Question

x is a normally distributed random variable with mean 26 and standard deviation 20.
what is the probability that x is between 6 and 46?
use the 0.68 - 0.95 - 0.997 rule and write your answer as a decimal. round to the nearest thousandth if necessary.

Explanation:

Step1: Calculate the difference from the mean

For \(x = 6\), \(26-6=20\). For \(x = 46\), \(46 - 26=20\).

Step2: Relate to the 0.68 - 0.95 - 0.997 rule

The standard deviation \(\sigma=20\). The interval \((6,46)\) is \(\mu-\sigma\) to \(\mu+\sigma\) (since \(\mu = 26\), \(\mu-\sigma=26 - 20=6\) and \(\mu+\sigma=26+20 = 46\)).
According to the 0.68 - 0.95 - 0.997 rule, the probability that a normally distributed random variable \(X\) lies within \(\mu\pm\sigma\) is \(0.68\).

Answer:

\(0.68\)