Confidence Interval of Proportion Calculator

Confidence Interval for a Proportion Calculator

90% 95% 99%

function calculateConfidenceInterval() { var sampleSize = parseFloat(document.getElementById('sampleSize').value); var numberOfSuccesses = parseFloat(document.getElementById('numberOfSuccesses').value); var confidenceLevel = parseFloat(document.getElementById('confidenceLevel').value); var resultDiv = document.getElementById('result'); // Input validation if (isNaN(sampleSize) || sampleSize 0).'; return; } if (isNaN(numberOfSuccesses) || numberOfSuccesses sampleSize) { resultDiv.innerHTML = 'Number of Successes cannot be greater than Sample Size.'; return; } if (isNaN(confidenceLevel) || confidenceLevel = 100) { resultDiv.innerHTML = 'Please select a valid Confidence Level.'; return; } // Calculate sample proportion (p-hat) var p_hat = numberOfSuccesses / sampleSize; // Determine Z-score based on confidence level var z_score; if (confidenceLevel === 90) { z_score = 1.645; } else if (confidenceLevel === 95) { z_score = 1.96; } else if (confidenceLevel === 99) { z_score = 2.576; } else { // Fallback or more advanced calculation for other levels could be added here // For simplicity, we'll stick to the common ones provided in the dropdown resultDiv.innerHTML = 'Unsupported Confidence Level selected.'; return; } // Calculate Standard Error (SE) // Handle edge case where p_hat is 0 or 1 to avoid sqrt(negative) or division by zero issues var standardError; if (p_hat === 0 || p_hat === 1) { standardError = 0; // If proportion is 0 or 1, there's no variability } else { standardError = Math.sqrt((p_hat * (1 – p_hat)) / sampleSize); } // Calculate Margin of Error (ME) var marginOfError = z_score * standardError; // Calculate Confidence Interval var lowerBound = p_hat – marginOfError; var upperBound = p_hat + marginOfError; // Ensure bounds are within [0, 1] for proportions lowerBound = Math.max(0, lowerBound); upperBound = Math.min(1, upperBound); // Display results resultDiv.innerHTML = '

Results:

' + 'Sample Proportion (p̂): ' + (p_hat * 100).toFixed(2) + '%' + 'Margin of Error (ME): ' + (marginOfError * 100).toFixed(2) + '%' + 'Confidence Interval: (' + (lowerBound * 100).toFixed(2) + '% , ' + (upperBound * 100).toFixed(2) + '%)' + 'We are ' + confidenceLevel + '% confident that the true population proportion lies between ' + (lowerBound * 100).toFixed(2) + '% and ' + (upperBound * 100).toFixed(2) + '%. '; }

Understanding the Confidence Interval for a Proportion

When you conduct a survey or an experiment, you often want to estimate the proportion of a certain characteristic in a larger population based on a smaller sample. For example, you might want to know the percentage of voters who support a particular candidate, or the proportion of products that are defective in a large batch.

A sample proportion (p̂) is a point estimate – a single value that estimates the true population proportion. However, because it's based on a sample, it's unlikely to be exactly equal to the true population proportion. This is where a confidence interval comes in.

What is a Confidence Interval?

A confidence interval for a proportion provides a range of values within which the true population proportion is likely to fall, along with a certain level of confidence. For instance, a "95% confidence interval" means that if you were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population proportion.

Key Components of the Calculation

The calculation of a confidence interval for a proportion typically involves these steps:

1. Sample Proportion (p̂): This is the proportion of "successes" (the characteristic you're interested in) in your sample. It's calculated as:

   p̂ = x / n

   Where:
   • x = Number of successes in the sample
   • n = Total sample size
2. Confidence Level: This is the probability that the confidence interval contains the true population proportion. Common confidence levels are 90%, 95%, and 99%.
3. Z-score (Critical Value): This value corresponds to your chosen confidence level and is derived from the standard normal distribution. It defines how many standard errors away from the mean you need to go to capture the desired percentage of the distribution.
   • For 90% confidence, Z ≈ 1.645
   • For 95% confidence, Z ≈ 1.96
   • For 99% confidence, Z ≈ 2.576
4. Standard Error (SE): This measures the typical distance between the sample proportion and the true population proportion. It's calculated as:

   SE = √[ p̂ * (1 – p̂) / n ]

5. Margin of Error (ME): This is the "plus or minus" amount that defines the width of the confidence interval. It's calculated by multiplying the Z-score by the Standard Error:

   ME = Z * SE

6. Confidence Interval: Finally, the interval is constructed by adding and subtracting the Margin of Error from the Sample Proportion:

   Confidence Interval = p̂ ± ME

Interpreting the Results

If your calculator output for a 95% confidence interval is (45%, 55%), it means you are 95% confident that the true proportion of the population with the characteristic you're studying lies between 45% and 55%. It does NOT mean there's a 95% chance the true proportion is within this specific interval, but rather that 95% of such intervals constructed from repeated sampling would contain the true proportion.

Factors Affecting the Confidence Interval

• Sample Size (n): A larger sample size generally leads to a smaller standard error and thus a narrower confidence interval, providing a more precise estimate.
• Confidence Level: A higher confidence level (e.g., 99% vs. 95%) will result in a wider confidence interval, as you need a larger range to be more certain that you've captured the true population proportion.
• Sample Proportion (p̂): The standard error is largest when p̂ is close to 0.5 (50%) and smaller when p̂ is closer to 0 or 1.

Example Usage

Imagine a political poll where 1000 randomly selected voters were asked if they approve of the current president. 550 of them said "yes". We want to calculate a 95% confidence interval for the true proportion of voters who approve.

• Sample Size (n): 1000
• Number of Successes (x): 550
• Confidence Level: 95%

Using the calculator above:

• Sample Proportion (p̂) = 550 / 1000 = 0.55 (55%)
• For 95% confidence, Z-score = 1.96
• Standard Error (SE) = √[ 0.55 * (1 – 0.55) / 1000 ] ≈ 0.0157
• Margin of Error (ME) = 1.96 * 0.0157 ≈ 0.0308
• Confidence Interval = 0.55 ± 0.0308 = (0.5192, 0.5808)

So, we are 95% confident that the true proportion of voters who approve of the president is between 51.92% and 58.08%.