Find the critical​ value(s) and rejection​ region(s) for the type of​ z-test with level of significance 

alphaα. 

Include a graph with your answer.

TwoTwo​-tailed 

​test, 

alphaαequals=0.01

To find the critical values and rejection regions for a two-tailed \( z \)-test with a significance level \(\alpha = 0.01\), follow these steps:

1. **Understand the Test:**
   - **Two-tailed test:** The critical values will be at both ends of the normal distribution.
   - **Significance level:** \(\alpha = 0.01\), which means 0.005 in each tail (since the test is two-tailed).

2. **Find the Critical Values:**
   - For a two-tailed test with \(\alpha = 0.01\), you split the significance level between the two tails, so each tail has an area of \( \alpha/2 = 0.005 \).
   - Using the standard normal distribution table (or a calculator), find the \( z \)-score that corresponds to the cumulative probability of \( 0.005 \) in the left tail and \( 0.995 \) in the right tail.
   
3. **Calculate the Critical \( z \)-Values:**
   - The \( z \)-score that corresponds to a left tail probability of 0.005 is approximately \( -2.576 \).
   - Similarly, the \( z \)-score that corresponds to a right tail probability of 0.995 is approximately \( 2.576 \).

   Therefore, the critical values are \( -2.576 \) and \( 2.576 \).

4. **Determine the Rejection Regions:**
   - For a two-tailed test, the rejection regions are the areas in the tails beyond the critical values.
   - If the test statistic \( z \) falls below \( -2.576 \) or above \( 2.576 \), we reject the null hypothesis \( H_0 \).

5. **Graph:**
   - To visualize the critical values and rejection regions, plot the standard normal distribution curve.
   - Mark the critical values \( -2.576 \) and \( 2.576 \) on the \( x \)-axis.
   - Shade the regions to the left of \( -2.576 \) and to the right of \( 2.576 \) to indicate the rejection regions.

Here is a graph illustrating the critical values and rejection regions:

```plaintext
      Rejection Region                     Rejection Region
          <-----|----------------|----------|----------------|----->
                -2.576           0        2.576
```

In this graph:
- The area to the left of \( -2.576 \) and the area to the right of \( 2.576 \) are shaded, representing the rejection regions.
- The area between \( -2.576 \) and \( 2.576 \) is the non-rejection region where the null hypothesis is not rejected.

Would you like more details or have any questions?

### Related Questions
1. What is the difference between a critical value and a test statistic?
2. How do you interpret the rejection region in hypothesis testing?
3. What are the steps to perform a hypothesis test?
4. Can you explain the significance level \(\alpha\) in hypothesis testing?
5. How do you use a standard normal distribution table?
6. What is the importance of a two-tailed test?
7. How do you calculate the \( P \)-value for a given test statistic?
8. What are the consequences of incorrectly rejecting the null hypothesis?

**Tip:** Always make sure to check whether your test is one-tailed or two-tailed, as this will affect the calculation of critical values and rejection regions.

Learn how to find critical values and rejection regions for a two-tailed z-test with a significance level α = 0.01. Understand the concepts of hypothesis testing, critical values, and the significance level in statistical analysis.

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