Question

x is a normally distributed random variable with mean 61 and standard deviation 13. what is the probability that x is between 22 and 87? 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

For $x = 22$, $z_1=\frac{22 - 61}{13}=\frac{- 39}{13}=-3$. For $x = 87$, $z_2=\frac{87 - 61}{13}=\frac{26}{13}=2$.

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that the probability within $z=-3$ to $z = 3$ is 0.997 and within $z=-2$ to $z = 2$ is 0.95. The probability within $z=-3$ to $z=-2$ is $\frac{0.997 - 0.95}{2}=0.0235$.
The probability $P(-3

Answer:

0.974