Question

A set of data items is normally distributed with a mean of 400 and a standard deviation of 60. Find the data item in this distribution that corresponds to the following z-score:

z=3

Answer 1

The data item is
`X = 580`

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 400 and a standard deviation of 60.

This means that
`\mu = 400, \sigma = 60`

z=3

We have to find X when Z = 3. So

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

`3 = (X - 400)/(60)`

`X - 400 = 3*60`

`X = 580`

The data item is
`X = 580`