An automobile battery manufacturer offers a 22/37 warranty on its batteries. The

Question

An automobile battery manufacturer offers a 22/37 warranty on its batteries. The first number in the warranty code free replacement period; the second number is the prorated-credit period. Under this warranty, if a battery falls within 22 months of purchase, the manufacturer replaces the battery at no charge to the consumer. If the battery falls 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. If the manufacturer's assumptions are correct, it would need to replace of its batteries free of charge. The company finds that it is replacing 1.54% of its batteries free of charge. It suspects that its assumption about the standard deviation of the life of its batteries is incorrect. A standard deviation of results in a 1.54% replacement rate. Using the revised standard deviation for battery life, what percentage of the manufacturer's batteries don't quality for free replacement but do qualify for the prorated credit? 2.94% 95.52% 47.06% 48.46%

Explanation / Answer

from normal distribution Z =(X-mean)/std deviation

for manufacturer's assumption to be correct

P(X<22) =P(Z<(22-30)/3.5) =P(Z<-2.2857) =0.0111

hence 1.11% batteries need to be replaced

2) fo 1.54% , from normal value table z =-2.15965

for this value of std deviation =(22-30)/-2.15965 =3.7043

3) for qualify for prorated credit

P(22<X<37) =P((22-30)/3.5<Z<(37-30)/3.5) =P(-2.1596<Z<1.8897) =0.9706-0.0154 =0.9552

hence option 95.52% is correct.

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