To construct a 99% confidence interval for the standard deviation of the pH of rainwater, we will use the chi-square distribution because the data is assumed to be normally distributed, and we are working with a sample standard deviation.

Given:

• Sample size $n = 12$
• Sample standard deviation $s = 0.332$
• pH values:
  4.66, 5.22, 5.10, 4.71, 4.77, 4.86, 5.82, 4.85, 5.10, 4.84, 4.76, 4.64

Steps to Calculate the Confidence Interval for the Standard Deviation:

1. Calculate the sample variance: $s^2 = (0.332)^2 = 0.110224$

2. Degrees of freedom: $df = n - 1 = 12 - 1 = 11$

3. Determine the chi-square critical values for a 99% confidence interval. Using the chi-square distribution table (or a calculator), we find the chi-square critical values for 11 degrees of freedom at the 0.5% and 99.5% percentiles (since we want 99% confidence, we look at the outer 0.5% tails):

   • $\chi^2_{0.005,11} \approx 3.053$
   • $\chi^2_{0.995,11} \approx 26.757$

4. Compute the confidence interval for the population variance $\sigma^2$: $\frac{(n - 1) s^2}{\chi^2_{0.995}} \leq \sigma^2 \leq \frac{(n - 1) s^2}{\chi^2_{0.005}}$ Substituting the values: $\frac{11 \times 0.110224}{26.757} \leq \sigma^2 \leq \frac{11 \times 0.110224}{3.053}$ $0.0453 \leq \sigma^2 \leq 0.3976$

5. Take the square root to find the confidence interval for the population standard deviation $\sigma$: $\sqrt{0.0453} \leq \sigma \leq \sqrt{0.3976}$ $0.213 \leq \sigma \leq 0.631$

Interpretation:

We are 99% confident that the true population standard deviation of the pH of rainwater in this region lies between 0.213 and 0.631.

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Would you like further details or have any questions on this process?

Here are 5 questions for you to explore based on this:

1. How do changes in sample size affect the width of the confidence interval?
2. What assumptions must hold for the chi-square distribution to be applicable in this context?
3. How would the interval change if the confidence level were reduced to 95%?
4. Can you compare this interval with one for the mean and discuss the differences?
5. What role do outliers play in computing confidence intervals for standard deviation?

Tip: Increasing the confidence level widens the confidence interval but also increases certainty in capturing the true parameter.