Final answer:
a. The probability that one selected subcomponent is longer than 114 cm is 0.6424. b. The probability that the mean length of 3 randomly selected subcomponents exceeds 114 cm is 0.2676. c. The probability that if 3 subcomponents are randomly selected, all 3 have lengths that exceed 114 cm is approximately 0.262.
Step-by-step explanation:
a. To find the probability that one selected subcomponent is longer than 114 cm, we need to calculate the z-score and then find the area under the standard normal curve to the right of the z-score. The z-score can be calculated using the equation z = (X - μ) / σ, where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. In this case, X = 114 cm, μ = 112 cm, and σ = 5.6 cm. Plugging these values into the equation, we get z = (114 - 112) / 5.6 = 0.357. Using a standard normal distribution table or a calculator, we can find that the probability to the right of z = 0.357 is approximately 0.6424. Therefore, the probability that one selected subcomponent is longer than 114 cm is 0.6424.
b. To find the probability that the mean length of 3 randomly selected subcomponents exceeds 114 cm, we first need to find the sampling distribution of the mean. The mean of the sampling distribution will still be 112 cm, but the standard deviation will be σ / sqrt(n), where n is the sample size. In this case, n = 3. Plugging in the values, we get σ / sqrt(3) = 5.6 / sqrt(3) ≈ 3.236. Next, we need to find the z-score for a mean of 114 cm using the equation z = (X - μ) / (σ / sqrt(n)). Plugging in the values, we get z = (114 - 112) / (3.236) ≈ 0.617. Using a standard normal distribution table or a calculator, we can find that the probability to the right of z = 0.617 is approximately 0.2676. Therefore, the probability that the mean length of 3 randomly selected subcomponents exceeds 114 cm is 0.2676.
c. To find the probability that if 3 subcomponents are randomly selected, all 3 have lengths that exceed 114 cm, we need to multiply the probabilities of each subcomponent having a length longer than 114 cm. Since the subcomponents are selected randomly and independently, the probability for each subcomponent will be the same as the probability calculated in part a, which is 0.6424. Therefore, the probability that all 3 subcomponents have lengths that exceed 114 cm is 0.6424 raised to the power of 3, which is approximately 0.262.