To find the probability [tex]\(P(x \leq 47)\)[/tex] for a normal distribution with a mean ([tex]\(\mu\)[/tex]) of 50 and a standard deviation ([tex]\(\sigma\)[/tex]) of 3, we can follow these steps:
1. Standardize the value (calculate the z-score):
Convert the value 47 to its corresponding z-score in the standard normal distribution.
The z-score formula is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
Substitute the values into the formula:
[tex]\[
z = \frac{47 - 50}{3} = \frac{-3}{3} = -1
\][/tex]
2. Find the cumulative probability:
The next step is to find the cumulative probability associated with the z-score of -1 in the standard normal distribution.
Using standard normal distribution tables or a cumulative distribution function (CDF), the cumulative probability for [tex]\(z = -1\)[/tex] is approximately 0.1587.
3. Interpret the result:
The cumulative probability [tex]\(P(z \leq -1)\)[/tex] represents the probability that the value of [tex]\(x\)[/tex] is less than or equal to 47 in the given normal distribution.
Therefore, the probability [tex]\(P(x \leq 47)\)[/tex] is approximately 0.1587, which corresponds to option A.
Thus, the correct answer is:
A. 0.16
