Confidence Intervals
A Confidence Interval (CI) is a range of values computed from sample statistics that is likely to contain the true population parameter. Rather than offering a single point estimate, it quantifies the margin of error and standard error to establish statistical confidence.
- estimation ranges: A confidence interval is an estimated range of values likely to contain the true population parameter.
- confidence level: The percentage (e.g., 90%, 95%, 99%) indicates the long-run probability of intervals capturing the parameter.
- sample dimension: Larger sample sizes (n) reduce the standard error, producing narrower, more precise intervals.
Confidence Interval Formula
| Symbol | Meaning & Description |
|---|---|
| CI | Confidence Interval range |
| x̄ | Sample Mean |
| z* | Critical value based on confidence level |
| s | Sample Standard Deviation |
| n | Sample Size |
SE = s / √n = 15 / √30 = 2.7386
E = z* × SE = 1.96 × 2.7386 = 5.3677
Lower = 100 - 5.3677 = 94.6323. Upper = 100 + 5.3677 = 105.3677. CI = [94.6323, 105.3677]
Confidence Interval Calculator Sandbox
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SE = s / √n = 15 / √30 = 2.7386
E = z* × SE = 1.96 × 2.7386 = 5.3677
Lower = 100 - 5.3677 = 94.6323. Upper = 100 + 5.3677 = 105.3677. CI = [94.6323, 105.3677]
Misinterpreting the Probability: Saying there is a 95% chance the true population mean lies inside the *calculated* interval is incorrect. The true mean is a fixed value. The correct interpretation is that 95% of similarly constructed sample intervals will contain the true population mean.
Confidence Interval Calculator
Calculate the confidence interval for a sample mean with standard deviation and size bounds.