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

The mean $\mu = 533$ and the standard deviation $\sigma=98$.
We want to find $k$ such that $\mu - k\sigma=337$.
Substitute the values: $533 - k\times98 = 337$.
Solve for $k$: $k\times98=533 - 337=196$, so $k = \frac{196}{98}=2$.

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

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

The interval from $\mu - 2\sigma$ to $\mu$ is half of the interval $\mu - 2\sigma$ to $\mu+ 2\sigma$.
Since the normal distribution is symmetric about the mean, the percentage of data from $\mu - 2\sigma$ to $\mu$ is $\frac{95\%}{2}=47.5\%$.

Answer:

$47.5$