Question

which z - values correspond to the middle 92% of the standard normal distribution?
round your answers to the nearest thousandth.
< z <

Explanation:

Step1: Calculate the tail area

The total area under the standard - normal curve is \(1\). If the middle area is \(0.92\), then the total area in the two tails is \(1 - 0.92=0.08\). The area in each tail is \(\frac{0.08}{2}=0.04\).

Step2: Find the \(z\) - value for the left - tail

We want to find \(z_1\) such that \(P(Z\lt z_1)=0.04\). Using a standard - normal table (or a calculator with a normal - distribution function, e.g., the inverse of the cumulative distribution function for a standard normal distribution \(\Phi^{-1}(p)\)), for \(p = 0.04\), \(z_1\approx - 1.751\).

Step3: Find the \(z\) - value for the right - tail

Because of the symmetry of the standard normal distribution (\(Z\sim N(0,1)\)), if \(P(Z\lt z_1)=0.04\), then \(P(Z\gt z_2)=0.04\) and \(z_2=-z_1\). So \(z_2 = 1.751\).

Answer:

\(-1.751