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 0.0026.
(round to four decimal places as needed.)
(b) find the area under the normal curve to the left of ( z = - 1.57 ) plus the area under the normal curve to the right of ( z = 2.57 ).
the combined area is
(round to four decimal places as needed.)

Explanation:

Step1: Find the area to the left of \(z = - 1.57\)

Using the standard normal table or a calculator with a normal - distribution function, the area to the left of \(z=-1.57\) is \(P(Z < - 1.57)\).
From the standard normal table, \(P(Z < - 1.57)=0.0582\)

Step2: Find the area to the right of \(z = 2.57\)

The area to the right of \(z = 2.57\) is \(P(Z>2.57)\). Since \(P(Z > z)=1 - P(Z\leq z)\), and from the standard normal table \(P(Z\leq2.57) = 0.9949\), then \(P(Z>2.57)=1 - 0.9949=0.0051\)

Step3: Calculate the combined area

The combined area \(A=P(Z < - 1.57)+P(Z>2.57)\)
\(A = 0.0582+0.0051=0.0633\)

Answer:

\(0.0633\)