Question

At a local university, your application will only be considered if you score in the top 75% (better than at least 25% of the population) of all SAT scores. You score 428 on your SAT. You know that SAT scores are normally distributed, have µ = 500, and σ = 100. Based on this information, do you need to retake the test?

Answer 1

Answer: The 75^th percentile is the SAT score that holds 75% of the scores below it and 25% above it.

To compute the 75^th percentile, we use the formula X=μ + Zσ, and we will use the standard normal distribution table. We begin by going into the interior of the standard normal distribution table to find the area under the curve closest to 0.75, and from this we can determine the corresponding Z score. Once we have this we can use the equation X=μ + Zσ, because we already know that the mean and standard deviation are 500 and 100, respectively.

The exact Z value holding 75% of the values below it is 0.6744898 which was determined from a table of standard normal probabilities with more precision.

Using Z= 0.6744898 the 75^th percentile of SAT scores is: X = 500 + 0.6744898(100) = 567.449, close to 567.

Thus if one scores 428, he has to retake the test. As it says the application will only be considered if you score in the top 75%.