Question

scores on the gre (graduate record examination) are normally distributed with a mean of 533 and a standard deviation of 98. use the empirical rule to find the percentage of people taking the test who score between 337 and 533. answer: %

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\mu\pm\sigma\)) of the mean.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\mu\pm2\sigma\)) of the mean.
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\mu\pm3\sigma\)) of the mean.

We are given \(\mu = 533\) and \(\sigma=98\). We want to find the percentage of data between \(x_1 = 337\) and \(x_2=533\).

First, find the number of standard deviations \(x_1\) is from the mean. Use the formula \(z=\frac{x-\mu}{\sigma}\).

For \(x = 337\), \(z=\frac{337 - 533}{98}=\frac{-196}{98}=- 2\)

Step2: Apply the Empirical Rule

The Empirical Rule says that the percentage of data within \(\mu\pm2\sigma\) (i.e., \(533-2\times98\) to \(533 + 2\times98\) or \(337\) to \(729\)) is \(95\%\).

The normal distribution is symmetric about the mean \(\mu\). The interval from \(337\) to \(533\) is half of the interval from \(337\) to \(729\).

So the percentage of data from \(337\) to \(533\) is \(\frac{95\%}{2}=47.5\%\)

Answer:

\(47.5\%\)