Question

If Z is a standard normal variable and A is a positive number, then P(Z < −A) = P(Z > A).

A) True
B) False

Answer 1

The statement is True. The probability P(Z < −A) is equal to P(Z > A) because the standard normal distribution is symmetric about the mean, with equal areas in the tail ends for equal distances from the mean.

Step-by-step explanation:

The statement is True. In a standard normal distribution, which is symmetrical about the mean, the probability of a variable Z being less than a negative number (−A) is equal to the probability of Z being greater than the positive number (A).

This reflects the symmetric property of the normal distribution, where values equidistant from the mean on both sides have the same probability.

For example, if we have a standard normal variable Z and a positive number A, we can say that P(Z < −A) = P(Z > A). This is because the total area under the curve of a normal distribution is equal to 1, and it is symmetrically split across the mean (which is 0 in a standard normal distribution).

Thus, the areas in the tail ends of the distribution are equal for the same distance from the mean on either side.To understand why this statement is true, we need to consider the properties of the standard normal distribution.

For any standard normal variable Z, the area to the left of a negative value -A is equal to the area to the right of the positive value A. This means that P(Z < -A) = P(Z > A).