Question

(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 0.0027
(round to four decimal places as needed.)
(b) find the area under the normal curve to the left of ( z=-1.58 ) plus the area under the normal curve to the right of ( z = 2.58 ).
the combined area is
(round to four decimal places as needed.)

Explanation:

Step1: Find the area to the left of \(z = - 1.58\)

Using the standard normal table (or a calculator with a normal - distribution function), the area to the left of \(z=-1.58\) is \(P(Z < - 1.58)\).
From the standard normal table, \(P(Z < - 1.58)=0.0571\).

Step2: Find the area to the right of \(z = 2.58\)

The area to the right of \(z = 2.58\) is \(P(Z>2.58)\).
Since \(P(Z > z)=1 - P(Z\leq z)\), for \(z = 2.58\), \(P(Z\leq2.58)=0.9951\) (from the standard normal table). So \(P(Z>2.58)=1 - 0.9951=0.0049\).

Step3: Calculate the combined area

The combined area \(A=P(Z < - 1.58)+P(Z>2.58)\).
Substitute the values: \(A = 0.0571+0.0049\).

Answer:

\(0.0620\)