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# Statistics Problem: Probability theory for BreatheCorp battery life

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Problem Set 2

BreatheCorp manufactures two types of Portable Oxygen Concentrator Batteries: lithium ion and NiMH. 56% of the batteries they produce are lithium ion. They offer a 5,000 hour warranty on each of the batteries they sell, and 93% of all batteries exceed 5,000 hours of useful life. Furthermore, 53% of all the batters sold are both all lithium ion AND exceed 5,000 hours of life. You can find this information filled out in the contingency table in the Data tab. Use this information to answer the following questions. Questions 2.1 and 2.2 relate to information from week 4 in the class, 2.3 comes from week 5 material, and 2.4 is based on material from week 6. Note that part a (and ONLY part a) of question 2.3 is an extra credit question. Should you encounter any difficulties with these problems, the optional problems below are very similar to the questions in this problem set, and the answers to the optional questions can be found in the back of the textbook. You can also request that the tutor work extensively with you on the optional problems.

Problem 2.1

"Use probability theory to answer the following questions:

(a) Give an example (related to the setting of the question) of a simple event.
(b) Give an example (related to the setting of the question) of a joint event.
(c) Use the information given to complete the contingency table.
(d) What is the probability that a battery selected at random is a NiMH battery?
(e) What is the probability that a battery selected at random fails to exceed 5,000 hours?
(f) What is the probability that a battery selected at random is either a NiMH or fails to exceed 5,000 hours?
(g) What is the probability that a battery selected at random is both an NiMH and exceeds 5,000 hours?
(h) What is the probability that an NiMH battery selected at random exceeds 5,000 hours of life?
(i) What is the probability that a lithium ion battery selected at random exceeds 5,000 hours of life?"

Problem 2.2

"Assume that BreatheCorp sells their lithium ion batteries for \$350 each and their NiMH batteries for \$285 each. Further assume that the cost of producing a lithium ion battery is \$280 and the cost of producing a NiMH battery is \$245. Finally, assume that if a battery does not last 5,000 hours, BreatheCorp will replace it free of charge to the consumer; BreatheCorp will incur the cost of replacement, but will not receive any additional revenue.

(a) Calculate the profit earned/loss incurred on; a lithium ion that exceeds 5,000 hours, a lithium ion that does not exceed 5,000 hours, a NiMH that exceeds 5,000 hours, and a NiMH that fails to last 5,000 ours.
(b) What is the expected value (expected profit) of producing a lithium ion battery? A NiMH battery?
(c) What is the variance and standard deviation (of profit, NOT of battery life) for producing an NiMH battery? Of producing a lithium ion battery? "

Problem 2.3

"Assume that BreatheCorp battery life is normally distributed and that battery types have have a mean life of 5,750 hours. Furthermore, the standard deviation of battery life for NiMH batteries is 561.72 hours and the standard deviation of battery life for lithium ion is 465.5 miles.
(a) EXTRA CREDIT-Using the probabilities you calculated in problem 2.1, show that the numbers given here for the standard deviation of each battery type are correct. Note: depending on rounding, the numbers you calculate may be slightly higher or lower than the ones given in the problem.
(b) If BreatheCorp wanted to ensure that 95% of their batteries exceeded the warrantee, what would they have to set their warrantee at for NiMH batteries? For lithium ion tires?

Excel Tips:
Excel has a trio of built in functions that are useful for calculating Z scores and dealing with the standard normal distribution. These are =STANDARDIZE, =NORMSINV, and =NORMSDIST. See the Excel helpfile for details on how to use these functions."

Problem 2.4

"In problem 2.3, you were given mean battery life and standard deviations of battery life for each battery type. Use this information to calculate:
(a) The probability that a sample of 16 NiMH will have an average battery life of more than 6,000 hours.
(b) The probability that a sample of 64 NiMH will have an average battery life of more than 6,000 hours.
(c) The probability that a sample of 5 lithium ion batteries will have an average life of less than 5,000 hours.
(d) The probability that a sample of 20 lithium ion batteries will have an average life of less than 5,000 hours."