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Let's analyze the problem step-by-step.

### Problem Summary:
- **Survey information**: 
  - Total adults surveyed = 2971
  - Adults who started paying bills online in the last year = 1476
- **Task**: Construct a 99% confidence interval for the population proportion.

### Steps to Solve:

1. **Sample Proportion (p-hat)**:
   The sample proportion is calculated as:
   \[
   \hat{p} = \frac{1476}{2971}
   \]

2. **Standard Error (SE)**:
   The standard error for a proportion is calculated as:
   \[
   SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
   \]
   where \( n = 2971 \) is the total sample size.

3. **Z-Score for 99% Confidence**:
   The critical value (Z-score) for a 99% confidence interval is approximately 2.576.

4. **Confidence Interval Formula**:
   The confidence interval is given by:
   \[
   CI = \hat{p} \pm Z \times SE
   \]

### Calculation:

Let's compute these values.

#### 1. Sample proportion (\(\hat{p}\)):
\[
\hat{p} = \frac{1476}{2971} \approx 0.496
\]

#### 2. Standard error (SE):
\[
SE = \sqrt{\frac{0.496(1 - 0.496)}{2971}} \approx 0.0091
\]

#### 3. Confidence Interval:
\[
CI = 0.496 \pm 2.576 \times 0.0091
\]
\[
CI = 0.496 \pm 0.0234
\]
So, the 99% confidence interval is approximately:
\[
(0.4726, 0.5194)
\]

### Interpretation of Results:
- **Answer C** is correct: "With 99% confidence, it can be said that the population proportion of adults who say they have started paying bills online in the last year is between the endpoints of the given confidence interval."

Do you want more details or have any other questions?

### Related Questions:
1. How is the Z-score for a confidence level calculated?
2. Why is the sample size important in determining the confidence interval?
3. What happens to the confidence interval if we increase the sample size?
4. How would the confidence interval change with a 95% confidence level?
5. What assumptions are necessary for constructing a confidence interval for a proportion?

**Tip:** Confidence intervals provide a range in which the true population parameter likely falls, helping to quantify uncertainty in sample data.

In a survey of 2971 adults, 1476 say they have started paying bills online in the last year. Construct a 99% confidence interval for the population proportion. Interpret the results.

Learn how to construct a 99% confidence interval for the population proportion of adults who started paying bills online. This problem involves calculating the sample proportion, standard error, and using the Z-score for a 99% confidence level. Includes detailed step-by-step solution and interpretation.

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