Question

which z-values correspond to the middle 90% of the standard normal distribution? round your answers to the nearest thousandth. < z <

Explanation:

Step1: Find the area in the tails

The total area under the standard - normal curve is \(1\). If the middle area is \(0.90\), then the area in the two tails is \(1 - 0.90=0.10\). The area in each tail is \(\frac{0.10}{2}=0.05\).

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

We want to find the \(z\) - value \(z_1\) such that \(P(Z\lt z_1)=0.05\). Using a standard - normal table (or a calculator with a normal - distribution function, e.g., the inverse of the cumulative distribution function for the standard normal distribution \(\Phi^{-1}(p)\) in a calculator), we find \(z_1\approx - 1.645\).

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

Since the standard normal distribution is symmetric about \(z = 0\), if \(P(Z\lt z_1)=0.05\), then \(P(Z\gt z_2)=0.05\) and \(z_2=-z_1\). So \(z_2 = 1.645\).

Answer:

\(-1.645