SAT Math scores have a Normal Distribution N(515, 114).

Question

 If a student wants a score in the top 20% of SAT Math scores, he must score at least ____________.

Answer 1

Question - is 114 the standard deviation or the variance of the distribution?  I'll assume the variance given its relative size to the mean 515.

 

Step 1) You will have to find the z-score such that 20% of the area under the Standard Normal Distribution Curve lies to the right of this z-score.

 

I call this a reverse look up because, depending on how your table is set up, you have to look up the probability inside the table and read out to get the z-score.  Also, this is a probability to the right and most tables only show probabilities to the left. 

 

Looking online, the z-score is +0.84.  You will need your table to get this value.

 

Step 2) Convert z=+0.84 to a raw score with a mean of 515 and standard deviation of sqrt(114)=10.677.

This is the same concept as your other question posted:

515+(+0.84)(10.677) = 515 + 8.96868 = 523.96868.  So he must score at least 524 on the SAT.

 

In other words:  The probability that a student scores above 524 on the SAT is 20%.  Therefore in order to be in the top 20%, he will need to score 524 or above. (if he scores 523 it won't be high enough).

Answer 2

Hello, thank you for taking the time to post your question!

The underlying formula that we want to use to solve this one is

Z = (x - mean) / (standard deviation) ... in this case we will be solving for the value of x, so

x = mean + z * standard deviation

for the inputs we have

mean = u = 515

standard deviation o = 114

top 20% means the 80^th percentile, meaning z = 0.84

x = u + z*o

x = 515 + 0.84(114)

x = 610.76

so they would need to score at least 611 to be in the top 20% of SAT Math scores

I hope that helps you get moving in a better direction on this type of question! Feel free to reach out if you have any additional questions beyond that :)