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 20.8 an the standard deviation was 5.3. Identify the test scores that are significantly low or significantly high.

What test scores are significantly​ low? Select the correct answer below and fill in the answer​ box(es) to complete your choice.

Answer 1

Test scores of 10.2 or lower are significantly low.

Test scores of 31.4 or higher are significantly high.

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 = 20.8, \sigma = 5.3`

Identify the test scores that are significantly low or significantly high.

Significantly low

Z = -2 and lower.

So the significantly low scores are thoses values that are lower or equal than X when Z = -2. So

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

`-2 = (X - 20.8)/(5.3)`

`X - 20.8 = -2*5.3`

`X = 10.2`

Test scores of 10.2 or lower are significantly low.

Significantly high

Z = 2 and higher.

So the significantly high scores are thoses values that are higherr or equal than X when Z = 2. So

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

`2 = (X - 20.8)/(5.3)`

`X - 20.8 = 2*5.3`

`X = 31.4`

Test scores of 31.4 or higher are significantly high.