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)=0.9164. carry your intermediate computations to at least four decimal places. round your answer to two decimal places.

Explanation:

Step1: Use symmetry of standard - normal

Since the standard - normal distribution is symmetric about \(z = 0\), we know that \(P(-c\leq Z\leq c)=2\Phi(c)- 1\), where \(\Phi(c)\) is the cumulative distribution function of the standard - normal distribution.
We are given that \(2\Phi(c)-1 = 0.9164\).

Step2: Solve for \(\Phi(c)\)

First, solve the equation \(2\Phi(c)-1 = 0.9164\) for \(\Phi(c)\).
Add 1 to both sides: \(2\Phi(c)=0.9164 + 1=1.9164\).
Then divide both sides by 2: \(\Phi(c)=\frac{1.9164}{2}=0.9582\).

Step3: Find \(c\) using the standard - normal table

Looking up the value \(0.9582\) in the standard - normal table (or using a calculator with a normal - distribution function), we find the \(z\) - value corresponding to this cumulative probability.
The \(z\) - value (i.e., \(c\)) such that \(\Phi(c)=0.9582\) is approximately \(1.73\).

Answer:

\(1.73\)