# batteries by liwenting

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```									                                                                            Problem Set 2
You work for BreatheCorp, a manufacturer of medical supplies, including portable oxygen concentrators. Your c
lithium ion and NiMH. 56% of the batteries you produce are lithium ion. You offer a 5,000 hour warrantee on ea
exceed 5,000 hours of useful life. Furthermore, 55% of all the batteries sold are both lithium ion AND exceed 5,0
out in the contingency table in the Data tab. Use this information to answer the following questions. Questions
class, 2.3 comes from week 5 material, and 2.4 is based on material from week 6. Note that part a (and ONLY par
you encounter any difficulties with these problems, the optional problems below are very similar to the questio
optional questions can be found in the back of the textbook. You can also request that the tutor work extensive

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 \$429 each and their NiMH batteries for \$339 each. F
ion battery is \$368 and the cost of producing a NiMH battery is \$277. Finally, assume that if a battery does not las
charge to the consumer; BreatheCorp will incur the cost of replacement, but will not receive any additional reve

(a) Calculate the profit earned/loss incurred on; a lithium ion that exceeds 5,000 hours, a lithium ion that does n
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? O

Problem 2.3

Assume that BreatheCorp battery life is normally distributed and that battery types have have a mean life of 5,7
battery life for NiMH batteries is 561.72 hours and the standard deviation of battery life for lithium ion is 357.11
(a) EXTRA CREDIT-Using the probabilities you calculated in problem 2.1, show that the numbers given here for th
Note: depending on rounding, the numbers you calculate may be slightly higher or lower than the ones given in
(b) If BreatheCorp wanted to ensure that 95% of their batteries exceeded the warrantee, what would they have
lithium ion batteries?

Excel Tips:
Excel has a trio of built in functions that are useful for calculating Z scores and dealing with the standard normal
and =NORMSDIST. See the Excel helpfile for details on how to use these functions.
lithium ion batteries?

Excel Tips:
Excel has a trio of built in functions that are useful for calculating Z scores and dealing with the standard normal
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. U
(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.

Optional Problems

Levine, questions 4.6, 4.12, 4.26, 5.2, 6.6, 6.10, 7.26
m Set 2
oxygen concentrators. Your concentrators can come with two types of batteries:
a 5,000 hour warrantee on each of the batteries they sell, and 95% of all batteries
h lithium ion AND exceed 5,000 hours of life. You can find this information filled
owing questions. Questions 2.1 and 2.2 relate to information from week 4 in the
ote that part a (and ONLY part a) of question 2.3 is an extra credit question. Should
are very similar to the questions in this problem set, and the answers to the
hat the tutor work extensively with you on the optional problems.

ours?
s to exceed 5,000 hours?
xceeds 5,000 hours?
hours of life?
000 hours of life?

MH batteries for \$339 each. Further assume that the cost of producing a lithium
e that if a battery does not last 5,000 hours, BreatheCorp will replace it free of

urs, a lithium ion that does not exceed 5,000 hours, a NiMH that exceeds 5,000

? A NiMH battery?
roducing an NiMH battery? Of producing a lithium ion battery?

s have have a mean life of 5,750 hours. Furthermore, the standard deviation of
y life for lithium ion is 357.11 miles.
he numbers given here for the standard deviation of each battery type are correct.
lower than the ones given in the problem.
antee, what would they have to set their warrantee at for NiMH batteries? For

ng with the standard normal distribution. These are =STANDARDIZE, =NORMSINV,
ng with the standard normal distribution. These are =STANDARDIZE, =NORMSINV,

y life for each battery type. Use this information to calculate:
re than 6,000 hours.
ore than 6,000 hours.
of less than 5,000 hours.
e of less than 5,000 hours.
Contingency Table
NiMH       Lithium Ion            Total
Exceeds 5,000 hours                              0.55           0.95
Does not Exceed
Total                                            0.56

Price and Cost Information
NiMH        Lithium Ion
Sale Price            \$       339 \$              429
Cost of Production    \$       277 \$              368

Mean Battery Life
NiMH                         5750
Lithium Ion                  5750

Standard Deviation of Battery Life
NiMH                   561.722967
Lithium Ion            357.114714

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