Question

find the critical value ( z_{alpha / 2} ) that corresponds to the confidence level 86%.

( z_{alpha / 2}=square )
(round to two decimal places as needed.)

Explanation:

Step1: Calculate \(\alpha\)

The confidence level \(C = 86\%=0.86\). Using the formula \(\alpha=1 - C\), we get \(\alpha=1 - 0.86 = 0.14\).

Step2: Calculate \(\frac{\alpha}{2}\)

\(\frac{\alpha}{2}=\frac{0.14}{2}=0.07\).

Step3: Find the \(z\) - value

We want to find \(z_{\alpha/2}\) such that \(P(Z>z_{\alpha/2}) = 0.07\), which is equivalent to \(P(Z\leq z_{\alpha/2})=1 - 0.07=0.93\). Looking up in the standard normal table (or using a calculator with a normal - distribution function, e.g., for a TI - 84: invNorm(0.93,0,1)), we find \(z_{\alpha/2}\approx1.48\).

Answer:

\(1.48\)