Quick Start Company makes 12-volt car batteries. From historical data, the compa
ID: 3131221 • Letter: Q
Question
Quick Start Company makes 12-volt car batteries. From historical data, the company knows that the life of such a battery is a normally distributed random variable with a mean life of 43 months and a standard deviation of 9 months.
(a) What percentage of Quick Start 12-volt batteries will last between 33 months and 48 months?
(b) If Quick Start does not want to make refunds for more than 10% of its batteries under a full-guarantee policy, how long should the company guarantee the 12-volt batteries?
(c) Seventy-five 12-volt batteries are randomly selected n=75. What is the probability that the mean lifetime of the batteries in this sample will be between 41 and 42 months?
Explanation / Answer
Quick Start Company makes 12-volt car batteries. From historical data, the company knows that the life of such a battery is a normally distributed random variable with a mean life of 43 months and a standard deviation of 9 months.
(a) What percentage of Quick Start 12-volt batteries will last between 33 months and 48 months?
z value for 33, z=(33-43)/9= -1.11
z value for 48, z=(48-43)/9= 0.56
P( 33<x<48) = P( -1.11<z <0.56)
=P(z <0.56) – P( z < -1.11) = 0.7123 - 0.1335
=0.5788
(b) If Quick Start does not want to make refunds for more than 10% of its batteries under a full-guarantee policy, how long should the company guarantee the 12-volt batteries?
Z value for bottom 10% = -1.282
X value = 43-1.282*9=31.462
Answer: 31.46 months
(c) Seventy-five 12-volt batteries are randomly selected n=75. What is the probability that the mean lifetime of the batteries in this sample will be between 41 and 42 months?
Standard error = sd/sqrt(n) =9/sqrt(75) =1.0392
z value for 41, z=(41-43)/ 1.0392= -1.92
z value for 42, z=(42-43)/ 1.0392= -0.96
P( 41<x<42) = P( -1.92<z <-0.96)
=P(z <-0.96) – P( z < -1.92)
= 0.1685 -0.0274
=0.1411
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