If X is normal distribution random variable with mean $μ= 10$ and standard deviation $σ= 2$, 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) $P(X < 13) = 0.9332$
(B) $P(X >16) = 0.9986$
(C) $P(12 <X < 14.5) = 0.1465$
(D) $P(X > 8) = 0.8413$
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
Given $X$ is normal with $\mu=10,\;\sigma=2$
Standardization formula
$Z=\frac{X-\mu}{\sigma}=\frac{X-10}{2}$
(A) $P(X<13)$
$Z=\frac{13-10}{2}=1.5$
$P(X<13)=F(1.5)=0.9332$
(A) is true
(B) $P(X>16)$
$Z=\frac{16-10}{2}=3$
$P(X>16)=1-F(3)=1-0.9986=0.0014$
(B) is false
(C) $P(12<X<14.5)$
$Z_1=\frac{12-10}{2}=1$
$Z_2=\frac{14.5-10}{2}=2.25$
$P(12<X<14.5)=F(2.25)-F(1)=0.9878-0.8413=0.1465$
(C) is true
(D) $P(X>8)$
$Z=\frac{8-10}{2}=-1$
$P(X>8)=1-F(-1)=F(1)=0.8413$
(D) is true
The correct options are (A), (C) and (D).