Assume that the random variable $X$ is normally distributed, with mean $\mu=48$ and standard deviation $\sigma=9$. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.
\[
\mathrm{P}(\mathrm{X} \leq 46)
\]
Which of the following shaded regions corresponds to $P(X \leq 46)$ ?
A.
B.
c.

Step 1 ：Assume that the random variable \(X\) is normally distributed, with mean \(\mu=48\) and standard deviation \(\sigma=9\).

Step 2 ：The question is asking for the probability that a normally distributed random variable \(X\) with mean 48 and standard deviation 9 is less than or equal to 46. This is equivalent to finding the area under the normal curve to the left of 46.

Step 3 ：To solve this, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF at a point \(x\) gives the probability that a random variable is less than or equal to \(x\).

Step 4 ：Compute the CDF at \(x = 46\) for a normal distribution with mean \(\mu = 48\) and standard deviation \(\sigma = 9\).

Step 5 ：The probability that the random variable \(X\) is less than or equal to 46 is approximately 0.412. This means that there is a 41.2% chance that a randomly selected value from this distribution will be less than or equal to 46.

Step 6 ：Final Answer: The probability that the random variable \(X\) is less than or equal to 46 is approximately \(\boxed{0.412}\).