Question

find the critical value(s) and rejection region(s) for the type of z - test with level of significance α. include a graph with your answer.
right - tailed test, α = 0.005
the critical value(s) is/are z = (round to two decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Determine the critical value

For a right - tailed \(z\) - test with significance level \(\alpha = 0.005\), we look up the \(z\) - value in the standard normal distribution table such that \(P(Z>z)=\alpha\). Since \(P(Z > z)=0.005\), then \(P(Z\leq z)=1 - 0.005=0.995\).
Looking up in the standard normal table (or using a calculator with a normal - distribution function, e.g., in Excel =NORM.S.INV(0.995)), we find the \(z\) - value.

Step2: Calculate the \(z\) - value

Using the standard normal distribution properties, the \(z\) - value corresponding to a cumulative probability of \(0.995\) is \(z = 2.58\) (rounded to two decimal places).

Answer:

The critical value is \(z = 2.58\). The rejection region is \(z>2.58\).
For the graph: Draw a standard normal curve (\(y = f(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2}}\)). Mark the \(z\) - axis. Shade the region to the right of \(z = 2.58\) (this is the rejection region). The non - shaded region is the non - rejection region.