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 992 and a standard deviation of 201. Scores on the ACT test are normally distributed with a mean of 19.1 and a standard deviation of 5.1. 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 67-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 1334, find the equivalent ACT score.
ACT score =  
Round answer to 1 decimal place.

Answer 1

Given that, scores on the SAT test are normally distributed with a mean of 992 and a standard deviation of 201.

Let X ~ Normal (992, 201)

Scores on the ACT test are normally distributed with a mean of 19.1 and a standard deviation of 5.1.

Let Y ~ Normal (19.1, 5.1)

a) We want to find, the value of x such that, P(X < x) = 0.67

Therefore, required SAT score = 1080

Now, we wanf to find, the value of y for Z = 0.44

y = (0.44 * 5.1) + 19.1

=> y = 2.244 + 19.1

=> y = 21.344

=> y ≈ 21.3

Therefore, required ACT score = 21.3

b) z-score for x = 1334

Z = (1334 - 992) / 201 = 342 / 201 = 1.70

We want to find, the value of y for Z = 1.70

y = (1.70 * 5.1) + 19.1

=> y = 8.67 + 19.1

=> y = 27.77

=> y = 27.8

Therefore, required ACT score = 27.8