It looks like you're working on constructing and interpreting a confidence interval for the mean pH of rainwater with a confidence level of 99.99% (although it seems there's a typo where you wrote "9999%"). Let's go over this step by step.

Constructing a Confidence Interval for the Mean

For a confidence interval for the mean, you need:

1. Sample mean ($\bar{x}$)
2. Standard deviation ($\sigma$) or sample standard deviation ($s$) if the population standard deviation is unknown.
3. Sample size ($n$)
4. Z-score or t-score depending on whether the population standard deviation is known and the sample size.

The general formula for a confidence interval for the population mean is:

$\bar{x} \pm Z \times \frac{\sigma}{\sqrt{n}}$ where $Z$ is the Z-score corresponding to the desired confidence level.

For a 99.99% confidence interval, the critical Z-score (from the standard normal distribution) is approximately 3.8906.

Now, let's go over the interpretation of the confidence interval.

Interpretation of Confidence Interval

1. Option A:

   • This choice is correct when interpreting confidence intervals. It states that we are 99.99% confident that the population mean pH of rainwater lies between two values. This is the standard interpretation of a confidence interval.

2. Option B:

   • This is incorrect. It implies that 99.99% of sample means will lie between two values, but that’s not how we interpret confidence intervals for population means.

3. Option C:

   • This is incorrect as well. It suggests that there is a 99.99% probability that the true mean lies in a specific interval, which is a misunderstanding of confidence intervals. The interval itself is fixed, and the confidence level refers to how often similarly constructed intervals will capture the population mean in the long run.

Final Answer:

• The correct choice is A:

  "There is 99.99% confidence that the population mean pH of rainwater is between ___ and ___."

To proceed, you would need to calculate the specific values for the interval using the formula provided.

Would you like further details on how to calculate this confidence interval, or do you have sample statistics? Here are 5 related questions:

1. How do you compute the Z-score for different confidence levels?
2. What is the difference between a confidence interval and a prediction interval?
3. What are the assumptions behind using the Z-distribution for confidence intervals?
4. How does increasing the confidence level affect the width of the confidence interval?
5. What steps can be taken if the sample size is small when constructing a confidence interval?

Tip: A higher confidence level will result in a wider confidence interval, providing more certainty that the interval contains the population mean.