Question

x is a normally distributed random variable with mean 15 and standard deviation 9. what is the probability that x is between 6 and 42? 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 bounds

For $x=6$: $z_1=\frac{6-15}{9}=-1$
For $x=42$: $z_2=\frac{42-15}{9}=3$

Step2: Apply 0.68-0.95-0.997 rule

Probability for $z=-1$ to $z=0$: $\frac{0.68}{2}=0.34$
Probability for $z=0$ to $z=3$: $\frac{0.997}{2}=0.4985$

Step3: Sum the two probabilities

$0.34 + 0.4985 = 0.8385$

Step4: Round to nearest thousandth

$0.8385 \approx 0.839$

Answer:

0.839