Let's tackle both parts step by step.

Part (a) - Finding Critical Values for the Two-Tailed Test

For a two-tailed hypothesis test, we need to find the critical values that correspond to the given levels of significance ($\alpha$). The critical values are the z-scores that correspond to the two tails of the standard normal distribution.

The critical values are based on the area in the tails, which is $\alpha/2$ on each side because it's a two-tailed test.

For each $\alpha$, we will find the corresponding z-scores:

1. $\alpha = 0.01$:
   The critical value for $\alpha = 0.01$ is the z-score that leaves 0.005 in each tail (since 0.01/2 = 0.005). We need to find the z-score for the area of $1 - 0.005 = 0.995$.

2. $\alpha = 0.05$:
   The critical value for $\alpha = 0.05$ is the z-score that leaves 0.025 in each tail (since 0.05/2 = 0.025). We need to find the z-score for the area of $1 - 0.025 = 0.975$.

3. $\alpha = 0.10$:
   The critical value for $\alpha = 0.10$ is the z-score that leaves 0.05 in each tail (since 0.10/2 = 0.05). We need to find the z-score for the area of $1 - 0.05 = 0.95$.

Using standard normal distribution tables or a calculator, we get the following critical values:

• For $\alpha = 0.01$, the critical z-scores are approximately $\pm 2.576$.
• For $\alpha = 0.05$, the critical z-scores are approximately $\pm 1.960$.
• For $\alpha = 0.10$, the critical z-scores are approximately $\pm 1.645$.

Thus, the critical values for each $\alpha$ are:

• $\alpha = 0.01$: $\pm 2.576$
• $\alpha = 0.05$: $\pm 1.960$
• $\alpha = 0.10$: $\pm 1.645$

Part (b) - Checking If the Null Hypothesis Is Rejected

The researcher found the test statistic $z \approx -1.731$. To determine if the null hypothesis should be rejected, we compare this test statistic to the critical values for each level of significance:

• For $\alpha = 0.01$:
  The critical values are $\pm 2.576$. Since $-1.731$ is not less than $-2.576$, the test statistic does not fall in the rejection region. So, we do not reject the null hypothesis at $\alpha = 0.01$.

• For $\alpha = 0.05$:
  The critical values are $\pm 1.960$. Since $-1.731$ is not less than $-1.960$, the test statistic does not fall in the rejection region. So, we do not reject the null hypothesis at $\alpha = 0.05$.

• For $\alpha = 0.10$:
  The critical values are $\pm 1.645$. Since $-1.731$ is less than $-1.645$, the test statistic does fall in the rejection region. Therefore, we reject the null hypothesis at $\alpha = 0.10$.

Conclusion:

• The null hypothesis is rejected at $\alpha = 0.10$.
• The null hypothesis is not rejected at $\alpha = 0.01$ or $\alpha = 0.05$.

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Would you like more details on any step or concept? Here are five related questions to deepen understanding:

1. What is the difference between one-tailed and two-tailed hypothesis tests?
2. How is the critical value related to the level of significance ($\alpha$) in hypothesis testing?
3. Why does lowering the significance level make it harder to reject the null hypothesis?
4. How do you interpret the rejection region in a two-tailed test?
5. What does the test statistic represent in hypothesis testing?

Tip: In hypothesis testing, always ensure that the test statistic falls outside the critical values (in the rejection region) before rejecting the null hypothesis.