A battery company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of its battery is normally distributed, with a mean of 46 months and a standard deviation of 9 months.
A button hyperlink to the SALT program that reads: Use SALT.
(a)
If the company 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 the company does not want to make refunds for more than 11% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

To determine the percentage of batteries the company expects to replace within the 36-month period, we need to calculate the area under the normal distribution curve for the range of values below 36 months.

Using the mean (μ = 46 months) and the standard deviation (σ = 9 months), we can standardize the value 36 using the formula z = (x - μ) / σ, where x is the value we want to standardize.

For x = 36, z = (36 - 46) / 9 = -10 / 9 = -1.11.

Using a standard normal distribution table or a statistical calculator, we can find the percentage associated with this z-value. From the table or calculator, we find that the area to the left of -1.11 is approximately 0.1335.

Since the company guarantees a full refund for batteries that fail within the 36-month period, the percentage of batteries the company expects to replace is 0.1335, or 13.35% (rounded to two decimal places).

(b) To find the duration of the guarantee that ensures refunds for no more than 11% of the batteries, we need to find the corresponding z-value for this desired percentage.

Using a standard normal distribution table or a statistical calculator, we find that the z-value associated with 11% is approximately -1.23.

Now, we can use the formula z = (x - μ) / σ to solve for x, where x is the duration of the guarantee we want to find.

Substituting the known values, -1.23 = (x - 46) / 9.

Solving for x, we get x = (-1.23 * 9) + 46 = 34.53.