SOLUTION: Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with

Algebra ->  Probability-and-statistics -> SOLUTION: Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with      Log On


   

Question 1169726: Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 45.4 months and a standard deviation of 8.1 months.
(a) If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of its batteries will the company expect to replace? (Round your answer to two decimal places.)
(b) If Quick Start does not want to make refunds for more than 6% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
mean = 45.4 months.
standard deviation = 8.1 months.
critical raw score is 36 months.

z-score is equal to (x-m)/s
x is the raw score
m is the mean
s is the standard deviation.

in this problem, z = (36-45.4)/8.1 = -1.160493827.

area to the left of that z-score is equal to .1229239579.

this means that approximately 12.29% of the batteries will live less than 36 months.

those are the ones that would need to be replaced if the warranty is for 36 months.

if the company wants less than 6% of its batteries be replaced for free, then the critical z-score would need to be -1.554773593.

use the z-score formula to solve for the raw score.

z = (x-m)/s becomes -1.554773593 = (x-45.4)/8.1

solve for x to get x = 8.1 * -1.554773593 + 45.4 = 32.80633389.

round to the next lowest month to remain below 6% rebates.

if you round to the nearest month without regard to whether it's less than 6%, then you would round to 33 months.

either one will get you close to the desired percent.

33 months would require rebates to 6.29% of the customers.

32 months would require rebates to 4.90% of the customers.

here are the results from using an online calculator.











the first one used area from a value to find the probability of getting a tire life less than a raw score of 36 when the mean was 45.4 and the standard deviation was 8.1.

the second one used value from an area to find the z-score required to have only 6% tire life less than the z-score indicated.

the third one used value from an area to find the raw score required to have only 6% tire life less than the raw score indicated.

when looking for or from raw score, the mean and standard deviation of the problem are used.

when looking for or from z-score, the mean is set to 0 and the standard deviation is set to 1.

the fourth one used area from a value to find the percent under the lower limit of 32 months.

the fifth one used area from a vlaue to find the percent under the lower limit of 33 months.

any questions, feel free to contact me and i'll explain whatever it is you need to know more about this based on my capability to do so.

theo