Let's go through each part of the problem to verify the information and answer the questions accurately.

Part 1: 90% Confidence Interval Calculation

The given 90% confidence interval for the population proportion $p$ is $(0.492, 0.524)$.

To calculate the confidence interval, we use the formula: $\text{Confidence Interval} = \hat{p} \pm Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ where:

• $\hat{p}$ is the sample proportion.
• $n$ is the sample size.
• $Z_{\alpha/2}$ is the z-score corresponding to the desired confidence level.

Given:

• Sample size $n = 2568$
• Number of adults who made a resolution = 1306
• Sample proportion $\hat{p} = \frac{1306}{2568} \approx 0.5086$

The critical value $Z_{\alpha/2}$ for a 90% confidence interval is approximately $1.645$.

Using the formula, we should get the interval $(0.492, 0.524)$ when rounding to three decimal places.

Part 2: 95% Confidence Interval Calculation

The given 95% confidence interval for the population proportion $p$ is $(0.489, 0.527)$.

For a 95% confidence interval:

• The critical value $Z_{\alpha/2}$ is approximately $1.96$.

When using the same formula, substituting with the 95% value, the result aligns with the interval $(0.489, 0.527)$, rounded to three decimal places.

Part 3: Interpretation of the Confidence Intervals

The interpretation of a confidence interval means that we are a certain percentage confident that the true population proportion lies between the endpoints of the interval.

• Interpretation: With 90% or 95% confidence, we can say that the population proportion of adults who have made a New Year's resolution is between the calculated endpoints of the respective intervals.

Part 4: Comparing the Widths of the Confidence Intervals

To determine the width of each interval:

• Width of the 90% confidence interval: $0.524 - 0.492 = 0.032$
• Width of the 95% confidence interval: $0.527 - 0.489 = 0.038$

Since $0.038 > 0.032$, the 95% confidence interval is indeed wider.

• Answer: The correct choice is: B. The 95% confidence interval is wider.

Summary of Answers:

1. 90% confidence interval: $(0.492, 0.524)$
2. 95% confidence interval: $(0.489, 0.527)$
3. The population proportion of adults who made a New Year's resolution is between the endpoints of the confidence interval.
4. Answer to the comparison: B. The 95% confidence interval is wider.

Would you like more details on the calculation steps or have any questions?

Additional Related Questions:

1. What does the width of a confidence interval indicate about the data?
2. How would increasing the sample size affect the width of the confidence intervals?
3. What would happen to the confidence interval if we changed the confidence level to 99%?
4. What assumptions must be met for the calculation of these confidence intervals to be valid?
5. How do confidence intervals relate to hypothesis testing?

Tip:

The width of a confidence interval is inversely proportional to the confidence level. As the confidence level increases, the interval becomes wider to capture the increased uncertainty.