If \(X\) is normal distribution random variable with mean \(\mu = 10\) and standard deviation \(\sigma = 2\), \(Z\) is standard normal variable and \(F(Z)\) is cumulative distribution function, then which of the following are true?
[Given that \(F(1.5) = 0.9332, F(3) = 0.9986, F(2.25) = 0.9878\) and \(F(1) = 0.8413\)]
(A) \(P(X \lt 13) = 0.9332\)
(B) \(P(X \gt 16) = 0.9986\)
(C) \(P(12 \lt X \lt 14.5) = 0.1465\)
(D) \(P(X \gt 8) = 0.8413\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
Each value of X is changed to the standard normal variable:
\( \displaystyle Z = \frac{X - \mu}{\sigma} = \frac{X - 10}{2} \)
F(z) = P(Z < z) is the area to the left of z, and by symmetry F(−z) = 1 − F(z).
Checking the statements
(A) \(P(X \lt 13) = 0.9332\) — Correct. At X = 13, Z = 1.5, so \(P(X \lt 13) = F(1.5) = 0.9332\).
(B) \(P(X \gt 16) = 0.9986\) — Incorrect. At X = 16, Z = 3, so \(P(X \gt 16) = 1 - F(3) = 1 - 0.9986 = 0.0014\); 0.9986 is the area to the left of Z = 3.
(C) \(P(12 \lt X \lt 14.5) = 0.1465\) — Correct. X = 12 and X = 14.5 give Z = 1 and Z = 2.25, so the probability is \(F(2.25) - F(1) = 0.9878 - 0.8413 = 0.1465\).
(D) \(P(X \gt 8) = 0.8413\) — Correct. At X = 8, Z = −1, so \(P(Z \gt -1) = P(Z \lt 1) = F(1) = 0.8413\).