Final answer:
The correct probability that x is less than or equal to 9 when x is normally distributed with mean 12 and standard deviation 2 is approximately 0.4332, after using a Z-score calculation and referring to the standard normal table.
Step-by-step explanation:
If x is normally distributed with mean 12 and standard deviation 2, then to find the probability that x is less than or equal to 9, we would use a standard normal distribution table, calculator, or software to find the Z-score followed by the corresponding probability.
To calculate the Z-score, we use the formula: Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. For x = 9, it becomes Z = (9 - 12) / 2 = -1.5. Referring to a Z-table or using software gives us the probability associated with Z = -1.5, which is approximately 0.0668. However, this needs to be subtracted from 0.5 (as the standard normal table gives the area from the mean to our z value for negative z-scores) to get the total area to the left of Z = -1.5. Therefore, the probability P(X ≤ 9) is approximately 0.5 - 0.0668 = 0.4332. None of the options provided by the student match this result, which may indicate a misinterpretation of the Z-table or an error in the options given.