Question

x is a normally distributed random variable with mean 36 and standard deviation 16. what is the probability that x is between 16 and 56? write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Standardize the values

For \(x = 16\), \(z_1=\frac{16 - 36}{16}=\frac{- 20}{16}=-1.25\)
For \(x = 56\), \(z_2=\frac{56 - 36}{16}=\frac{20}{16}=1.25\)

Step2: Find the probability using the standard normal table

\(P(16<X<56)=P(-1.25 < Z < 1.25)\)
Since \(P(-1.25 < Z < 1.25)=\Phi(1.25)-\Phi(-1.25)\)
From the standard - normal table, \(\Phi(1.25)=0.8944\) and \(\Phi(-1.25)=1 - \Phi(1.25)=1 - 0.8944 = 0.1056\)
\(P(-1.25 < Z < 1.25)=0.8944-(1 - 0.8944)=0.8944 - 0.1056=0.7888\)

Answer:

\(0.789\)