Question

find the area of the shaded region under the standard normal curve if convenient, use technology to find the area. the area of the shaded region is (round to four decimal places as needed.)

Explanation:

Step1: Use the property of the standard normal distribution

The total area under the standard normal curve is \(1\). The standard normal distribution is symmetric about \(z = 0\). The area to the left of \(z = 0\) is \(0.5\). We need to find the area between \(z=-2.25\) and \(z = 0\).
We know that the area to the left of \(z=-2.25\) is \(P(Z<-2.25)\). Using the standard - normal table (or a calculator with a normal - distribution function, e.g., in Excel: NORM.S.DIST(-2.25, TRUE)), we find \(P(Z < - 2.25)=0.0122\).

Step2: Calculate the area of the shaded region

The area of the shaded region \(A\) (between \(z=-2.25\) and \(z = 0\)) can be calculated as \(A=\Phi(0)-\Phi(-2.25)\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution. Since \(\Phi(0) = 0.5\) and \(\Phi(-2.25)=0.0122\), then \(A = 0.5-0.0122=0.4878\).

Answer:

\(0.4878\)