Question

Consider a value to be significantly low if its z score less than or equal to minus−2 or consider a value to be significantly high if its z score is greater than or equal to 2. A test is used to assess readiness for college. In a recent​ year, the mean test score was 20.620.6 and the standard deviation was 5.25.2. Identify the test scores that are significantly low or significantly high.

Answer 1

Test scores of 10.2 or lower are significantly low.

Test scores of 31 or higher are significantly high

Explanation:

Z-score:

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 = 20.6, \sigma = 5.2`

Significantly low:

Z-scores of -2 or lower

So scores of X when Z = -2 or lower

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

`-2 = (X - 20.6)/(5.2)`

`X - 20.6 = -2*5.2`

`X = 10.2`

Test scores of 10.2 or lower are significantly low.

Significantly high:

Z-scores of 2 or higher

So scores of X when Z = 2 or higher

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

`2 = (X - 20.6)/(5.2)`

`X - 20.6 = 2*5.2`

`X = 31`

Test scores of 31 or higher are significantly high