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 43.8 months and a standard deviation of 6.5 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?


(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)?

Answer 1

To calculate the percentage of batteries that will be expected to be replaced within the 36-month period, we need to find the area under the normal distribution curve from 0 to 36.

Step-by-step explanation:

In this question, we are given information about the average life of a Quick Start car battery, which follows a normal distribution with a mean of 43.8 months and a standard deviation of 6.5 months.

(a) To calculate the percentage of batteries that will be expected to be replaced within the 36-month period, we need to find the area under the normal distribution curve from 0 to 36. We can use the z-score formula to standardize the value of 36 and then use a standard normal distribution table to find the corresponding area. The percentage of batteries that will be expected to be replaced is the same as the percentage of batteries that fall within the range of 0 to 36 months.

1. Subtract the mean from 36: 36 - 43.8 = -7.8
2. Divide the result by the standard deviation: -7.8 / 6.5 = -1.2
3. Using the z-score -1.2, find the corresponding area under the standard normal distribution curve using a standard normal distribution table or a calculator with standard normal distribution capabilities.

Answer 2

a) The company should expect to replace 11.51% of its batteries.

b) 35 months.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 43.8, \sigma = 6.5`

(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?

This is the pvalue of Z when X = 36. Then

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

`Z = (36 - 43.8)/(6.5)`

`Z = -1.2`

`Z = -1.2` has a pvalue of 0.1151.

The company should expect to replace 11.51% 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 warranty should be the 10th percentile, which is X when Z has a pvalue of 0.1. So it is X when Z = -1.28.

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

`-1.28 = (X - 43.8)/(6.5)`

`X - 43.8 = -1.28*6.5`

`X = 35.48`

To the nearest month, 35 months.