Question

x is a normally distributed random variable with mean 40 and standard deviation 10. what is the probability that x is between 10 and 70? 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 the number of standard deviations from the mean

For \(x = 10\): \(z_1=\frac{10 - 40}{10}=\frac{- 30}{10}=-3\)
For \(x = 70\): \(z_2=\frac{70 - 40}{10}=\frac{30}{10}=3\)

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that for a normal distribution, approximately \(P(\mu - 3\sigma<X<\mu + 3\sigma)=0.997\)

Answer:

\(0.997\)