This is a hypothesis test for a population proportion, where we are testing the claim that the population proportion $p \geq 0.25$.

Here's a step-by-step solution to finding the critical value(s):

1. Step 1: Identify the null and alternative hypotheses.

   • Null hypothesis, $H_0: p = 0.25$
   • Alternative hypothesis, $H_1: p < 0.25$ (This is a left-tailed test because we are testing if the proportion is less than 0.25.)

2. Step 2: Set the significance level, $\alpha$.

   • $\alpha = 0.01$

3. Step 3: Determine the critical value for a left-tailed test.

   • Since it's a left-tailed test, we need to find the $z$-score that corresponds to a cumulative probability of $1 - \alpha = 0.99$ in the standard normal distribution.

4. Step 4: Find the critical value for $z$.

   • Using a $z$-table or a calculator, the $z$-score that corresponds to a cumulative probability of $0.99$ is approximately $z = -2.33$.

5. Step 5: State the rejection region.

   • For a left-tailed test with $\alpha = 0.01$, the rejection region is: $z < -2.33$

Answer: The critical value for this test is $z = -2.33$. The rejection region is $z < -2.33$. If the test statistic falls within this region, we would reject the null hypothesis.

Would you like further assistance with calculating the test statistic or interpreting the result?

Here are some questions that could extend your understanding:

1. How do you calculate the test statistic for a hypothesis test on a population proportion?
2. What steps are involved in determining if the test statistic falls within the rejection region?
3. What are Type I and Type II errors in hypothesis testing, and how do they relate to the significance level?
4. How would this hypothesis test change if it were a right-tailed test instead?
5. What practical significance would rejecting the null hypothesis have in this context?

Tip: Always double-check your $z$-table or calculator to ensure you’re using the correct critical value for the desired confidence level.