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 21.6 and the standard deviation was 5.4. Identify the test scores that are significantly low or significantly high. What test scores are significantly​ low?

Answer 1

The scores that are less than or equal to 10.8 are considered significantly low.

The scores that are greater than or equal to 32.4 are considered significantly high.

Explanation:

We are given the following information in the question:

Mean, μ = 21.6

Standard Deviation, σ = 5.4

We are given that the distribution of score is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

Significantly low score:

`z \leq -2\\z = \displaystyle(x-21.6)/(5.4) \leq -2\\\\\displaystyle(x-21.6)/(5.4) \leq -2\\\\x\leq -2(5.4) + 21.6\\\Rightarrow x \leq 10.8`

Thus, scores that are less than or equal to 10.8 are considered significantly low.

Significantly high score:

`z \geq 2\\z = \displaystyle(x-21.6)/(5.4) \geq 2\\\\\displaystyle(x-21.6)/(5.4) \geq 2\\\\x\geq 2(5.4) + 21.6\\\Rightarrow x \geq 32.4`

Thus, scores that are greater than or equal to 32.4 are considered significantly high.