Final answer:
a. The probability that one selected subcomponent is longer than 121 cm is approximately 0.3594. b. The probability that the mean length of 3 randomly selected subcomponents exceeds 121 cm is approximately 0.2686. c. The probability that all 3 randomly selected subcomponents have lengths that exceed 121 cm is approximately 0.046.
Step-by-step explanation:
a. To find the probability that one selected subcomponent is longer than 121 cm, we need to find the area under the normal distribution curve to the right of 121 cm. We can use the Z-score formula: Z = (X - mean) / standard deviation. Plugging in the values, we have Z = (121 - 119) / 5.6 ≈ 0.36. Using a Z-table or calculator, we can find that the probability is approximately 0.3594.
b. To find the probability that the mean length of 3 randomly selected subcomponents exceeds 121 cm, we first need to find the distribution of the sample mean. The mean of the sample mean is the same as the population mean, which is 119 cm. The standard deviation of the sample mean, also known as the standard error, is given by: standard deviation / sqrt(sample size). Plugging in the values, we have standard error = 5.6 / sqrt(3) ≈ 3.23. Now, we can find the Z-score using the formula Z = (X - mean) / standard error. Plugging in X = 121, mean = 119, and standard error = 3.23, we have Z ≈ 0.619. Using a Z-table or calculator, we can find that the probability is approximately 0.2686.
c. To find the probability that all 3 randomly selected subcomponents have lengths that exceed 121 cm, we need to find the probability that one selected subcomponent has a length longer than 121 cm and raise it to the power of the sample size (3 in this case). From part (a), we know that the probability of one subcomponent being longer than 121 cm is approximately 0.3594. Therefore, the probability that all 3 subcomponents have lengths that exceed 121 cm is approximately 0.3594^3 ≈ 0.046.