Question

if z is a standard normal variable, find the probability that z lies between - 2.41 and 0. round to four decimal places.
a. 0.4910
b. 0.5080
c. 0.0948
d. 0.4920

Explanation:

Step1: Recall standard - normal table property

The standard - normal distribution is symmetric about \(z = 0\). The cumulative distribution function of a standard - normal variable \(Z\), denoted as \(\varPhi(z)\), gives \(P(Z\leq z)\). We know that \(P(Z\leq0)=0.5\) and we want to find \(P(- 2.41<Z<0)\).

Step2: Use the standard - normal table

From the standard - normal table, \(P(Z\leq - 2.41)\) can be found. The table gives values of \(P(Z\leq z)\) for non - negative \(z\) values. Since the standard - normal distribution is symmetric, \(P(Z\leq - 2.41)=1 - P(Z\leq2.41)\). Looking up \(P(Z\leq2.41)\) in the standard - normal table, we get \(P(Z\leq2.41) = 0.9920\), so \(P(Z\leq - 2.41)=1 - 0.9920=0.0080\).

Step3: Calculate the probability

We use the formula \(P(-2.41 < Z<0)=P(Z < 0)-P(Z\leq - 2.41)\). Since \(P(Z < 0)=0.5\) and \(P(Z\leq - 2.41)=0.0080\), then \(P(-2.41 < Z<0)=0.5 - 0.0080 = 0.4920\).

Answer:

D. 0.4920