A study was conducted on students from a particular high school over the last 8

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 1027 and a standard deviation of 201. Scores on the ACT test are normally distributed with a mean of 19.5 and a standard deviation of 4.9. 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 35-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 1449, find the equivalent ACT score.
ACT score =  
Round answer to 1 decimal place.

Explanation / Answer

a) P(X < x) = 0.35

or, P((X - mean)/sd < (x - 1027)/201) = 0.35

or, P(Z < (x - 1027)/201) = 0.35

or, (x - 1027)/201 = -0.39

or, x = -0.39 * 201 + 1027

or, x = 948.61

b) P(X < x) = 0.35

or, P((X - mean)/sd < (x - 19.5)/4.9) = 0.35

or, P(Z < (x - 19.5)/4.9) = 0.35

or, (x - 19.5)/4.9 = -0.39

or, x = -0.39 * 4.9 + 19.5

or, x = -0.39 * 4.9 + 19.5

or, x = 17.589

For SAT score

Z-score = (X - mean)/sd

             = (1449 - 1027)/201

             = 2.1

For ACT score

z = 2.1

( X - 19.5)/4.9 = 2.1

or, X = 2.1 * 4.9 + 19.5

or, X = 29.79

So the equivalent ACT score is 29.79

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