Question

numeric 2 points
given a normally distributed data set with the mean = 80 and the standard deviation = 8, according to the empirical rule, what percent of the data points are below 64?
type your answer without using the percent symbol.

Explanation:

Step1: Calculate the number of standard deviations from the mean

We know that \(\mu = 80\) and \(\sigma=8\). We want to find out how many standard deviations \(64\) is from the mean.
We use the formula \(z=\frac{x - \mu}{\sigma}\). Substitute \(x = 64\), \(\mu = 80\) and \(\sigma = 8\) into the formula:
\(z=\frac{64 - 80}{8}=\frac{- 16}{8}=-2\)

Step2: Use the empirical rule

The empirical rule states that for a normal distribution:

• Approximately \(95\%\) of the data lies within \(\mu\pm2\sigma\) (i.e., \(z=- 2\) to \(z = 2\)).
• The percentage of data less than \(\mu - 2\sigma\) (i.e., \(z=-2\)) is \(\frac{100\% - 95\%}{2}=2.5\%\)

Answer:

\(2.5\)