Final answer:
a) The probability that a randomly chosen bulb will burn longer than 1400 hours is 7.64%. b) The probability that a randomly chosen bulb will last between 1000 and 1050 hours is 10.03%. c) The probability that a randomly chosen bulb will last no longer than 1060 hours is 15.87%.
Step-by-step explanation:
a) To find the probability that a randomly chosen bulb will burn longer than 1400 hours, we need to find the area under the normal curve to the right of 1400. We can use z-scores to find this probability. The z-score is calculated as (x - mean) / standard deviation. Substituting the values, we get a z-score of (1400 - 1200) / 140 = 1.43. Using a z-table or calculator, we find that the probability is approximately 0.0764, or 7.64%.
b) To find the probability that a randomly chosen bulb will last between 1000 and 1050 hours, we need to find the area under the normal curve between these two values. Since the distribution is symmetrical, we can find the area to the left of 1050 and subtract the area to the left of 1000. Using z-scores, we calculate the z-score for 1050 as (1050 - 1200) / 140 = -1.07, and the z-score for 1000 as (1000 - 1200) / 140 = -1.43. Using a z-table or calculator, we find the areas corresponding to these z-scores and subtract to get the probability, which is approximately 0.1003, or 10.03%.
c) To find the probability that a randomly chosen bulb will last no longer than 1060 hours, we need to find the area under the normal curve to the left of 1060. Using z-scores, we calculate the z-score for 1060 as (1060 - 1200) / 140 = -1.00. Using a z-table or calculator, we find the area corresponding to this z-score, which is approximately 0.1587, or 15.87%.