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Suppose a normal distribution has a mean of 50 and a standard deviation of 3. What is [tex]P(x \leq 47)[/tex]?

A. 0.16
B. 0.84
C. 0.025
D. 0.975


Sagot :

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