Let's break down the problem you've shared.

(a) Critical Values for Different Significance Levels

In the first part, the researcher is determining the critical values for a two-tailed hypothesis test at three different levels of significance:

• At $\alpha = 0.01$, the critical z-value is $\pm 2.576$
• At $\alpha = 0.05$, the critical z-value is $\pm 1.960$
• At $\alpha = 0.10$, the critical z-value is $\pm 1.645$

These values come from the standard normal distribution, where the critical values correspond to the boundaries beyond which the null hypothesis would be rejected.

(b) Rejection of the Null Hypothesis

For part (b), the researcher chooses a significance level and finds that the test statistic is $z = 1.089$. You are asked to determine whether the null hypothesis would be rejected at each of the significance levels:

• At $\alpha = 0.01$: The critical z-values are $\pm 2.576$. Since the test statistic $1.089$ is less than $2.576$, the null hypothesis is not rejected at this significance level.

• At $\alpha = 0.05$: The critical z-values are $\pm 1.960$. Since the test statistic $1.089$ is also less than $1.960$, the null hypothesis is not rejected at this significance level either.

• At $\alpha = 0.10$: The critical z-values are $\pm 1.645$. Since the test statistic $1.089$ is still less than $1.645$, the null hypothesis is not rejected at this significance level either.

Thus, the correct answer would be None of the above because at all levels of significance ($\alpha = 0.01, 0.05, 0.10$), the test statistic does not exceed the critical value, and the null hypothesis is not rejected.

Would you like further explanation or any clarifications on these steps? Here are some related questions to explore:

1. How do we calculate critical values for different significance levels?
2. What is the significance of a two-tailed hypothesis test?
3. How does the value of the test statistic relate to rejecting or failing to reject the null hypothesis?
4. Why do we use the standard normal distribution for hypothesis testing?
5. What would happen if the test statistic were greater than the critical value?

Tip: Remember that the critical value in a hypothesis test sets the threshold. If the test statistic exceeds this value (in absolute terms), the null hypothesis is rejected.