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: Calculate the number of standard deviations from the mean

The mean $\mu = 533$, the standard deviation $\sigma=98$.
We want to find how many standard deviations $337$ is from $533$.
Let $x = 337$. The formula for the number of standard deviations $z=\frac{x-\mu}{\sigma}$.
Substitute the values: $z=\frac{337 - 533}{98}=\frac{-196}{98}=- 2$.

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately $68\%$ of the data lies within $\mu\pm\sigma$ ($z = \pm1$)
• Approximately $95\%$ of the data lies within $\mu\pm2\sigma$ ($z=\pm2$)
• Approximately $99.7\%$ of the data lies within $\mu\pm3\sigma$ ($z = \pm3$)

The interval from $z=-2$ to $z = 0$ (since $\mu = 533$ corresponds to $z = 0$).
Since the normal distribution is symmetric about the mean, the percentage of data between $z=-2$ and $z = 0$ is $\frac{95\%}{2}=47.5\%$

Answer:

$47.5$