Question

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

True or False?

Answer 1

True

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

As the normal distribution is symmetric, we have that
`P(z < -A) = P(z > A)`

For example:

`z = -2` has a pvalue of 0.0228, which means that
`P(z < -2) = 0.0228`

`z = 2` has a pvalue of 0.9772, which means that
`P(z > 2) = 1 - 0.9772 = 0.0228`

True or False?

As shown above, this statement is True, due to the symmetry of the normal distribution.