Question

x is a normally distributed random variable with mean 69 and standard deviation 15. what is the probability that x is between 28 and 74? write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Standardize the values

The formula for standardization is \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 69\) (mean) and \(\sigma=15\) (standard deviation).
For \(x = 28\):
\(z_1=\frac{28 - 69}{15}=\frac{-41}{15}\approx - 2.73\)
For \(x = 74\):
\(z_2=\frac{74 - 69}{15}=\frac{5}{15}\approx0.33\)

Step2: Use the standard normal distribution table

We want to find \(P(-2.73<Z<0.33)\)
By the property \(P(a < Z < b)=P(Z < b)-P(Z < a)\)
From the standard - normal table:
\(P(Z < 0.33)=0.6293\)
\(P(Z<-2.73) = 0.0032\)

Step3: Calculate the probability

\(P(-2.73 < Z < 0.33)=P(Z < 0.33)-P(Z < - 2.73)\)
\(P(-2.73 < Z < 0.33)=0.6293-0.0032 = 0.6261\approx0.626\)

Answer:

\(0.626\)