Question

x is a normally distributed random variable with mean 20 and standard deviation 2. what is the probability that x is between 16 and 18? 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 $x = 16$, $z_1=\frac{16 - 20}{2}=\frac{- 4}{2}=-2$.
For $x = 18$, $z_2=\frac{18 - 20}{2}=\frac{-2}{2}=-1$.

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that about 68% of the data lies within 1 standard deviation of the mean ($z=-1$ to $z = 1$), about 95% lies within 2 standard deviations of the mean ($z=-2$ to $z = 2$), and about 99.7% lies within 3 standard deviations of the mean ($z=-3$ to $z = 3$).
The area between $z=-2$ and $z = 2$ is 0.95, and the area between $z=-1$ and $z = 1$ is 0.68.
The area between $z=-2$ and $z=-1$ is $\frac{0.95 - 0.68}{2}$.

Step3: Calculate the probability

$\frac{0.95 - 0.68}{2}=\frac{0.27}{2}=0.135$

Answer:

0.135