Question

find the area of the indicated region under
the standard normal curve.
(round to four decimal places as needed.)
the area to the left of ( z = - 2.05 ) under the standard normal curve is (square).
(round to four decimal places as needed.)
the area to the right of ( z = 1.18 ) under the standard normal curve is (square).
(round to four decimal places as needed.)

Explanation:

Step1: Find the area for the first part

We know that the area under the standard - normal curve \(A = \varPhi(z_2)-\varPhi(z_1)\), where \(z_1=-1.5\) and \(z_2 = 1.4\).
From the standard - normal table, \(\varPhi(-1.5)=0.0668\) and \(\varPhi(1.4)=0.9192\).
So \(A_1=\varPhi(1.4)-\varPhi(-1.5)=0.9192-(1 - 0.9332)=0.9192 - 0.0668=0.8524\)

Step2: Find the area to the left of \(z=-2.05\)

Using the standard - normal table, for \(z=-2.05\), \(\varPhi(-2.05)\).
Looking up in the table, \(\varPhi(-2.05)=0.0202\)

Step3: Find the area to the right of \(z = 1.18\)

The area to the right of \(z\) is \(A=1-\varPhi(z)\).
For \(z = 1.18\), from the standard - normal table, \(\varPhi(1.18)=0.8810\)
So \(A_3=1 - 0.8810=0.1190\)

Answer:

The area of the first - indicated region is \(0.8524\).
The area to the left of \(z=-2.05\) is \(0.0202\).
The area to the right of \(z = 1.18\) is \(0.1190\).