Question

x is a normally distributed random variable with mean 65 and standard deviation 10. what is the probability that x is between 85 and 95? 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 $\mu = 65$ is the mean and $\sigma = 10$ is the standard deviation.
For $x = 85$, $z_1=\frac{85 - 65}{10}=\frac{20}{10}=2$.
For $x = 95$, $z_2=\frac{95 - 65}{10}=\frac{30}{10}=3$.

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that about 95% of the data lies within 2 standard deviations of the mean ($z=\pm2$) and about 99.7% of the data lies within 3 standard deviations of the mean ($z = \pm3$).
The proportion of data between $z = 2$ and $z = 3$ is $\frac{0.997-0.95}{2}$.
$0.997$ is the proportion of data within $z=\pm3$ and $0.95$ is the proportion of data within $z=\pm2$. So $\frac{0.997 - 0.95}{2}=\frac{0.047}{2}=0.0235$.

Answer:

$0.024$