Question

A testing bureau reports that the mean for the population of Graduate Record Exam (GRE) scores is 500 with a standard deviation of 90. The scores are normally distributed. The proportion of scores between 300 and 400 is ___.

Answer 1

Answer:

The proportion of scores between 300 and 400 is 12.03%.

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:

$\mu = 500, \sigma = 90$

The proportion of scores between 300 and 400 is

This is the pvalue of Z when X = 400 subtracted by the pvalue of Z when X = 300. So

X = 400

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

$Z = \frac{400 - 500}{90}$

$Z = -1.11$

$Z = -1.11$ has a pvalue of 0.1335.

X = 300

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

$Z = \frac{300 - 500}{90}$

$Z = -2,22$

$Z = -2.22$ has a pvalue of 0.0132.

0.1335 - 0.0132 = 0.1203

The proportion of scores between 300 and 400 is 12.03%.