Final answer:
a. The probability that one selected subcomponent is longer than 114 cm is approximately 0.642. b. The probability that the mean length of 3 randomly selected subcomponents exceeds 114 cm is approximately 0.269. c. The probability that all 3 randomly selected subcomponents have lengths that exceed 114 cm is approximately 0.263.
Step-by-step explanation:
a. To find the probability that a selected subcomponent is longer than 114 cm, we need to calculate the z-score. The z-score formula is: z = (x - µ) / σ, where z is the z-score, x is the value we want to find the probability for, µ is the mean, and σ is the standard deviation. Plugging in the values, we have: z = (114 - 112) / 5.6 = 0.357. Using the z-table or a calculator, we can find that the probability of a z-score of 0.357 or greater is approximately 0.642.
b. To find the probability that the mean length of 3 randomly selected subcomponents exceeds 114 cm, we need to calculate the standard error of the mean (SE). The formula for SE is: SE = σ / √n, where σ is the standard deviation and n is the sample size. Plugging in the values, we have: SE = 5.6 / √3 = 3.23. We can then calculate the z-score using the formula: z = (x - µ) / SE, where x is the value we want to find the probability for, µ is the mean, and SE is the standard error. Plugging in the values, we have: z = (114 - 112) / 3.23 = 0.619. Using the z-table or a calculator, we can find that the probability of a z-score of 0.619 or greater is approximately 0.269.
c. To find the probability that all 3 randomly selected subcomponents have lengths that exceed 114 cm, we can simply multiply the probability of one subcomponent being longer than 114 cm by itself 3 times, since the events are independent. Using the probability from part a, we have: probability = 0.642 * 0.642 * 0.642 = 0.263.