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 with a normal - distribution function (e.g., in Excel NORM.S.DIST(-3,TRUE) or in R pnorm(-3)), the area to the left of \(z=-3\) is \(P(Z < - 3)\).

$$P(Z < - 3)=0.00135$$

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

The area to the right of \(z = 3\) is \(P(Z>3)\). Since the total area under the standard - normal curve is \(1\), and \(P(Z > 3)=1 - P(Z\leqslant3)\). Using the standard normal table or a calculator (e.g., in Excel 1 - NORM.S.DIST(3,TRUE) or in R 1 - pnorm(3)), \(P(Z\leqslant3)=0.99865\). So \(P(Z > 3)=1 - 0.99865 = 0.00135\)

Step3: Calculate the combined area

The combined area \(A\) is the sum of the area to the left of \(z=-3\) and the area to the right of \(z = 3\).

$$A=P(Z < - 3)+P(Z > 3)$$

Substitute the values from Step 1 and Step 2:

$$A=0.00135 + 0.00135=0.0027$$

Answer:

\(0.0027\)