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:

plaintextCopy code
      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.