Statistics

Problem 2
Acme Tires manufactures two types of tires: high performance tires and all weather tires. 45% of the tires they produce are high performance tires. They offer a 40,000 mile warrantee on each of the tires they sell, and 89% of all tires exceed 40,000 miles of tread life. Furthermore, 37% of all the tires sold are both all high performance tires AND exceed 40,000 of tread life. You can find this information filled out in the contingency table in tiressu10.xlsx. 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 tire selected at random is an all weather tire?
(e) What is the probability that a tire selected at random fails to last for 40,000 miles?
(f) What is the probability that a tire selected at random is either a high performance tire or fails to last for 40,000 miles?
(g) What is the probability that a tire selected at random is both an all weather tire and exceeds 40,000 miles of tread life?
(h) What is the probability that an all weather tire selected at random exceeds 40,000 miles of tread life?
(i) What is the probability that a high performance tire selected at random exceeds 40,000 miles of tread life?

Problem 2.2

Assume that Acme Tires sells their high performance tires for $220 each and their all weather tires for $140 each. Further assume that the cost of producing a high performance tire is $180 and the cost of producing an all weather tire is $125. Finally, assume that if a tire does not last 40,000 miles, Acme tires will replace it free of charge to the consumer; Acme will incur the cost of replacement, but will not receive any additional revenue.

(a) Calculate the profit earned/loss incurred on; a high performance tire that exceeds 40,000 miles, a high performance tire that does not exceed 40,000 miles, an all weather tire that exceeds 40,000 miles, and an all weather tire that fails to last 40,000 miles.
(b) What is the expected value (expected profit) of producing an all weather tire? Of producing a high performance tire?
(c) What is the variance and standard deviation of producing an all weather tire? Of producing a high performance tire?

Problem 2.3

Assume that Acme tire life is normally distributed and that both all weather tires and high performance tires have have a mean life of 50,000 miles. Furthermore, the standard deviation of tire life for all weather tires is 6,241.06 miles and the standard deviation of tire life for high performance tires is 10,824.07 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 tire 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 Acme wanted to ensure that 95% of their tires exceeded the warrantee, what would they have to set their warrantee at for all weather tires? For high performance 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 tire life and standard deviations of tire life for each tire type. Use this information to calculate:
(a) The probability that a sample of 16 all weather tires will have an average tire life of more than 52,000 miles.
(b) The probability that a sample of 64 all weather tires will have an average life of more than 52,000 miles.
(c) The probability that a sample of 5 high performance tires will have an average life of less than 40,000 miles.
(d) The probability that a sample of 20 high performance tires will have an average life of less than 40,000 miles.