Question

x is a normally distributed random variable with mean 14 and standard deviation 3. what is the probability that x is between 5 and 8? 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 5 and 8

For $x=5$: $z_1=\frac{5-14}{3}=-3$
For $x=8$: $z_2=\frac{8-14}{3}=-2$

Step2: Apply 0.68-0.95-0.997 rule

The rule states:

• 99.7% of data is within $\mu\pm3\sigma$ (z=-3 to 3)
• 95% of data is within $\mu\pm2\sigma$ (z=-2 to 2)

The area left of z=-3 is $\frac{1-0.997}{2}=0.0015$
The area left of z=-2 is $\frac{1-0.95}{2}=0.025$

Step3: Find the area between z=-3 and z=-2

Subtract the two areas: $0.025 - 0.0015$

Answer:

0.0235 rounded to 0.024