X is a normal variate with mean 80 and standard deviation 5, and Z is the standard normal variate. Which of the following statements are correct? (Given \(P(0 \le Z \le 1) = 0.3413\) and \(P(0 \le Z \le 2) = 0.4772\))
(A) \(P(X \le 80) = 0.5\)
(B) \(P(X \ge 85) = 0.3413\)
(C) \(P(X \ge 90) = 0.4772\)
(D) \(P(70 \le X \le 80) = 0.4772\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
Here \(\mu = 80\) and \(\sigma = 5\). Using \(Z = \frac{X - \mu}{\sigma}\), the values 70, 85 and 90 give Z = −2, 1 and 2. The normal curve is symmetric about the mean, and the area on each side of the mean is 0.5:
\(P(X \ge 85) = 0.5 - P(0 \le Z \le 1) = 0.5 - 0.3413 = 0.1587\)
\(P(X \ge 90) = 0.5 - P(0 \le Z \le 2) = 0.5 - 0.4772 = 0.0228\)
\(P(70 \le X \le 80) = P(-2 \le Z \le 0) = 0.4772\)
So P(X ≤ 80) = 0.5, P(X ≥ 85) = 0.1587, P(X ≥ 90) = 0.0228 and P(70 ≤ X ≤ 80) = 0.4772.
Checking the statements
(A) \(P(X \le 80) = 0.5\) — Correct. 80 is the mean, and half the area lies to its left.
(B) \(P(X \ge 85) = 0.3413\) — Incorrect. P(X ≥ 85) = 0.5 − 0.3413 = 0.1587; 0.3413 is P(80 ≤ X ≤ 85).
(C) \(P(X \ge 90) = 0.4772\) — Incorrect. P(X ≥ 90) = 0.5 − 0.4772 = 0.0228.
(D) \(P(70 \le X \le 80) = 0.4772\) — Correct. Z runs from −2 to 0, which by symmetry has area 0.4772.