Which of the following are true?(A) If the mean of a normal variate i…

The correct answer is Option (4) → (A), (C) and (D) only

For (A), the Z-score is

    \( \displaystyle Z=\frac{X-\mu}{\sigma}=\frac{20-12}{4}=2 \)

For (B), the die shows 1 with probability \( \frac{3}{6} \), 2 with probability \( \frac{2}{6} \) and 5 with probability \( \frac{1}{6} \):

    \( \displaystyle \text{Mean}=1\cdot\frac{3}{6}+2\cdot\frac{2}{6}+5\cdot\frac{1}{6} \)

    \( \displaystyle =\frac{3+4+5}{6} \)

    \( \displaystyle =2 \)

For (C), in a Poisson distribution the mean equals the variance, so λ = 2:

    \( \displaystyle P(X=0)=\frac{e^{-\lambda}\lambda^0}{0!}=e^{-2} \)

For (D), in a binomial distribution,

    \( \displaystyle np=9 \)

    \( \displaystyle npq=\left(\frac{3}{2}\right)^2=\frac{9}{4} \)

    \( \displaystyle q=\frac{9/4}{9}=\frac{1}{4} \)

    \( \displaystyle p=\frac{3}{4} \)

    \( \displaystyle n=\frac{9}{3/4}=12 \)

So the Z-score is 2, the mean of the die is 2, \( P(X=0)=\frac{1}{e^2} \) and the number of trials is 12.

Checking the statements

(A) If the mean of a normal variate is 12 and standard deviation is 4, then the Z-score of data point 20 is 2. — Correct. Z = (20 − 12) ÷ 4 = 2.

(B) Mean of the numbers obtained on throwing a die having ‘1’ on three faces, ‘2’ on two faces and ‘5’ on one face is 3. — Incorrect. The mean is 2, not 3.

(C) If the variance of a Poisson’s distribution is 2, then \( P(X=0)=\frac{1}{e^2} \). — Correct. λ = 2 gives \( e^{-2} \).

(D) For a binomial distribution, mean = 9 and the standard deviation is \( \frac{3}{2} \), then the number of trials is 12. — Correct. q = \( \frac{1}{4} \), p = \( \frac{3}{4} \), n = 12.