Question

find the area under the standard normal curve that lies either to the left of -1.96 or to the right of 1.96. use four decimal places.

Explanation:

Step1: Use the property of standard normal distribution

The standard normal distribution is symmetric about \(z = 0\). The area to the left of \(z=-1.96\) is \(P(Z < - 1.96)\), and the area to the right of \(z = 1.96\) is \(P(Z>1.96)\). By symmetry, \(P(Z < -1.96)=P(Z>1.96)\)

Step2: Find the area using standard normal table

From the standard - normal table (or using a calculator with a normal - distribution function, e.g., in Excel: =NORM.S.DIST(-1.96,TRUE) for \(P(Z < -1.96)\)), we know that \(P(Z < -1.96)=0.025\) and \(P(Z>1.96)=1 - P(Z\leqslant1.96)\). Since \(P(Z\leqslant1.96)=0.975\) (from the standard - normal table: \(P(Z\leqslant z)\) where \(z = 1.96\)), then \(P(Z>1.96)=1 - 0.975=0.025\)

Step3: Calculate the total area

The total area \(A=P(Z < -1.96)+P(Z>1.96)\). Substitute the values: \(A = 0.025+0.025\)

Answer:

\(0.0500\)