A population confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. It provides a measure of uncertainty around a sample estimate.

1. Sample Proportion (\(p\)): \[p = \frac{x}{n}\] Where \(x\) is the number of successes and \(n\) is the sample size.

2. Standard Error (\(SE\)): \[SE = \sqrt{\frac{p(1-p)}{n}}\] This measures the variability of the sample proportion.

3. Margin of Error (\(ME\)): \[ME = z \times SE\] Where \(z\) is the z-score for the chosen confidence level.

4. Confidence Interval (\(CI\)): \[CI = p \pm ME\] This gives the lower and upper bounds of the interval.

- Calculate the sample proportion.
- Determine the z-score for the chosen confidence level.
- Calculate the standard error.
- Compute the margin of error.
- Determine the lower and upper bounds of the confidence interval.

Let's calculate a 95% confidence interval for a sample size of 1000 with 600 successes.

- Sample proportion: \(p = \frac{600}{1000} = 0.6\)
- Z-score for 95% confidence: \(z = 1.96\)
- Standard Error: \(SE = \sqrt{\frac{0.6(1-0.6)}{1000}} = 0.0155\)
- Margin of Error: \(ME = 1.96 \times 0.0155 = 0.0304\)
- Confidence Interval: \(0.6 \pm 0.0304 = (0.5696, 0.6304)\)

This diagram illustrates the 95% confidence interval for the example calculation. The blue area represents the confidence interval, and the red line shows the sample proportion.

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