Question

using the standard normal distribution, find the two z - scores that form the middle shaded region. the shaded region is symmetric about ( z = 0 ). round your z - scores to two decimal places. shaded area ( = 0.22 ) negative z - score ( = ) positive z - score ( = ) question help: message instructor

Explanation:

Step1: Calculate the area in the two tails

Since the total area under the standard - normal curve is \(1\), and the shaded area (middle area) is \(A = 0.22\). The area in the two tails is \(1 - 0.22=0.78\). The area in each tail is \(\frac{1 - 0.22}{2}=\frac{0.78}{2}=0.39\).

Step2: Find the \(z\) - score

We want to find \(z\) such that \(P(Z\lt z)=1 - 0.39 = 0.61\) (for the positive \(z\) - score). Looking up in the standard - normal table (or using a calculator with a normal - distribution function, e.g., for a TI - 84: invNorm(0.61)).
Using the standard - normal table (the \(z\) - table), we scan the body of the table for the value closest to \(0.61\). The \(z\) - score corresponding to an area of \(0.61\) is approximately \(z = 0.28\). Due to the symmetry of the standard - normal distribution (\(Z\sim N(0,1)\)), the negative \(z\) - score is \(-z=- 0.28\).

Answer:

Negative \(z\) - score \(=-0.28\)
Positive \(z\) - score \(=0.28\)