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 X_{1}

So, X_{1} ~ N(1001,
199^{2})

and the ACT test scores be denoted by
a Random Variable X_{2}

So, X_{2} ~ N(19.7,
4^{2})

For 58^{th} Percentile the
Z-score is 0.202

[We know, X = µ + Zσ ]

- Therefore, X
_{1}= 1001 + 0.202 × 199 =1041.198

So, the SAT score is: 1041.198 ≈1041

- Corresponding value of X
_{2}

X_{2} = 19.7 + 0.202 ×4
=20.508

Corresponding ACT score is 20.508 ≈ 20.5

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

We know if X ~ N(µ,σ^{2})

Here, X_{1} ~ N(1001,
199^{2})

Here, X_{2} ~ N(19.7,
4^{2})

Comparing (i) and (ii) we get,

* *

So, 1300 = 49.75 X_{2} +
20.295

=> X_{2} = 25.7227 ≈
25.7

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