Question

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 21.5 and a standard deviation of 4.7.

i) type of distribution: _sampling distributions of mean______________
ii) find the mean: __21.5 points________ and standard deviation: ____4.7 points______
iii) find the following:

a. Find the lowest test score that a student could get and still meet the colleges requirement.
b. If 1300 students are randomly selected, how many would be expected to have a test score that would meet the colleges requirement?
c. How does the answer to part (a) change if the college decided to accept the top 15% of all test scores?

Answer 1

a) The lowest test score that a student could get and still meet the colleges requirement is 27.0225.

b) 156 would be expected to have a test score that would meet the colleges requirement

c) The lowest score that would meet the colleges requirement would be decreased to 26.388.

Explanation:

Problems of normally distributed samples are 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 = 21.5, \sigma = 4.7`

a. Find the lowest test score that a student could get and still meet the colleges requirement.

This is the value of X when Z has a pvalue of 1 - 0.12 = 0.88. So it is X when Z = 1.175.

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

`1.175 = (X - 21.5)/(4.7)`

`X - 21.5 = 1.175*4.7`

`X = 27.0225`

The lowest test score that a student could get and still meet the colleges requirement is 27.0225.

b. If 1300 students are randomly selected, how many would be expected to have a test score that would meet the colleges requirement?

Top 12%, so 12% of them.

0.12*1300 = 156

156 would be expected to have a test score that would meet the colleges requirement

c. How does the answer to part (a) change if the college decided to accept the top 15% of all test scores?

It would decrease to the value of X when Z has a pvalue of 1-0.15 = 0.85. So X when Z = 1.04.

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

`1.04 = (X - 21.5)/(4.7)`

`X - 21.5 = 1.04*4.7`

`X = 26.388`

The lowest score that would meet the colleges requirement would be decreased to 26.388.