Answer to Question #113705 in Statistics and Probability for Katenda

Question

Question 11

The value of the mean of a normal distribution corresponds to which values on the standard normal curve?

(1) 0
(2) 1
(3) 100
(4) 1000
(5) None of the above.

Question 12

For any normal distribution (according to the empirical rule), approximately what percentage of
values lies within three standard deviations of the mean?

(1) 2%
(2) 68.3%
(3) 95.4%
(4) 99.7%
(5) 100%

Consider the standard normal distribution and answer questions 13 and 14.

Question 13

Which one of the following statements is incorrect?

(1) P(Z = 0) = 0
(2) P(Z ≥ 0) = 0.5
(3) P(Z ≤ 0) = 0.5
(4) P(Z ≤ -1) = 0.1587
(5) P(Z ≥ 1) = 1 - P(Z ≤ -1)

Question 14

P(-1 ≤ Z ≤ 2) =?

(1) 0.1359
(2) 0.4772
(3) 0.8185
(4) 0.8415
(5) None of the above

Answer 1

Question 11.

The value of mean (expectation) of standard normal distribution is (1) 0.

Question 12.

For any normal distribution (according to the empirical rule), approximately (4) 99.7% of values lies within three standard deviations of the mean.

Question 13.

(5) $P\{Z\geq 1\}=1-P\{Z\leq -1\}$ is incorrect.

$P\{Z\geq 1\}=1-P\{Z<1\}$

Question 14.

$P\{-1\leq Z\leq 2\}=F(2)-F(-1)\\ F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{t^2}{2}}dt\\ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{0}^x e^{-\frac{t^2}{2}}dt\text{ --- Laplace function}\\ F(2)=F(0)+\Phi(2)=0.5+0.4772=0.9772\\ F(-1)=F(0)-\Phi(1)=0.5-0.3413=0.1587\\ P\{-1\leq Z\leq 2\}=F(2)-F(-1)=0.9772-0.1587=0.8185 (3)$