Question

x is a normally - distributed random variable with mean 35 and standard deviation 15. what is the probability that x is between 5 and 80? 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 z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation.
For $x = 5$, $z_1=\frac{5 - 35}{15}=\frac{- 30}{15}=-2$.
For $x = 80$, $z_2=\frac{80 - 35}{15}=\frac{45}{15}=3$.

Step2: Use the empirical rule

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:
The area between $z=-2$ and $z = 2$ is approximately 0.95, and the area between $z=-3$ and $z = 3$ is approximately 0.997.
The area between $z=-2$ and $z = 3$ is the area between $z=-3$ and $z = 3$ minus the area outside $z=-2$ and $z = 3$ on the left - hand side.
The area outside $z=-2$ and $z = 2$ on each side is $\frac{1 - 0.95}{2}=0.025$, and the area outside $z=-3$ and $z = 3$ on each side is $\frac{1 - 0.997}{2}=0.0015$.
The area between $z=-2$ and $z = 3$ is $0.997-(0.0015 + 0.025)=0.9705$.

Answer:

0.971