Acid rain, caused by the reaction of certain air pollutants with rainwater, appears to be a growing problem in Eastern Canada. (Acid rain affects the soil and causes corrosion on exposed metal surfaces.) Pure rain falling through clean air registers a pH value of 5.7 (pH is a measure of acidity: 0 is acid; 14 is alkaline). Suppose water samples from 40 rainfalls are analyzed for pH, and x and s are equal to 3.1 and 0.6, respectively. Find a 99% confidence interval for the mean pH in rainfall. (Round your answers to three decimal places.)

To find a 99% confidence interval for the mean pH in rainfall based on the sample data, we can use the following formula for the confidence interval:

\[
\text{CI} = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}}
\]

where:
- \(\bar{x}\) is the sample mean,
- \(s\) is the sample standard deviation,
- \(n\) is the sample size,
- \(t\) is the critical value from the \(t\)-distribution with \(n - 1\) degrees of freedom.

### Step 1: Identify Given Information
- \(\bar{x} = 3.1\)
- \(s = 0.6\)
- \(n = 40\)
- Confidence level = 99%

### Step 2: Find the Critical Value \(t\)
Since we need a 99% confidence interval, the significance level is \( \alpha = 1 - 0.99 = 0.01 \).
With \(n - 1 = 39\) degrees of freedom, we look up the critical \(t\)-value for a two-tailed test with \(\alpha/2 = 0.005\).

Using a \(t\)-table or calculator:
- \(t_{0.005, 39} \approx 2.708\).

### Step 3: Calculate the Margin of Error
The margin of error is calculated as:
\[
\text{Margin of Error} = t \cdot \frac{s}{\sqrt{n}}
\]

Substituting in our values:
\[
\text{Margin of Error} = 2.708 \cdot \frac{0.6}{\sqrt{40}} \approx 2.708 \cdot 0.0949 \approx 0.257
\]

### Step 4: Calculate the Confidence Interval
Now, we can find the confidence interval:
\[
\text{CI} = \bar{x} \pm \text{Margin of Error} = 3.1 \pm 0.257
\]

So, the 99% confidence interval for the mean pH is approximately:
\[
(3.100 - 0.257, 3.100 + 0.257) = (2.843, 3.357)
\]

### Answer
The 99% confidence interval for the mean pH in rainfall is:
\[
(2.843, 3.357)
\]

Would you like further explanation on any part? 

Here are five related questions you might consider:
1. How does changing the confidence level affect the width of the confidence interval?
2. Why is the \(t\)-distribution used instead of the normal distribution in this case?
3. What would happen to the confidence interval if we had a larger sample size?
4. How do different levels of pH in rain impact various environments?
5. What factors contribute most to acid rain formation?

**Tip:** For greater accuracy in estimating population parameters, increase the sample size whenever possible.

Calculate the 99% confidence interval for the mean pH of rainfall samples in Eastern Canada, with a sample mean of 3.1 and a standard deviation of 0.6. Understand confidence intervals, margin of error, and t-distribution with step-by-step guidance. Suitable for students in Grades 11-12.

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