# A college requires applicants to have an ACT score in the top 12% of all test scores. The ACT scores are normally distributed, with a mean of 21 and a standard deviation of 4.7. Find the lowest...

A college requires applicants to have an ACT score in the top 12% of all test scores. The ACT scores are normally distributed, with a mean of 21 and a standard deviation of 4.7. Find the lowest test score that a student could get and still meet the colleges requirement.

embizze | High School Teacher | (Level 2) Educator Emeritus

Posted on

Your questionwas edited so that there is one question per guidelines. The same general method applies to each of your questions:

We are given a mean `mu=21` ,and a standard deviation `sigma=4.7` .

We are asked to find the lowest score that is in the top 12% of all scores.

First we find the z-score such that 12% are above it -- this corresponds to a z-score with 88% below it. From a standard normal table we find that a z-score of 1.17 has 87.9% below it, while a z-score of 1.18 has 88.1% below it. Thus we could use either 1.17,1.18, or split the difference to get 1.1754. (My calculator gives z=1.174986791 so I will use z=1.175)

Now we want to find the x that corresponds to a z of 1.175. To compute z we use: `z=(x-mu)/sigma` . Multiplying both sides by `sigma` and adding `mu` we get `x=zsigma+mu`

The cut score will be `x=1.175(4.7)+21=26.52`

Thus the lowest composite score will be 27.