Question

x is a normally distributed random variable with mean 34 and standard deviation 14.
what is the probability that x is between 18 and 42?
write your answer as a decimal rounded to the nearest thousandth.
save answer

Explanation:

Step1: Calculate the z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 34\) (mean) and \(\sigma=14\) (standard deviation).
For \(x = 18\):
\(z_1=\frac{18 - 34}{14}=\frac{- 16}{14}\approx - 1.14\)
For \(x = 42\):
\(z_2=\frac{42-34}{14}=\frac{8}{14}\approx0.57\)

Step2: Use the standard normal distribution table

We want to find \(P(18<X<42)=P(-1.14 < Z < 0.57)\)
Since \(P(-1.14 < Z < 0.57)=P(Z < 0.57)-P(Z<-1.14)\)
From the standard - normal table: \(P(Z < 0.57)=0.7157\) and \(P(Z<-1.14) = 0.1271\)

Step3: Calculate the probability

\(P(-1.14 < Z < 0.57)=0.7157-0.1271 = 0.5886\approx0.589\)

Answer:

\(0.589\)