Final answer:
The student's question involves applying probability concepts to the lifespan of light bulbs, assuming an exponential distribution and using mean lifetimes and standard deviation to estimate probabilities and make business decisions.
Step-by-step explanation:
The question provided requires the application of probability theory to determine the likelihood of events related to the lifespan of light bulbs. In this context, the longevity of a light bulb, which follows an exponential distribution with a mean lifetime of eight years, serves as the random variable of interest. The student is asked about the probability that light bulbs will last beyond a certain number of years, as well as the duration for a warranty based on the lower percentile of that distribution.
For example, to find the probability that a light bulb lasts more than 700 hours, one would typically first verify that the distribution follows a normal or exponential pattern. However, with the given mean and standard deviation, alongside the use of the empirical rule or z-scores, we could estimate the probability for a normal distribution, which is a common approach in textbooks, assuming the lifetime of light bulbs is normally distributed. It is important to interpret the results in the context of the problem since the calculations dictate the business decisions, such as warranty periods.