Question

Suppose x is a normally distributed random variable with a mean of 22 and a standard deviation of 5. The probability that x is less than 9.7 is _____. a. .4931 b. 0 c. .0069 d. .9931

Answer 1

c. .0069

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.

Mean of 22 and a standard deviation of 5.

This means that
`\mu = 22, \sigma = 5`

The probability that x is less than 9.7 is

This is the p-value of Z when X = 9.7. So

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

`Z = (9.7 - 22)/(5)`

`Z = -2.46`

`Z = -2.46` has a p-value of 0.0069, and thus, the correct answer is given by option c.