Let's go through this problem step-by-step.

Given Information:

1. Test Statistic: $z = 2.83$
2. Claim: $p > 0.46$ (indicating a right-tailed test)
3. Significance Level: $\alpha = 0.01$

Part (a): Finding the Critical Value(s)

Since this is a right-tailed test with a 0.01 significance level, we need to find the critical value that corresponds to the upper 1% of the standard normal distribution.

Using the standard normal table or Z-scores, we find that:

• The critical value for a right-tailed test at $\alpha = 0.01$ is approximately 2.33.

So the answer for part (a) is:

• Option A: There is one critical value; the critical value is 2.33.

Part (b): Should we reject $H_0$ or fail to reject $H_0$?

The test statistic $z = 2.83$ is compared to the critical value of 2.33:

• Since $2.83 > 2.33$, the test statistic falls in the critical region for a right-tailed test.

Therefore, we should reject $H_0$.

The answer for part (b) is:

• Option B: $H_0$ should be rejected since the test statistic is in the critical region.

Final Answers:

• (a) Option A: The critical value is 2.33.
• (b) Option B: Reject $H_0$.

Would you like further clarification on any step, or do you have any other questions?

Additional Questions for Exploration:

1. What would be the critical value if the significance level was 0.05 instead of 0.01?
2. How would the decision change if the test statistic was 2.00 instead of 2.83?
3. Why does the direction of the test (right-tailed vs. left-tailed) affect the critical region?
4. What is the interpretation of rejecting $H_0$ in this context?
5. How would you find the critical value for a two-tailed test with the same significance level?

Tip:

For hypothesis testing, always check the direction of the test (right-tailed, left-tailed, or two-tailed) to correctly identify the critical value and interpret the test statistic.