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
(round to two decimal places as needed. use a comma to separate answers as needed.)
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
(round to two decimal places as needed.)
○ a. the rejection region is z <
○ b. the rejection regions are z < and z >
○ c. the rejection region is z > 2.58
choose the correct graph of the rejection region below.
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Recall the property of right - tailed z - test

For a right - tailed z - test with significance level \(\alpha\), we find the z - value \(z_{\alpha}\) such that \(P(Z>z_{\alpha})=\alpha\).
We know that the total area under the standard normal curve is \(1\). So, \(P(Z\leq z_{\alpha}) = 1-\alpha\).
Given \(\alpha = 0.005\), then \(1-\alpha=1 - 0.005=0.995\).

Step2: Use the standard normal table (z - table)

Looking up the value in the standard normal table (the cumulative distribution function of the standard normal distribution \(\varPhi(z)=P(Z\leq z)\)), we find the z - value corresponding to an area of \(0.995\).
From the standard normal table, \(z = 2.58\) (since \(\varPhi(2.58)\approx0.9951\)).
The rejection region for a right - tailed test is \(z>z_{\alpha}\).

Answer:

The critical value is \(z = 2.58\) and the rejection region is \(z>2.58\). The correct graph is the one that has the shaded (rejection) region on the right - hand side of \(z = 2.58\) (option C among the graph options if we assume option C is the right - tailed shaded graph).