Question

A set of data values is normally distributed with a mean of 90 and a standard deviation of 4. Give the standard score and approximate percentile for a data value of 89.

Answer 1

Standard score of -0.25.

A data value of 89 is approximately in the 40th percentile.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 90, \sigma = 4`

Give the standard score and approximate percentile for a data value of 89.

Z when X = 89

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

`Z = (89 - 90)/(4)`

`Z = -0.25`

`Z = -0.25` has a pvalue of 0.4013.

So a data value of 89 is approximately in the 40th percentile.