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 998 and a standard deviation of 203. Scores on the ACT test are normally distributed with a mean of 20.7 and a standard deviation of 4.3. 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 51-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 1424, find the equivalent ACT score. ACT score = Round answer to 1 decimal place.

Answer #1

For normal distribution, P(X < A) = P(Z < (A - mean)/standard deviation)

Let the 51-percentile SAT score be S

P(X < S) =0.51

P(Z < (S - 998)/203) = 0.51

From standard normal distribution table, take Z score corresponding to probability of 0.51

(S - 998)/203 = 0.025

S = 1003.075

Actual SAT score = 1003

Let the exquivalent ACT score be A

(A - 20.7)/4.3 = 0.025

A = 20.8

For SAT score of 1424, corresponding Z score is

(1424 - 998)/203 = 2.10

Corresponding ACT score = 20.7 + 2.10x4.3

= 29.7

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