Question

suppose z follows the standard normal distribution. use the calculator provided, or this table, to determine the value of c so that the following is true. p(-c ≤ z ≤ c)=0.9371. carry your intermediate computations to at least four decimal places. round your answer to two decimal places.

Explanation:

Step1: Use symmetry of standard - normal distribution

Since the standard - normal distribution is symmetric about \(z = 0\), we know that \(P(-c\leq Z\leq c)=2P(0\leq Z\leq c)\). So, \(2P(0\leq Z\leq c)=0.9371\), then \(P(0\leq Z\leq c)=\frac{0.9371}{2}=0.46855\).

Step2: Find the \(z\) - value in the standard - normal table

We know that \(P(Z\leq c)=P(Z\leq0)+P(0\leq Z\leq c)\). Since \(P(Z\leq0) = 0.5\), then \(P(Z\leq c)=0.5 + 0.46855=0.96855\). Looking up this value in the standard - normal table (the \(z\) - table), we find the \(z\) - value corresponding to a cumulative probability of \(0.96855\).

Answer:

\(c\approx1.86\)