4. It is known that the life of a particular auto transmission follows a normal

Question

4. It is known that the life of a particular auto transmission follows a normal distribution with mean 72,000 miles and standard deviation of 12,000 miles.

a. A manufacturer warranties the transmission up to 40,000 miles. What percent of the transmissions will fail before the end of the warranty period? Would it be unusual for a transmission to expire before the warranty period? Explain.

b. What percent of the transmissions will last longer than 65,000 miles?

c. What percent of the transmissions last longer than 100,000 miles? Is this unusual?

d. A transmission in the top 10% has been running for how many miles?

Explanation / Answer

X = + Z

X = 72000 + 12000Z

=> Z = (X - 72000) / 12000

a. X = 40000

=> Z = (40000 - 72000) / 12000

=> Z = -32000/12000 = -2.67

The probability is 0.0038 or 0.38%.

0.38% will fail before warranty period.

As the probability is too low, it would be unusual for the transmission to expire.

b. Z = (65000 - 72000) / 12000

= -17000 / 12000

= -1.4167

The probability is 0.0783 or 7.83%.

c. Z = (100000 - 72000) / 12000 = 28000/12000

= 2.33

The probability is 0.9901 or 99.01%. This is a bit unusual.

d. The z value for top 10% = 1.28

=> X = 72000 + 12000 * 1.28 = 87360 miles.

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