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 score was 22.5 and the standard deviation was 5.1. Identify the scores that are significantly low or significantly high.

Answer 1

Significantly low scores are ≤ 12.3 and significantly high scores are ≥ 32.7.

Step-by-step explanation:

Based on the given information, a value can be considered significantly low if its z-score is less than or equal to -2. On the other hand, a value can be considered significantly high if its z-score is greater than or equal to 2.

To find the z-scores for the given test scores, we can use the formula z = (x - μ) / σ, where z is the z-score, x is the value, μ is the mean, and σ is the standard deviation.

Let's calculate the z-scores using the mean score of 22.5 and the standard deviation of 5.1:

1. Significantly low scores:
z-score ≤ -2
z ≤ -2
(x - 22.5) / 5.1 ≤ -2
(x - 22.5) ≤ -10.2
x ≤ 12.3

Therefore, any score less than or equal to 12.3 can be considered significantly low.

2. Significantly high scores:
z-score ≥ 2
z ≥ 2
(x - 22.5) / 5.1 ≥ 2
(x - 22.5) ≥ 10.2
x ≥ 32.7

Therefore, any score greater than or equal to 32.7 can be considered significantly high.

Answer 2

the z-score is given by:
z-score=(x-μ)/σ
where:
x=score
μ=mean=22.5
σ=std deviation=5.1
given that the z-score is -2, then the lowest cost will be:
-2=(x-22.5)/5.1
solving for x we get:
-10.2=x-22.5
x=-10.2+22.5
x=12.3
therefor the lowest score is 12.3

Given that the z-score is 2 when the score is highest, then highest score will be:
2=(x-22.5)/5.1
10.2=x-22.5
x=10.2+22.5
x=32.7
therefore the highest score is 32.7