Question

The mean of a certain set of measurements is 27 with a standard deviation of 14. The distribution of the

data from which the sample was drawn is mound-shaped. The proportion of measurements that is less

than 13 is


Less than at least 3

4

.
Exactly 16%.

At least 16%.

At most 16%.

Answer 1

Exactly 16%.

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.

The mean of a certain set of measurements is 27 with a standard deviation of 14.

This means that
`\mu = 27, \sigma = 14`

The proportion of measurements that is less than 13 is

This is the p-value of Z when X = 13, so:

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

`Z = (13 - 27)/(14)`

`Z = -1`

`Z = -1` has a p-value of 0.16, and thus, the probability is: Exactly 16%.