Let's solve each part of the problem step-by-step.

Part 1 (a)

Question: Assume that a 0.10 significance level will be used to test the claim that $p_1 < p_2$. Which is better: a hypothesis test or a confidence interval?

Answer: A hypothesis test is better.

Reasoning: A hypothesis test is more appropriate when testing a specific claim about the population proportions, in this case, $p_1 < p_2$.

Part 2 (b)

Question: In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method, P-value method, critical value method?

Answer: P-value method and critical value method are equivalent.

Reasoning: Both the P-value method and the critical value method use a standard deviation based on the assumption that the two population proportions are equal, and they will always lead to the same conclusion. The confidence interval method uses a standard deviation based on estimated values of the population proportions.

Part 3 (c)

Question: If a 0.10 significance level is to be used to test the claim that $p_1 < p_2$, what confidence level should be used?

Answer: 90%

Reasoning: For a one-tailed test at the 0.10 significance level, the corresponding confidence level for the confidence interval is 90%.

Part 4 (d)

Question: If the claim in part (c) is tested using this sample data, we get this confidence interval: $-0.000551 < p_1 - p_2 < -0.000385$. What does this confidence interval suggest about the claim?

Answer:

• Because the confidence interval does not contain zero, there is a significant difference between the two proportions.
• Because the confidence interval consists of only negative values, it appears that the first proportion ($p_1$) is less than the second proportion ($p_2$).
• There is sufficient evidence to support the claim that the rate of the disease is less for children given the vaccine than it is for children given a placebo.

Summary:

To summarize the answers for each part:

• (a) A hypothesis test is better.
• (b) P-value method and critical value method are equivalent.
• (c) 90%
• (d) There is sufficient evidence to support the claim that the rate of the disease is less for children given the vaccine than it is for children given a placebo.

Would you like more details or have any questions about these parts?

Here are 5 more questions you might ask:

1. How do you calculate the confidence interval for two population proportions?
2. What are the assumptions required for conducting a hypothesis test for two proportions?
3. Can you explain the difference between a one-tailed and a two-tailed test?
4. How do you interpret the P-value in the context of a hypothesis test?
5. What is the relationship between the significance level and the confidence level?

Tip: When interpreting confidence intervals, always check whether zero is included in the interval. If zero is not included, it suggests a significant difference between the proportions.