SOLUTION: 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.

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Question 1206701: 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 1092 and a standard deviation of 197. Scores on the ACT test are normally distributed with a mean of 21.6 and a standard deviation of 4.4. It is assumed that the two tests measure the same aptitude, but use different scales. Round all answers to 1 decimal place.
If a student gets an SAT score that is the 49-percentile, find the actual SAT score. SAT score =
What would be the equivalent ACT score for the student?ACT score=
If a student gets an SAT score of 1367.8, find the equivalent ACT score. ACT score=
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
you would use the z-scores to make these measurements comparable between the two normal distributions.

the SAT mean is 1092 and standard deviation is 197.

the ACT mean is 21.6 and the standard deviation is 4.4.

the 49th percentile means that 49% of all scores are below that.
under the normal distribution curve, that would be the area to the left of the equivalent z-score.

you would use the z-score table or a z-score calculator to find the z-score associated with an area of .49 to the left of it.

the calculator that i used can be found at https://davidmlane.com/hyperstat/z_table.html

the calculator is set to getting a z-score with an area to the left of it equal to .49.

the results of that is shown below.

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the calculator says that the z-score is -.025.

you can use the z-score formula to find the equivalent raw scores for both the SAT scores and the ACT scores.

the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation.

for the SAT, you get z = (x - m) / s becomes -.025 = (x - 1092) / 197.
solve for x to get x = -.025 * 197 + 1092 = 1087.075.

for the ACT, you get z = (x - m) / s becomes -.025 = (x - 21.6) / 4.4.
solve for x to get x = -.025 * 4.4 + 21.6 = 21.49.

the z-score for a SAT score of 1367.8 is found by solving the z-score formula of z = (x - m) / s becoming z = (1367.8 - 1092) / 197 = 1.4.

that z-score formula is used to find the comparable ACT score.

z = (x - m) / s becomes 1.4 = (x - 21.6) / 4.4.
solve for x to get x = 1.4 * 4.4 + 21.6 = 27.76.

answers to your questions are:

If a student gets an SAT score that is the 49-percentile, find the actual SAT score.

the SAT score is 1087.1.

What would be the equivalent ACT score for the student?

the equivalent ACT score is 21.5.

If a student gets an SAT score of 1367.8, find the equivalent ACT score.

the equivalent ACT score is 27.8.

the calculator at was used to calculate the z-scores.

it can also be used to calculate the raw scores directly.

with z-score, you set the mean and standard deviation to 0 and 1 resprecively.
with raw scores, you set the mean and standard deviation to what they are.

the z-score analysis results are shown below:

the comparable raw score results are shown below.

let me know if you have any questions.

theo