Question

determine the total area under the standard normal curve in parts (a) through (c) below.
(a) find the area under the normal curve to the left of ( z = - 3 ) plus the area under the normal curve to the right of ( z = 3 ).
the combined area is
(round to four decimal places as needed.)

Explanation:

Step1: Find the area to the left of \(z = - 3\)

Using the standard normal table or a calculator, \(P(Z<-3)=\Phi(-3)\). From the standard - normal distribution table, \(\Phi(-3)=0.0013\)

Step2: Find the area to the right of \(z = 3\)

Since the total area under the standard - normal curve is \(1\), \(P(Z > 3)=1-\Phi(3)\). And because of the symmetry of the standard - normal distribution \(\Phi(-z)=1 - \Phi(z)\), so \(P(Z>3)=\Phi(-3)\)

Step3: Calculate the combined area

The combined area \(A=P(Z < - 3)+P(Z>3)\). Substitute the values from Step 1 and Step 2: \(A = 0.0013+0.0013\)

Answer:

\(0.0026\)