If $y$ 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) $P(X < 13) = 0.9332$

(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:

Given a normal distribution variable $X$ with mean $\mu = 10$ and standard deviation $\sigma = 2$.

To convert any value to standard normal:

$z = \frac{x - \mu}{\sigma}$

$z = \frac{x - 10}{2}$

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Checking statement (A): $P(X < 13) = 0.9332$

Converting $X = 13$ to z-score:

$z = \frac{13 - 10}{2}$

$z = \frac{3}{2}$

$z = 1.5$

Therefore:

$P(X < 13) = P(Z < 1.5)$

$P(X < 13) = F(1.5)$

$P(X < 13) = 0.9332$

Statement (A) is TRUE.

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Checking statement (B): $P(X > 16) = 0.9986$

Converting $X = 16$ to z-score:

$z = \frac{16 - 10}{2}$

$z = \frac{6}{2}$

$z = 3$

Therefore:

$P(X > 16) = P(Z > 3)$

$P(X > 16) = 1 - F(3)$

$P(X > 16) = 1 - 0.9986$

$P(X > 16) = 0.0014$

Statement (B) is FALSE.

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Checking statement (C): $P(12 < X < 14.5) = 0.1465$

Converting $X = 12$ to z-score:

$z = \frac{12 - 10}{2}$

$z = 1$

Converting $X = 14.5$ to z-score:

$z = \frac{14.5 - 10}{2}$

$z = 2.25$

Therefore:

$P(12 < X < 14.5) = P(1 < Z < 2.25)$

$P(12 < X < 14.5) = F(2.25) - F(1)$

$P(12 < X < 14.5) = 0.9878 - 0.8413$

$P(12 < X < 14.5) = 0.1465$

Statement (C) is TRUE.

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Checking statement (D): $P(X > 8) = 0.8413$

Converting $X = 8$ to z-score:

$z = \frac{8 - 10}{2}$

$z = \frac{-2}{2}$

$z = -1$

Therefore:

$P(X > 8) = P(Z > -1)$

By symmetry of the standard normal distribution:

$P(Z > -1) = P(Z < 1)$

$P(Z > -1) = F(1)$

$P(Z > -1) = 0.8413$

Statement (D) is TRUE.

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Statements (A), (C), and (D) are TRUE.

Statement (B) is FALSE.