Question

A standardized​ exam's scores are normally distributed. In a recent​ year, the mean test score was 1521 and the standard deviation was 314. The test scores of four students selected at random are 1920​, 1290​, 2220​, and 1420. Find the​ z-scores that correspond to each value and determine whether any of the values are unusual

Answer 1

A score of 1920 has a z-score of 1.27.

A score of 1290 has a z-score of -0.74.

A score of 2220 has a z-score of 2.23.

A score of 1420 has a z-score of -0.32.

The score of 2220 is more than two standard deviations from the mean, so it is unusual.

Explanation:

When the distribution is normal, we use 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.

If X is 2 or more standard deviations from the mean, it is considered unusual.

In this question, we have that:

`\mu = 1521, \sigma = 314`

Score of 1920:

X = 1920. Then

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

`Z = (1920 - 1521)/(314)`

`Z = 1.27`

A score of 1920 has a z-score of 1.27.

Score of 1290:

X = 1290. Then

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

`Z = (1290 - 1521)/(314)`

`Z = -0.74`

A score of 1290 has a z-score of -0.74.

Score of 2220:

X = 1290. Then

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

`Z = (2220 - 1521)/(314)`

`Z = 2.23`

A score of 2220 has a z-score of 2.23.

Since it is more than 2 standard deviations of the mean, the score of 2220 is unusual.

Score of 1420:

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

`Z = (1420 - 1521)/(314)`

`Z = -0.32`

A score of 1420 has a z-score of -0.32.