Question

If Z is standard Normal, which of the following statements is correct? Support your choices.
A) P(Z=2)-P(Z=-2) = 0.95

B) P(Z>2) = 0.025

C) P(Z<-1) > P(Z<-2)

D) P(2Z<2) = 2P(Z<1)

For the following statements, identify whether they are true or false, explain
(a) When the distribution of a random variable is skewed, mean and standard deviation are better
measures for center and spread than median and IQR.
(b) Two independent evets are always disjoint.
(c) Probability of an event is always greater than zero and less than 1.
(d) Two disjoint events are always independent.

Answer 1

a) From the standard normal tables, we get:

P(Z = 2 ) - P(z = -2) is not 0.95 but P(Z < 2) - P(Z < -2) is 0.95

Therefore this is incorrect.

b) From standard normal tables, we have P(Z < 2) = 0.975, therefore P(Z > 2 ) = 1- 0.975 = 0.025

Therefore this statement is correct.

c) Clearly P(Z < -1) has greater area under the curve than P(Z < -2)

Therefore, P(Z < -1) > P(Z < -2)

Therefore this statement is correct.

d) P(2Z < 2) = P(Z < 1) and not equal to 2P(Z < 1)

Therefore this is incorrect.

Question 2:

a) This is false. For non skewed distributions, we use mean and standard deviation and for skewed we use median and IQR.

b) This is false. For independent events P(A)P(B) = P(A and B) but not necessarily 0. For disjoint events P(A and B) = 0

c) This is true.

d) This is false. For disjoint events P( A and B) = 0 but this does not mean P(A)P(B) = 0 as P(A) and P(B) could be non zero and so A and B wont be independent.