Let's go through the problem step by step.

Part (a)

The researcher is performing a right-tailed hypothesis test, and you're asked to find the critical values for the following significance levels ($\alpha$):

• $\alpha = 0.01$
• $\alpha = 0.05$
• $\alpha = 0.10$

For a right-tailed test, the critical value corresponds to the $z$-score that leaves a probability of $\alpha$ in the right tail of the standard normal distribution.

Using a Z-table or a statistical calculator, the critical values are:

• For $\alpha = 0.01$, the critical value is approximately 2.326.
• For $\alpha = 0.05$, the critical value is approximately 1.645.
• For $\alpha = 0.10$, the critical value is approximately 1.282.

Part (b)

The test statistic is $z = 2.535$. We need to determine for which significance levels ($\alpha = 0.01$, $\alpha = 0.05$, $\alpha = 0.10$) the null hypothesis would not be rejected.

The null hypothesis is rejected if the test statistic exceeds the critical value. The test statistic $z = 2.535$ is compared with the critical values:

• For $\alpha = 0.01$, the critical value is 2.326. Since $2.535 > 2.326$, the null hypothesis is rejected.
• For $\alpha = 0.05$, the critical value is 1.645. Since $2.535 > 1.645$, the null hypothesis is rejected.
• For $\alpha = 0.10$, the critical value is 1.282. Since $2.535 > 1.282$, the null hypothesis is rejected.

Therefore, the null hypothesis is rejected for all significance levels, meaning None of the above is the correct answer for part (b).

Would you like a more detailed explanation of any part? Here are some related questions to deepen your understanding:

1. What is the relationship between the significance level and the critical value in hypothesis testing?
2. How does the direction of the hypothesis test (left-tailed, right-tailed, two-tailed) affect critical values?
3. How is the Z-score calculated in a hypothesis test?
4. What are the implications of rejecting or not rejecting the null hypothesis in practical terms?
5. Can critical values be calculated for other distributions, such as the t-distribution?

Tip: Always verify the direction of the test (right-tailed, left-tailed, or two-tailed) before determining critical values, as it affects the placement of the rejection region.