Question

Suppose SAT Writing scores are normally distributed with a mean of 488488 and a standard deviation of 111111. A university plans to award scholarships to students whose scores are in the top 8%8%. What is the minimum score required for the scholarship? Round your answer to the nearest whole number, if necessary.

Answer 1

Answer:

The minimum score required for the scholarship is 644.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 488, \sigma = 111$

What is the minimum score required for the scholarship?

Top 8%, which means that the minimum score is the 100-8 = 92th percentile, which is X when Z has a pvalue of 0.92. So it is X when Z = 1.405.

$Z = \frac{X - \mu}{\sigma}$

$1.405 = \frac{X - 488}{111}$

$X - 488 = 1.405*111$

$X = 644$

The minimum score required for the scholarship is 644.