Question

Assume that the random variable X is normally distributed, with mean u = 80 and standard
deviation o = 15. Compute the probability P(X > 92).
A) 0.7881
B) 0.1977
С) 0.2420
D) 0.2119
Show work

Answer 1

To compute the probability \( P(X > 92) \) for a normally distributed random variable \( X \) with mean \( \mu = 80 \) and standard deviation \( \sigma = 15 \), we can follow these steps:

1. **Standardize the random variable**: Convert \( X \) to a standard normal variable \( Z \) using the formula
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is the value we are interested in (in this case, 92).

2. **Calculate \( Z \)**:
\[
Z = \frac{92 - 80}{15} = \frac{12}{15} = 0.8
\]

3. **Find the probability**: We want to find \( P(X > 92) \), which is equivalent to finding \( P(Z > 0.8) \).

4. **Use the standard normal distribution table**: First, we find the cumulative probability \( P(Z < 0.8) \).

From standard normal distribution tables, we have:
\[
P(Z < 0.8) \approx 0.7881
\]

5. **Calculate \( P(Z > 0.8) \)**:
\[
P(Z > 0.8) = 1 - P(Z < 0.8)
\]
\[
P(Z > 0.8) = 1 - 0.7881 \approx 0.2119
\]

Thus, the probability \( P(X > 92) \) is approximately \( 0.2119 \).

The answer is:
**D) 0.2119**