SOLUTION: There are two major tests of readiness for college, the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores for more than 1 million stu

Question 1038548: There are two major tests of readiness for college, the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores for more than 1 million students in a recent high school graduating class was roughly normal with mean μ = 20.8 and standard deviation σ = 4.8. SAT scores are reported on a scale from 400 to 1600. The SAT scores for 1.4 million students in the same graduating class were roughly normal with mean μ = 1026 and standard deviation σ = 209.
How well must Abigail do on the SAT in order to place in the top 32% of all students?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
SAT scores are reported on a scale from 400 to 1600.
The SAT scores for 1.4 million students in the same graduating class were roughly normal with mean = 1026 and standard deviation = 209.
How well must Abigail do on the SAT in order to place in the top 32% of all students?

the mean is 1026 and the standard deviation is 209.

if she wants to be in the top 32% of all students, then she will have to have a score that is greater than or equal to the score of at least 68% of the students.

you would look in the z-score table for a z-score that has 68% or more of the area under the distribution curve to the left of it.

the z-score table i looked at shows:

a z-score of .46 has .6772 of the area under the normal distribution curve to the left of it.

a z-score of .47 has .6808 of the area under the normal distribution curve to the left of it.

the cutoff z-score from this table would be a z-score of .47 since more than 68% of the z-scores would have areas under the normal distribution curve less than it.

this means that less than 32% of the z-scores would have areas under the normal distribution curve greater than it.

you would then need to translate this z-score to a raw score.

the formula for z-score is:

z = (x-m)/s

z is the z-score
x is the raw score you are comparing against the mean.
m is the mean.
s is the standard deviation.

you know the following:
z = .47
m = 1026
s = 209

the formula becomes:

.47 = (x - 1026) / 209

solve for x to get x = .47 * 209 + 1026 = 1124.23

she would have to get a SAT score greater than or equal to 1124.23 in order to be placed in the top 32% of the students taking the test with a little room to spare.

if you used a calculator instead of the table, it would tell you that the z-score needed to be .4676988012.

this would have generated a raw score of 1123.749049.

that's slightly more lenient than 1124.23.

you would probably round your requirement to a score of greater than or equal to 1124.

the table i used can be found here:

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

an online z-score calculator that can be used can be found here:

http://davidmlane.com/hyperstat/z_table.html

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