Question

Writing on the SAT Exam It has been found 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.

Required:
a. What is the estimated percentile for a student who scores 425 on Writing?
b. What is the approximate score for a student who is at the 87th percentile for Writing?

Answer 1

a) The estimated percentile for a student who scores 425 on Writing is the 30.5th percentile.

b) The approximate score for a student who is at the 87th percentile for Writing is 613.5.

Explanation:

Problems of normally distributed distributions are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 484, \sigma = 115`

a. What is the estimated percentile for a student who scores 425 on Writing?

This is the pvalue of Z when X = 425. So

`Z = (X - \mu)/(\sigma)`

`Z = (425 - 484)/(115)`

`Z = -0.51`

`Z = -0.51` has a pvalue of 0.3050.

The estimated percentile for a student who scores 425 on Writing is the 30.5th percentile.

b. What is the approximate score for a student who is at the 87th percentile for Writing?

We have to find X when Z has a pvalue of 0.87. So X for Z = 1.126.

`Z = (X - \mu)/(\sigma)`

`1.126 = (X - 484)/(115)`

`X - 484 = 1.126*115`

`X = 613.5`

The approximate score for a student who is at the 87th percentile for Writing is 613.5.