Let's calculate the confidence intervals with results rounded to four decimal places.

### Step-by-Step Calculations:

1. **Calculate the Standard Error (SE):**
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{7}{\sqrt{13}} \approx 1.9403
   \]

2. **Determine the critical value \( Z_{\alpha/2} \) for different confidence levels:**
   - For a 90% confidence level, \( Z_{\alpha/2} \approx 1.645 \)
   - For a 95% confidence level, \( Z_{\alpha/2} \approx 1.960 \)
   - For a 99% confidence level, \( Z_{\alpha/2} \approx 2.576 \)

3. **Calculate the confidence intervals:**

   - **90% Confidence Interval:**
     \[
     40 \pm 1.645 \times 1.9403 = 40 \pm 3.1891
     \]
     \[
     \text{90% CI} = (36.8109, 43.1891)
     \]

   - **95% Confidence Interval:**
     \[
     40 \pm 1.960 \times 1.9403 = 40 \pm 3.8022
     \]
     \[
     \text{95% CI} = (36.1978, 43.8022)
     \]

   - **99% Confidence Interval:**
     \[
     40 \pm 2.576 \times 1.9403 = 40 \pm 4.9975
     \]
     \[
     \text{99% CI} = (35.0025, 44.9975)
     \]

### Summary with Four Decimal Places:
- **90% CI:** \( (36.8109, 43.1891) \)
- **95% CI:** \( (36.1978, 43.8022) \)
- **99% CI:** \( (35.0025, 44.9975) \)

Would you like more details on any of these calculations, or do you have any other questions?

### Follow-up Questions:
1. How does the sample size impact the precision of confidence intervals?
2. What would happen if we increased the confidence level to 99.9%?
3. Why is it important to round to a consistent number of decimal places?
4. How do you interpret the results of these confidence intervals in practical terms?
5. What assumptions are made when calculating these confidence intervals?
6. What if the sample size were smaller, how would that affect the intervals?
7. Can confidence intervals overlap, and what does that signify?
8. How does the concept of confidence intervals apply to real-world scenarios?

### Tip:
When performing calculations involving decimals, always maintain consistency in rounding to avoid errors in reporting and interpretation.

Learn how to calculate confidence intervals with results rounded to four decimal places. Includes step-by-step calculations for 90%, 95%, and 99% confidence levels. Suitable for advanced learners and professionals.

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