FC 3 Aa Aa E 3. The normal distribution An automobile battery manufacturer offer

Question

FC 3

Aa Aa E 3. The normal distribution An automobile battery manufacturer offers a 22/37 warranty on its batteries. The first number in the warranty code is the free-replacement period; the second number is the prorated credit period. Under this warranty, if a battery fails within 22 months of purchase, the manufacturer replaces the battery at no charge to the consumer. If the battery fails after 22 months but within 37 months, the manufacturer provides a prorated credit toward the purchase of a new battery. The manufacturer assumes that x, the lifetime of its auto batteries, is normally distributed with a mean of 30 months and a standard deviation of 3.5 months. Use the following Distributions tool to help you answer the questions that follow. (Hint: When you adjust the parameters of a distribution, you must reposition the vertical line (or lines) for the correct areas to be displayed.) Select a Distribution Distributions 2 3 of its batteries free of charge. If the manufacturer's assumptions are correct, it would need to replace

Explanation / Answer

Probabiltiy that battery will fail within 22 months
P(X<22) = P(z<(22-30)/3.5) = P(z<-2.29) = 0.0111

This means manufacturer need to replace 1.11% of its batteries free of charge.

As it is found that 1.54% of batteries are being replaced, lets find the standard deviation.

Z-value of 0.0154 is -2.1596

using central limit theorem
-2.1596 = (22 - 30)/sigma
sigma = 3.7043

For the last question, we need to find the probability that life of battery will be between 22 and 37
P(22 < X < 37) = P((22 - 30)/3.7043 < z < (37 - 30)/3.7043) = P(-2.1596 < z < 1.8896) = 0.9552

Hence 95.52% of the batteries don't qualify for free replacement but do qualify for the prorated credit.

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