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

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 982 and a standard deviation of 199. Scores on the ACT test are normally distributed with a mean of 19.2 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 60-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 1320, find the equivalent ACT score.

ACT score = 

Round answer to 1 decimal place.

Answer 1

Hi Ashley,

a) Go to a z-table and find the z-score that corresponds to 0.60 (60th percentile). Doing so, we find that a z-score of 0.25 = 0.5987 and a z-score of 0.26 = 0.6026. We can average these two z-scores to find the z-score of 0.60.

(0.25 + 0.26) / 2 = 0.255 (Z-score for 0.60, the 60th percentile)

So, we are looking for an SAT score that is 0.60 standard deviations above the mean. Using the standard deviation given to us for the SAT exam, we will solve for this problem like this.

SAT Score at 60th percentile = mean + (SD)*(z-score at 60th percentile)

SAT Score at 60th percentile = 982 + (199 * 0.255) = 1032.745 = 1033

b) For this problem, we're doing the very similar thing as we did in problem A but we'll be using the average and the standard deviation of the ACT exam. Since we're working with the same percentile (the 60th percentile), we'll be using the same z-score of 0.255 again.

ACT Score at 60th percentile = mean + (SD)*(z-score at 60th percentile)

ACT Score at 60th percentile = 19.2 + (5.1 * 0.255) = 20.5005 = 20.5

c) Given a SAT score of 1320, we are asked to find the equivalent ACT score. We first need to know how many points above the mean a score of 1320 is.

1320 - mean = 1320 - 982 = 338

Our score of 1320 is 338 points above the mean. Now we must compute how many standard deviations above the mean 338 points is.

Points / SD = 338 / 199 = 1.698

Our score of 1320 is 1.698 standard deviations above the mean. Taking the numbers from the ACT test, we will compute how many points above the mean 1.698 standard deviations is.

x / 5.1 = 1.698

x = 8.662

This tells us that 1.698 standard deviations above the mean for the ACT exam means that we are 8.662 points above the mean. To find our equivalent score, we will add 8.662 to the average.

8.662 + 19.2 = 27.862 = 27.9

Hope this helps! Have a good one!