A study on the effects of acid rain on trees in a particular forest found that 32 out of 77 sampled trees showed damage from acid rain.
If you were to create a 98% confidence interval for the true proportion of trees in the forest damaged by acid rain, what is the margin of error?
[Giver your answer to 3 decimal places]

To calculate the margin of error for a 98% confidence interval for the true proportion of trees damaged by acid rain, we need to use the formula for the margin of error for a proportion:

\[
\text{Margin of Error (ME)} = Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

where:

- \(\hat{p}\) is the sample proportion.
- \(n\) is the sample size.
- \(Z\) is the Z-score corresponding to the desired confidence level (98% in this case).

### Step 1: Calculate the Sample Proportion \(\hat{p}\)

\[
\hat{p} = \frac{\text{Number of damaged trees}}{\text{Total number of trees sampled}} = \frac{32}{77}
\]

\[
\hat{p} \approx 0.416
\]

### Step 2: Find the Z-Score for a 98% Confidence Level

The Z-score for a 98% confidence level can be found using a standard normal distribution table or calculator. For a 98% confidence level, the Z-score is approximately 2.326.

### Step 3: Calculate the Margin of Error

Now, plug the values into the margin of error formula:

\[
\text{ME} = 2.326 \times \sqrt{\frac{0.416 \times (1 - 0.416)}{77}}
\]

Let's calculate this step-by-step.The margin of error for the 98% confidence interval for the true proportion of trees damaged by acid rain is **0.131** (rounded to three decimal places).

Would you like more details or have any questions? Here are some related topics:

1. How to calculate confidence intervals for proportions.
2. Understanding Z-scores and their role in confidence intervals.
3. Difference between 95%, 98%, and 99% confidence levels.
4. The importance of sample size in statistical inference.
5. Applications of confidence intervals in environmental studies.

**Tip:** Always round the margin of error to the required number of decimal places, as specified in statistical problems.

Learn how to calculate the margin of error for a 98% confidence interval in a study on the effects of acid rain on trees. This problem involves statistical concepts, including proportions and confidence intervals, and provides a step-by-step solution. Suitable for high school students studying statistics.

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