n a survey of 2568 adults in a recent​ year, 1306 say they have made a New​ Year's resolution.
Construct​ 90% and​ 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.
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Part 1
The​ 90% confidence interval for the population proportion p is left parenthesis 0.492 comma 0.524 right parenthesis
  
  
.
​(Round to three decimal places as​ needed.)
Part 2
The​ 95% confidence interval for the population proportion p is left parenthesis 0.489 comma 0.527 right parenthesis
  
  
.
​(Round to three decimal places as​ needed.)
Part 3
With the given​ confidence, it can be said that the 
population proportion
 of adults who say they have made a New​ Year's resolution is 
between the endpoints
 of the given confidence interval.
Part 4
Compare the widths of the confidence intervals. Choose the correct answer below.
A.
The​ 90% confidence interval is wider.
B.
The​ 95% confidence interval is wider.
C.
The confidence intervals are the same width.
D.
The confidence intervals cannot be compared.

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.

In a survey of 2568 adults in a recent year, 1306 say they have made a New Year's resolution. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals.

This problem demonstrates how to calculate and interpret 90% and 95% confidence intervals for a population proportion. Using a sample of 2568 adults, where 1306 reported making a New Year's resolution, we calculate the intervals and compare their widths. Ideal for college-level statistics or advanced high school students.

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