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We are tasked with finding a 98% confidence interval for the population mean amount of ground toxins based on the provided sample data. The steps to calculate this confidence interval are as follows:

1. **Determine the sample mean (\(\bar{x}\)):** 
   First, we calculate the mean of the given data points.

2. **Determine the sample standard deviation (\(s\)):** 
   We need the standard deviation of the sample to estimate the population standard deviation.

3. **Find the critical value for 98% confidence level:** 
   Since the population standard deviation is not given and the sample size is less than 30, we'll use the t-distribution. The degrees of freedom for the t-distribution will be \(df = n - 1 = 17\).

4. **Calculate the margin of error (ME):** 
   The margin of error is calculated as:
   \[
   \text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
   \]
   where \(t_{\alpha/2}\) is the t-critical value for 98% confidence, \(s\) is the sample standard deviation, and \(n\) is the sample size.

5. **Construct the confidence interval:** 
   The confidence interval is given by:
   \[
   \left( \bar{x} - \text{ME}, \bar{x} + \text{ME} \right)
   \]

Let me now calculate the sample mean, sample standard deviation, and the confidence interval.The calculations yield the following results:

- **Sample Mean (\(\bar{x}\))**: 20.1 ppm
- **Sample Standard Deviation (s)**: 11.2 ppm
- **T-critical value** (for 98% confidence with 17 degrees of freedom): 2.567
- **Margin of Error (ME)**: 6.8 ppm

Thus, the 98% confidence interval for the population mean amount of ground toxins is:

\[
13.3 \, \text{ppm} < \mu < 26.9 \, \text{ppm}
\]

Would you like more details on any step or have questions?

Here are 5 related questions you could explore:
1. How does increasing the sample size affect the confidence interval?
2. What if the confidence level was 95% instead of 98%? How would the interval change?
3. How does the t-distribution differ from the normal distribution?
4. Why do we use the t-distribution instead of the z-distribution in this case?
5. How is the margin of error related to the sample standard deviation?

**Tip**: The margin of error decreases as the sample size increases, providing a more precise estimate of the population mean.

Scientists gather 18 different soil samples from a region to test for ground toxins in the water supply. The amount of toxins present in ppm (parts per million) are given in the data below. Find the 98% confidence interval for the population mean amount of ground toxins. You may assume the population is normally distributed.

Learn how to calculate a 98% confidence interval for the mean amount of ground toxins from 18 soil samples. This problem involves key statistical concepts like sample mean, sample standard deviation, and using the t-distribution for small sample sizes.

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