Question

If \(X\) is a normal random variable with mean 50 and variance 25, then \(P(X \le 55)\) is

• A. 0.1587
• B. 0.8413
• C. 0.9772
• D. 0.5

Hint:

Remember that \(P(Z \le 1) = 0.8413\).
The standard normal table is essential for such problems.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a normal distribution probability problem.

Step 2: Key Formula or Approach:
Convert to standard normal: \(Z = \frac{X - \mu}{\sigma}\).

Step 3: Detailed Explanation:
Given \(\mu = 50\), variance = 25, so \(\sigma = 5\).
For \(X = 55\): \[ Z = \frac{55 - 50}{5} = 1. \]
We need \(P(X \le 55) = P(Z \le 1)\).
From standard normal table, \(P(Z \le 1) = 0.8413\).
Thus, the probability is 0.8413, which is option (B).
This is a direct application of the standard normal distribution.
The value 0.8413 is standard for \(Z = 1\).