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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).

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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.

A researcher is going to perform a right-tailed hypothesis test. The test statistic will follow a standard normal distribution. The researcher might choose a significance level of 0.01, 0.05, or 0.10. Given these significance levels, find the critical values and determine at which significance level a test statistic of z = 2.535 would not reject the null hypothesis.

Learn how to calculate critical values for a right-tailed hypothesis test based on significance levels of 0.01, 0.05, and 0.10. Discover how to determine whether a test statistic of z = 2.535 would lead to rejecting the null hypothesis.

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