If \(x\) is a normal variate with mean 12 and standard deviation 4, then \(P[X \ge 20]\) is:
[Given that: \(P[0 \le Z \le 2] = 0.4772\)]
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.0228
Converting to the standard normal variate:
\( \displaystyle Z = \frac{X - \mu}{\sigma} = \frac{20 - 12}{4} = 2 \)
\(P[X \ge 20] = P[Z \ge 2]\)
The normal curve is symmetric about 0, so the area to the right of Z = 0 is 0.5:
\(P[Z \ge 2] = 0.5 - P[0 \le Z \le 2]\)
\(P[Z \ge 2] = 0.5 - 0.4772 = 0.0228\)
Why the other options are incorrect
Option (1) 0.9772 — Incorrect. This is \(0.5 + 0.4772 = P[X \le 20]\), the complement of the required probability.
Option (3) 0.00135 — Incorrect. This is the area beyond Z = 3 (X = 24), not beyond Z = 2.
Option (4) 0.7563 — Incorrect. The area beyond Z = 2 is less than 0.5; it equals 0.0228.