Question

Scores on a college entrance exam are normally distributed with a mean of 550 and a standard deviation of 100. Find the value that represents the 90th percentile of scores. Answer with a whole number.

Answer 1

Answer:

The value that represents the 90th percentile of scores is 678.

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:

$\mu = 550, \sigma = 100$

Find the value that represents the 90th percentile of scores.

This is the value of X when Z has a pvalue of 0.9. So X when Z = 1.28.

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

$1.28 = \frac{X - 550}{100}$

$X - 550 = 100*1.28$

$X = 678$

The value that represents the 90th percentile of scores is 678.