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 admittance. Scores on the SAT test are normally distributed with a mean of 1001 and a standard deviation of 199. Scores on the ACT test are normally distributed with a mean of 19.7 and a standard deviation of 4. It is assumed that the two tests measure the same aptitude, but use different scales.

1. If a student gets an SAT score that is the 58-percentile, find the actual SAT score. Round answer to a whole number.

2. What would be the equivalent ACT score for this student? Round answer to 1 decimal place.

3. If a student gets an SAT score of 1200, find the equivalent ACT score. Round answer to 1 decimal place.

Answer 1

Let the SAT test scores be denoted by a Random Variable X₁

So, X₁ ~ N(1001, 199²)

and the ACT test scores be denoted by a Random Variable X₂

So, X₂ ~ N(19.7, 4²)

For 58^th Percentile the Z-score is 0.202

[We know, X = µ + Zσ ]

1. Therefore, X₁ = 1001 + 0.202 × 199 =1041.198

So, the SAT score is: 1041.198 ≈1041

2. Corresponding value of X₂

X₂ = 19.7 + 0.202 ×4 =20.508

Corresponding ACT score is 20.508 ≈ 20.5

3. We know the Relation Between the two scores can be calculated as below:

We know if X ~ N(µ,σ²)

Here, X₁ ~ N(1001, 199²)

Here, X₂ ~ N(19.7, 4²)

Comparing (i) and (ii) we get,

  

So, 1300 = 49.75 X₂ + 20.295

=> X₂ = 25.7227 ≈ 25.7