Question

A study was conducted on students from a particular high school over the last 8 years. The following information was found regarding standardized tests used for college admitance. Scores on the SAT test are normally distributed with a mean of 1117 and a standard deviation of 194. Scores on the ACT test are normally distributed with a mean of 21.9 and a standard deviation of 3.8. It is assumed that the two tests measure the same aptitude, but use different scales.

If a student gets an SAT score that is the 77-percentile, find the actual SAT score.
SAT score =
Round answer to a whole number.
What would be the equivalent ACT score for this student?
ACT score =
Round answer to 1 decimal place.
If a student gets an SAT score of 1544, find the equivalent ACT score.
ACT score =
Round answer to 1 decimal place.

Answer 1

The actual SAT score is 1260.6.

The equivalent ACT score for this student(77th percentile) is 24.7.

The equivalent ACT score to a SAT score of 1544 is 30.26.

Explanation:

When the distribution is normal, we use 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.

SAT:

`\mu = 1117, \sigma = 194`

ACT:

`\mu = 21.9, \sigma = 3.8`

If a student gets an SAT score that is the 77-percentile, find the actual SAT score.

This is X when Z has a pvalue of 0.77. So X when Z = 0.74.

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

`0.74 = (X - 1117)/(194)`

`X - 1117 = 0.74*194`

`X = 1260.6`

The actual SAT score is 1260.6.

What would be the equivalent ACT score for this student?

X when Z = 0.74, using
`\mu = 21.9, \sigma = 3.8`

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

`0.74 = (X - 21.9)/(3.8)`

`X - 21.9 = 0.74*3.8`

`X = 24.7`

The equivalent ACT score for this student(77th percentile) is 24.7.

If a student gets an SAT score of 1544, find the equivalent ACT score.

First we find the z-score of the SAT, then we find the equivalent ACT score.

Z-score of SAT:

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

`Z = (1544 - 1117)/(194)`

`Z = 2.2`

Equivalent ACT:

X when Z = 2.2. So

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

`2.2 = (X - 21.9)/(3.8)`

`X - 21.9 = 2.2*3.8`

`X = 30.26`

The equivalent ACT score to a SAT score of 1544 is 30.26.