Final answer:
To calculate the probability that a component is at least 12 centimeters long, we first find the z-score for X=12 given the normal distribution with a mean of 14 and a variance of 9. Using this z-score, we then determine the probability from a standard normal distribution table or calculator and find that the probability P(X ≥ 12) is approximately 0.7475.
Step-by-step explanation:
The question asks to calculate the probability that a component produced by a machine is at least 12 centimeters long, given that the length of components X follows a normal distribution with a mean (μ) of 14 centimeters and a variance of 9 (standard deviation σ is the square root of the variance, so σ=3). To find the probability, we first need to standardize the value of 12 centimeters using the z-score formula:
Z = (X - μ) / σ
Here, X is 12, μ is 14, and σ is 3. Thus, the z-score is:
Z = (12 - 14) / 3 = -2/3 ≈ -0.6667
Next, we look up this z-score in the standard normal distribution table or use a calculator with statistical functions to find the probability associated with that z-score. This gives us the probability that a component is shorter than 12 centimeters. By subtracting this value from 1, we get the probability of a component being at least 12 centimeters long.
Assuming the use of a statistical tool, we find that the probability for Z ≥ -0.6667 is approximately 0.7475. Therefore, the probability that a component is at least 12 centimeters long is:
P(X ≥ 12) ≈ 0.7475 (rounded to four decimal places).