To determine how long the company should guarantee its batteries to keep refunds under 11%, we need additional information, such as a model of battery failure over time.

A common way to approach this question involves assuming a continuous probability distribution for battery life (like an exponential or normal distribution) and using it to predict when 11% of the batteries would fail. For example:

1. Exponential Model: This is often used in reliability studies, assuming the probability of failure at any point in time is constant. The probability that a battery will last less than time $t$ is given by $P(T < t) = 1 - e^{-\lambda t}$, where $\lambda$ is the failure rate. To keep refunds under 11%, we would set $P(T < t) = 0.11$ and solve for $t$ given $\lambda$.

2. Normal Distribution Model: If the failure times are normally distributed with a mean $\mu$ and standard deviation $\sigma$, we could use the cumulative distribution function to determine the time $t$ at which 11% of the batteries have failed, i.e., finding $t$ such that $P(T < t) = 0.11$.

Could you provide any specific details on the failure rate, mean battery life, or type of distribution we should assume for battery failure times? With that, I can compute the precise guarantee period.

Do you want further assistance on specific models, or do you have any more details to add?

Here are 5 follow-up questions for consideration:

1. What is the mean expected lifespan of the batteries?
2. Do we have data on how quickly battery performance typically declines over time?
3. Has the company used any model in the past to estimate failure rates for products?
4. Are there historical records of battery refunds that show failure patterns over time?
5. Would the company consider a partial refund policy to extend the guarantee period?

Tip: For businesses, using historical data to refine the failure distribution model often yields more accurate predictions for warranty terms.