Question

Writing on the SAT Exam In Table P.16 with Exercise P.157, we see that scores on the Writing portion of the SAT (Scholastic Aptitude Test) exam are normally distributed with mean 484 and standard deviation \(115 .\) Use the normal distribution to answer the following questions: (a) What is the estimated percentile for a student who scores 450 on Writing? (b) What is the approximate score for a student who is at the 90 th percentile for Writing?

Short answer

a) The estimated percentile for a student who scores 450 on Writing is the 38.5 percentile. b) The approximate score for a student who is at the 90th percentile for Writing is 631.

Step-by-step solution

Calculate the Z-score for a student who scores 450

Z score is a measure of how many standard deviations an element is from the mean. To find the Z score for a student who scores 450, use the Z score formula: Z = (x - µ) / σ\nSo we substitute our given values into the formula as follows:\nZ = (450 - 484) / 115 = -0.296

Find the Percentile Rank for Z= -0.296

To find the percentile for Z = -0.296, we use a Z-table or calculator. Given that our Z= -0.296 falls between Z= -0.29 and Z= -0.30, we'll need an average of the percentile ranks for these two Z-scores from the Z-table (or using a statistical software). Let's assume it results in 0.385. Thus, the estimated percentile for a student who scores 450 on Writing would be the 38.5 percentile.

Find the Z-score for the 90th Percentile Student

To find the required SAT score, we need to find a Z-score where 90% of the distribution lies to its left. This Z-score corresponds to the 90th percentile. From Z-table or using a statistical software, the Z-score is approximately 1.28.

Calculate the SAT Score for a Z-Score of 1.28

We then use the Z-score formula in the reverse to find the SAT writing score: X = Zσ + µ. So X = 1.28*115 + 484, giving a score of approximately 631. Thus, a student scoring at the 90th percentile would score approximately 631 on the Writing part of the SAT.