Final answer:
This question involves using the properties of the normal distribution to answer specific questions about a sample of fluorescent light bulbs.
Step-by-step explanation:
In this question, we are given a random sample of 16 fluorescent light bulbs with a mean life of 645 hours and a standard deviation of 31 hours. Since the population is assumed to have a normal distribution, we can use the properties of the normal distribution to answer specific questions.
(a) To find the probability that a light bulb lasts less than one year, we need to convert one year (365 days) to hours. Assuming 24 hours per day, one year is equal to 365 * 24 = 8760 hours. Using the given mean and standard deviation, we can calculate the z-score for 8760 hours and then use a standard normal distribution table or calculator to find the corresponding probability.
(b) To find the probability that a light bulb lasts between six and ten years, we need to convert six and ten years to hours, and then use the same calculation as in part (a) to find the corresponding probability.
(c) To find the minimum duration that 70% of all light bulbs last, we can use the concept of percentiles. We need to find the z-score corresponding to the 70th percentile of the normal distribution. Once we have the z-score, we can convert it to the corresponding duration in hours using the given mean and standard deviation.
(d) To find the cutoff lifetime for the warranty to take place, we need to find the z-score corresponding to the lowest two percent of all bulbs. Once we have the z-score, we can convert it to the corresponding duration in months using the given mean and standard deviation.
(e) To find the probability that a light bulb fails within the 8th year, we need to find the z-score corresponding to the end of the 8th year, and then use the same calculation as in part (a) to find the corresponding probability.