digit? Explain how this information could be used to detect a fake expense report.
(c) What鈥檚 ? According to Benford鈥檚 law, what proportion of first digits in the employee鈥檚 expense amounts should be greater than ? How could this information be used to detect a fake expense report?
Kids and toys Refer to Exercise 4. Calculate and interpret the standard deviation of the random variable . Show your work.
Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X.
| Value | |||||
| Probability | ? |
(a) Find
(b) If two patients who received artificial knees are chosen at random, what鈥檚 the probability that both of them report pain scores of or ? Show your work.
(c) Compute the mean and standard deviation of . Show your work.
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