Question

x is a normally distributed random variable with mean 16 and standard deviation 10.
what is the probability that x is between 10 and 22?
write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Standardize the values

For \(x = 10\), \(z_1=\frac{10 - 16}{10}=\frac{- 6}{10}=-0.6\)
For \(x = 22\), \(z_2=\frac{22 - 16}{10}=\frac{6}{10}=0.6\)

Step2: Find the probabilities

Using the standard normal distribution table or a calculator, \(P(Z < 0.6)\) and \(P(Z < - 0.6)\)
\(P(Z<0.6)\approx0.7257\), \(P(Z < - 0.6)\approx0.2743\)

Step3: Calculate the required probability

\(P(10<X<22)=P(-0.6 < Z < 0.6)=P(Z < 0.6)-P(Z < - 0.6)\)
\(P(-0.6 < Z < 0.6)=0.7257-0.2743 = 0.4514\approx0.451\)

Answer:

\(0.451\)