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 z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
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 standard normal distribution properties

We know that the total area under the standard normal curve is \(1\).
The area to the left of \(z=-1\) is \(0.1587\) (from the standard normal table), and the area to the left of \(z = 2\) is \(0.9772\).
The area between \(z=-1\) and \(z = 2\) is \(A=0.9772-0.1587\).

Step3: Calculate the percentage

\(A=0.9772 - 0.1587=0.8185\).
To convert this to a percentage, we multiply by \(100\), so the percentage is \(0.8185\times100 = 81.85\%\).

Answer:

\(81.85\)