Question

a normal distribution has mean $\mu = 505$ and standard deviation $\sigma = 38$. approximately what percent of the data fall between 467 and 581?
approximately $\square\\%$ of the data fall between 467 and 581.
(type an integer or a decimal.)

Explanation:

Step1: Calculate the number of standard deviations from the mean

For \(x = 467\):
\(z_1=\frac{467 - 505}{38}=\frac{- 38}{38}=-1\)
For \(x = 581\):
\(z_2=\frac{581 - 505}{38}=\frac{76}{38}=2\)

Step2: Use the empirical rule for normal distributions

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(z=-1\) and \(z = 1\)
• Approximately \(95\%\) of the data lies within \(z=-2\) and \(z = 2\)
• The percentage of data between \(z=-1\) and \(z = 2\) is \(\frac{68\%}{2}+95\%\div2\)

\(34\%+47.5\% = 81.5\%\)

Answer:

\(81.5\)