Confidence Interval
A confidence interval (CI) is a range of plausible values for an unknown population parameter, constructed from sample data so that the procedure captures the true parameter with a specified probability (the confidence level, e.g., 95%). Crucially, the confidence level refers to the long-run success rate of the procedure — not the probability that a particular interval contains the parameter. Confidence intervals are used throughout science, medicine, and engineering to quantify estimation uncertainty.
Key Formula
CI = x̄ ± z_(α/2) × (σ / √n)
LaTeX: \bar{x} \pm z_{\alpha/2} \cdot \dfrac{\sigma}{\sqrt{n}}
| Symbol | Meaning | Unit |
|---|---|---|
| \bar{x} | Sample mean | same as data |
| z_{\alpha/2} | Critical Z-value for confidence level (e.g., 1.96 for 95%) | dimensionless |
| \sigma | Population standard deviation | same as data |
| n | Sample size | count |
Worked Example
Problem
A survey of 100 students finds a mean study time of 3.2 hours/day with σ = 1.0 hour. Construct a 95% confidence interval for the population mean.
Solution
Step 1: x̄ = 3.2, σ = 1.0, n = 100, z₀.₀₂₅ = 1.96. Step 2: Standard error = σ/√n = 1.0/√100 = 0.10. Step 3: Margin of error = 1.96 × 0.10 = 0.196. Step 4: CI = 3.2 ± 0.196 = (3.004, 3.396).
Answer
95% CI: (3.00 hours, 3.40 hours) — the true mean study time likely lies in this range
Critical Z-values for Common Confidence Levels
| Confidence Level | α | α/2 | z_(α/2) |
|---|---|---|---|
| 90% | 0.10 | 0.05 | 1.645 |
| 95% | 0.05 | 0.025 | 1.960 |
| 98% | 0.02 | 0.010 | 2.326 |
| 99% | 0.01 | 0.005 | 2.576 |
| 99.9% | 0.001 | 0.0005 | 3.291 |
Interactive Tools
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Related Terms
Z-score
A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of its distribution. It standardises values from different distributions, enabling direct comparison by placing them on a common scale. Z-scores are widely used in quality control, hypothesis testing, and the construction of standard normal tables.
Hypothesis Testing
Hypothesis testing is a formal statistical procedure for making decisions about a population parameter based on sample data, by evaluating evidence against a null hypothesis (H₀) in favour of an alternative hypothesis (H₁). A test statistic is computed and compared to a critical value or converted to a p-value; if the result is statistically significant (p < α), the null hypothesis is rejected. It underpins scientific research, clinical trials, quality assurance, and data-driven decision-making across all quantitative disciplines.
Central Limit Theorem
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size n increases, regardless of the shape of the underlying population distribution, provided the population has a finite mean and variance. For most practical purposes, normality is achieved when n ≥ 30. The CLT is the theoretical foundation for Z-tests, t-tests, confidence intervals, and virtually all classical inferential statistics.
The concept and terminology of "confidence interval" was introduced by Jerzy Neyman in 1937, from the Latin "confidere" (to trust fully). Neyman deliberately avoided Bayesian language to distinguish frequency-based intervals from probability statements about fixed but unknown parameters.