Calculating 95 Confidence Interval

95% Confidence Interval Calculator

function calculateConfidenceInterval() { var sampleMean = parseFloat(document.getElementById("sampleMean").value); var sampleStdDev = parseFloat(document.getElementById("sampleStdDev").value); var sampleSize = parseInt(document.getElementById("sampleSize").value); var resultDiv = document.getElementById("result"); if (isNaN(sampleMean) || isNaN(sampleStdDev) || isNaN(sampleSize)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } if (sampleStdDev < 0) { resultDiv.innerHTML = "Sample Standard Deviation cannot be negative."; return; } if (sampleSize <= 1) { resultDiv.innerHTML = "Sample Size must be greater than 1 for standard deviation calculation."; return; } // For a 95% confidence interval, the Z-score is approximately 1.96 var zScore = 1.96; // Calculate Standard Error var standardError = sampleStdDev / Math.sqrt(sampleSize); // Calculate Margin of Error var marginOfError = zScore * standardError; // Calculate Lower and Upper Bounds var lowerBound = sampleMean – marginOfError; var upperBound = sampleMean + marginOfError; resultDiv.innerHTML = "

95% Confidence Interval:

" + "Lower Bound: " + lowerBound.toFixed(4) + "" + "Upper Bound: " + upperBound.toFixed(4) + "" + "We are 95% confident that the true population mean lies between " + lowerBound.toFixed(4) + " and " + upperBound.toFixed(4) + "."; } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 500px; margin: 30px auto; border: 1px solid #e0e0e0; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 25px; font-size: 1.8em; } .calc-input-group { margin-bottom: 18px; display: flex; flex-direction: column; } .calc-input-group label { margin-bottom: 8px; color: #555; font-size: 1em; font-weight: bold; } .calc-input-group input[type="number"] { padding: 12px; border: 1px solid #ccc; border-radius: 6px; font-size: 1.1em; width: 100%; box-sizing: border-box; transition: border-color 0.3s ease; } .calc-input-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25); } .calc-button { background-color: #007bff; color: white; padding: 14px 25px; border: none; border-radius: 6px; cursor: pointer; font-size: 1.15em; font-weight: bold; width: 100%; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 15px; } .calc-button:hover { background-color: #0056b3; transform: translateY(-2px); } .calc-button:active { transform: translateY(0); } .calc-result { margin-top: 30px; padding: 20px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 8px; font-size: 1.1em; color: #155724; line-height: 1.6; } .calc-result h3 { color: #0f5132; margin-top: 0; margin-bottom: 15px; font-size: 1.4em; text-align: center; } .calc-result p { margin-bottom: 8px; } .calc-result p strong { color: #0f5132; } .calc-result .error { color: #dc3545; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 5px; text-align: center; }

Understanding the 95% Confidence Interval

In statistics, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unobserved population parameter. The 95% confidence interval is one of the most commonly used intervals, indicating that if you were to take many samples and compute a confidence interval for each, approximately 95% of those intervals would contain the true population parameter.

What Does 95% Confidence Mean?

When we say we are "95% confident" that the true population mean falls within a certain range, it means that if we were to repeat our sampling process many times, 95% of the confidence intervals constructed would contain the actual population mean. It does NOT mean there is a 95% probability that the true mean falls within the calculated interval from a single sample. Instead, it reflects the reliability of the estimation procedure.

Components of the 95% Confidence Interval Calculation

The calculation of a 95% confidence interval for a population mean (when the population standard deviation is unknown but the sample size is sufficiently large, or when the population standard deviation is known) typically involves three key pieces of information:

1. Sample Mean (x̄): This is the average value of your observed data points from the sample. It serves as our best point estimate for the true population mean.
2. Sample Standard Deviation (s): This measures the amount of variation or dispersion of the data points within your sample. A larger standard deviation indicates more spread-out data.
3. Sample Size (n): This is the total number of observations or data points in your sample. A larger sample size generally leads to a narrower (more precise) confidence interval, assuming other factors remain constant.

The Formula

The general formula for a confidence interval for the mean is:

CI = Sample Mean ± (Z * (Sample Standard Deviation / sqrt(Sample Size)))

Where:

• Z is the Z-score corresponding to the desired confidence level. For a 95% confidence interval, the Z-score is approximately 1.96. This value comes from the standard normal distribution and represents the number of standard deviations from the mean that encompass 95% of the data.
• sqrt(Sample Size) is the square root of the sample size.
• The term (Sample Standard Deviation / sqrt(Sample Size)) is known as the Standard Error of the Mean (SE). It estimates the standard deviation of the sample mean's distribution.
• The term (Z * SE) is called the Margin of Error (ME). It defines the range above and below the sample mean that forms the confidence interval.

How to Interpret the Results

Once you calculate the lower and upper bounds of the 95% confidence interval, you can state your conclusion. For example, if your calculator yields a 95% CI of (169.02, 170.98) for the average height of students, you would say: "We are 95% confident that the true average height of all students in the population lies between 169.02 cm and 170.98 cm."

Example Calculation

Let's consider a scenario where a researcher wants to estimate the average weight of a certain species of fish in a large lake. They catch and weigh a sample of fish:

• Sample Mean (x̄): 1.5 kg
• Sample Standard Deviation (s): 0.3 kg
• Sample Size (n): 60 fish

Using the formula:

1. Standard Error (SE): SE = s / sqrt(n) = 0.3 / sqrt(60) ≈ 0.3 / 7.746 ≈ 0.0387 kg
2. Margin of Error (ME): ME = Z * SE = 1.96 * 0.0387 ≈ 0.07585 kg
3. Lower Bound: x̄ - ME = 1.5 - 0.07585 = 1.42415 kg
4. Upper Bound: x̄ + ME = 1.5 + 0.07585 = 1.57585 kg

Therefore, the 95% confidence interval for the average weight of fish in the lake is approximately (1.424 kg, 1.576 kg).

This calculator provides a quick and easy way to determine these bounds, helping you make informed statistical inferences from your sample data.