Confidence Interval Calculator

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain an unknown population parameter. It is calculated from a sample of data and provides a measure of the uncertainty associated with an estimate.

Formula and Its Components

The formula for a confidence interval for a population proportion is:

\[CI = p \pm z \times \sqrt{\frac{p(1-p)}{n}} \times \sqrt{\frac{N-n}{N-1}}\]

Where:

• \(p\) is the sample proportion
• \(z\) is the z-score for the desired confidence level
• \(n\) is the sample size
• \(N\) is the population size

Calculation Steps

1. Calculate the sample proportion (\(p\)) from your data.
2. Determine the z-score based on your desired confidence level.
3. Calculate the standard error using the formula above.
4. Multiply the standard error by the z-score to get the margin of error.
5. Subtract and add the margin of error to the sample proportion to get the lower and upper bounds of the confidence interval.

Example Calculation

Let's calculate a 95% confidence interval for a population proportion with these parameters:

• Sample size (\(n\)) = 100
• Population size (\(N\)) = 10,000
• Sample proportion (\(p\)) = 0.3 (30%)
• Confidence level = 95% (z-score = 1.96)

Plugging these values into our formula:

\[CI = 0.3 \pm 1.96 \times \sqrt{\frac{0.3(1-0.3)}{100}} \times \sqrt{\frac{10000-100}{10000-1}}\]

\[CI = 0.3 \pm 1.96 \times 0.0458 \times 0.9950\]

\[CI = 0.3 \pm 0.0891\]

\[CI = (0.2109, 0.3891)\]

Therefore, we can be 95% confident that the true population proportion falls between 21.09% and 38.91%.

Visual Representation

This diagram illustrates the 95% confidence interval for our example. The blue shaded area represents the interval (21.09% to 38.91%), while the red dashed line shows the sample proportion (30%).