Question

determine the area under the standard normal curve that lies between (a) ( z = - 1.14 ) and ( z = 1.14 ), (b) ( z = - 1.05 ) and ( z = 0 ), and (c) ( z = 0.92 ) and ( z = 1.48 ).
(a) the area that lies between ( z = - 1.14 ) and ( z = 1.14 ) is 0.7458
(round to four decimal places as needed.)
(b) the area that lies between ( z = - 1.05 ) and ( z = 0 ) is
(round to four decimal places as needed.)

Explanation:

Step1: Recall the property of standard normal distribution

The standard normal distribution is symmetric about \(Z = 0\). The area to the left of \(Z = 0\) is \(0.5\). We use the standard - normal table (or z - table) to find the area to the left of a given \(Z\) value.

Step2: Find the area to the left of \(Z=-1.05\)

Using the standard - normal table, for \(Z=-1.05\), the area to the left of \(Z =-1.05\) (denoted as \(P(Z\lt - 1.05)\)) is \(0.1469\).

Step3: Calculate the area between \(Z=-1.05\) and \(Z = 0\)

We know that the area between \(Z=-1.05\) and \(Z = 0\) is \(P(-1.05\lt Z\lt0)\). Since \(P(Z\lt0)=0.5\) and \(P(Z\lt - 1.05)=0.1469\), then \(P(-1.05\lt Z\lt0)=P(Z\lt0)-P(Z\lt - 1.05)\)

$$P(-1.05\lt Z\lt0)=0.5 - 0.1469=0.3531$$

Answer:

\(0.3531\)