Question

x is a normally distributed random variable with mean 62 and standard deviation 19.
what is the probability that x is between 17 and 40?
write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Standardize the values

For \(x = 17\), the \(z\)-score is \(z_1=\frac{17 - 62}{19}=\frac{- 45}{19}\approx - 2.37\).
For \(x = 40\), the \(z\)-score is \(z_2=\frac{40 - 62}{19}=\frac{-22}{19}\approx - 1.16\).

Step2: Use the standard normal distribution table

We want \(P(-2.37<Z<-1.16)\).
Since \(P(-2.37 < Z<-1.16)=P(Z < - 1.16)-P(Z < - 2.37)\).
From the standard - normal table, \(P(Z < - 1.16)=0.1230\) and \(P(Z < - 2.37)=0.0089\).

Step3: Calculate the probability

\(P(-2.37 < Z<-1.16)=0.1230 - 0.0089=0.1141\approx0.114\)

Answer:

\(0.114\)