Question

If the random variable x is distributed normally with a mean of zero and a standard deviation of one, which of the following probabilities is not correct? a. P(x < 2) = .9772 b. P(x ≥ 1) = .1587 c. P(x ≤ 1) = .8413 d. P(x = 0) = .50

Answer 1

d. P(x = 0) = .50

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.

a. P(x < 2) = .9772

This is the p-value of Z = 2.

Looking at the z-table, Z = 2 has a p-value of 0.9772, and thus, this probability is correct.

b. P(x ≥ 1) = .1587

This is 1 subtracted by the p-value of Z = 1.

Looking at the z-table, Z = 1 has a p-value of 0.8413.

1 - 0.8413 = 0.1587, and so, this probability is correct.

c. P(x ≤ 1) = .8413

This is the p-value of Z = 1, that is, 0.8413, so this is correct.

d. P(x = 0) = .50

The probability of an exact value on the normal distribution is 0, and thus, option d is wrong and is the answer to this question.