Question

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 45.0 months and a standard deviation of 8.1 months.

a. If Quick start guarantees a full refund on any battery that failswithin the 36 month period after purchase, what percentage of itsbatteries will the company expect to replace?
b. If quick Start does not want to make refunds for more than 10% ofits batteries under the full refund guarantee policy, for how longshould the company guarantee the batteries (to the nearestmonth)?

Answer 1

a. The company will be expected to replace 13.35% of its batteries.

b. The company should guarantee the batteries for 35 months.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 45.0 months and a standard deviation of 8.1 months.

This means that
`\mu = 45, \sigma = 8.1`

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?

The proportion is the p-value of Z when X = 36. So

`Z = (X - \mu)/(\sigma)`

`Z = (36 - 45)/(8.1)`

`Z = -1.11`

`Z = -1.11` has a p-value of 0.1335

0.1335*100% = 13.35%

The company will be expected to replace 13.35% of its batteries.

b. If quick Start does not want to make refunds for more than 10% of its batteries under the full refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

The guarantee should be the 10th percentile, that is, X when Z has a p-value of 0.1, so X when Z = -1.28.

`Z = (X - \mu)/(\sigma)`

`-1.28 = (X - 45)/(8.1)`

`X - 45 = -1.28*8.1`

`X = 34.6`

Rounding to the nearest month:

The company should guarantee the batteries for 35 months.