To determine the decision rule for this hypothesis test, let's go step by step:

1. Null and Alternative Hypotheses:

• $H_0: p = 0.40$
• $H_A: p \neq 0.40$

This is a two-tailed hypothesis test because the alternative hypothesis states that $p$ is not equal to 0.40.

2. Significance Level (α):

• $\alpha = 0.01$

This means we have a 1% significance level, and since it's a two-tailed test, we will split the significance level equally between both tails (0.005 in each tail).

3. Critical Values for a Two-Tailed Test:

For a two-tailed test at $\alpha = 0.01$, we need to find the critical z-values that correspond to the lower and upper tails.

• Using a standard normal distribution table, the z-value that corresponds to the cumulative probability of $0.005$ in the lower tail is approximately −2.575, and the z-value for $0.995$ (upper tail) is +2.575.

4. Decision Rule:

• We reject the null hypothesis $H_0$ if the calculated test statistic $z$ is less than −2.575 or greater than +2.575.
• Otherwise, we do not reject $H_0$.

5. Correct Answer:

Thus, the correct decision rule is:

A. The decision rule is: reject the null hypothesis if the calculated value of the test statistic $z$ is greater than 2.575 or less than −2.575. Otherwise, do not reject.

Would you like a detailed explanation of how to compute the z-test statistic for this test?

Here are five related questions to help deepen your understanding:

1. How do we calculate the z-test statistic in a hypothesis test for proportions?
2. Why do we split the significance level in a two-tailed test?
3. What is the importance of using the standard normal distribution in hypothesis testing?
4. How would the decision rule change if the significance level $\alpha$ were 0.05?
5. What are the implications of rejecting $H_0$ in this context?

Tip: Always check whether the test is one-tailed or two-tailed before determining the critical values!