Question

x is a normally distributed random variable with mean 44 and standard deviation 10.
what is the probability that x is between 14 and 74?
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 \(x = 14\): \(z_1=\frac{14 - 44}{10}=\frac{- 30}{10}=-3\)
For \(x = 74\): \(z_2=\frac{74 - 44}{10}=\frac{30}{10}=3\)

Step2: Apply the 0.68 - 0.95 - 0.997 rule

The 0.68 - 0.95 - 0.997 rule (empirical rule) states that for a normal distribution, approximately \(P(\mu - 3\sigma<X<\mu + 3\sigma)=0.997\)

Answer:

\(0.997\)