Question

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.

(a) What is the estimated percentile for a student who scores 450 on Writing? Round your answer to the nearest integer.

(b) What is the approximate score for a student who is at the 90th percentile for Writing? Round your answer to the nearest integer.

Answer 1

a) 38th percentile.

b) 631

Explanation:

Problems of normally distributed samples 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 problem, we have that:

`\mu = 484, \sigma = 115`

(a) What is the estimated percentile for a student who scores 450 on Writing? Round your answer to the nearest integer.

This is the pvalue of Z when X = 450. So

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

`Z = (450 - 484)/(115)`

`Z = -0.3`

`Z = -0.3` has a pvalue of 0.3821.

So the estimated percentile for a student who scores 450 on Writing is the 38th percentile.

(b) What is the approximate score for a student who is at the 90th percentile for Writing? Round your answer to the nearest integer.

Value of X when Z has a pvalue of 0.9. So X when Z = 1.28.

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

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

`X - 484 = 1.28*115`

`X = 631.2`

Rounded to the nearest integer, 631