Population Confidence Interval Calculator

What is a Population Confidence Interval?

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.

Formulas and Their Meanings

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.

Calculation Steps

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

Example Calculation

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

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

Visual Representation

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.