Question

x is a normally distributed random variable with mean 45 and standard deviation 18.
what is the probability that x is between 9 and 81?
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 number of standard deviations from the mean

For \(x = 9\): \(z_1=\frac{9 - 45}{18}=\frac{- 36}{18}=-2\)
For \(x = 81\): \(z_2=\frac{81 - 45}{18}=\frac{36}{18}=2\)

Step2: Apply the 0.68 - 0.95 - 0.997 rule

The 0.68 - 0.95 - 0.997 rule states that for a normal distribution, approximately \(P(\mu - 2\sigma<X<\mu + 2\sigma)=0.95\)

Answer:

\(0.95\)