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Interval estimation

Interval estimation is a fundamental technique in statistical inference that involves constructing a range of plausible values, known as a confidence interval, for an unknown population parameter based on sample data, accompanied by a confidence level, which indicates the proportion of such intervals from repeated sampling that would contain the true parameter value.^[1]^[2] Unlike point estimation, which provides a single best guess for the parameter, interval estimation quantifies the uncertainty around that estimate by incorporating a margin of error, often derived from the standard error of the estimator and critical values from probability distributions.^[1]^[2] The concept of interval estimation, particularly through confidence intervals, was formalized by Polish statistician Jerzy Neyman in 1934 as part of his work on the representative method in survey sampling, building on earlier contributions to hypothesis testing by collaborators Egon Pearson and William Gosset (Student).^[3] Neyman's approach emphasized using the sampling distribution of a statistic to define intervals with a guaranteed coverage probability, avoiding probabilistic statements about fixed parameters by focusing on long-run frequency properties across repeated samples.^[3]^[2] This framework distinguishes classical frequentist interval estimation from Bayesian alternatives, which produce credible intervals based on posterior probabilities rather than pre-data coverage guarantees.^[2] Common methods for constructing confidence intervals include the z-interval for population means when the variance is known, given by \bar{x} \pm z_{(1-\phi)/2} \sigma / \sqrt{n}, where \bar{x} is the sample mean, \sigma is the population standard deviation, n is the sample size, and z_{(1-\phi)/2} is the critical value from the standard normal distribution for confidence level \phi.^[1]^[2] For unknown variance, the t-interval substitutes the sample standard deviation s and uses t-distribution critical values: \bar{x} \pm t_{n-1,(1-\phi)/2} s / \sqrt{n}.^[2] Intervals for proportions, such as in binomial settings, follow \hat{p} \pm z_{(1-\phi)/2} \sqrt{\hat{p}(1-\hat{p})/n}, with adjustments for small samples like the Wilson score method.^[1]^[2] These methods rely on assumptions like normality or large sample sizes via the central limit theorem, and their width decreases with increasing sample size, reflecting greater precision.^[1]^[2] Interval estimation plays a crucial role in applied fields such as polling, quality control, and scientific research, where it provides not only an estimate but also a measure of reliability; for instance, during World War II, sequential estimation techniques akin to interval methods were used in the German tank problem to infer production numbers from captured serial data.^[1] Modern extensions include bootstrap resampling for non-parametric intervals and multivariable confidence sets, where the "size" is measured by volume rather than length, ensuring comprehensive uncertainty assessment in complex models.^[2]

Fundamentals

Definition and purpose

Interval estimation is a fundamental technique in statistical inference that estimates an unknown population parameter by constructing a range of values, or interval, within which the true parameter is likely to lie, based on data from a random sample. Unlike point estimation, which provides a single value as the best approximation of the parameter, interval estimation incorporates a measure of uncertainty, typically accompanied by a confidence level, which is the probability that the procedure would produce an interval containing the true parameter value over repeated random sampling from the same population. This approach is essential for understanding the precision of estimates derived from limited data.^[4] The primary purpose of interval estimation is to quantify the variability and reliability of statistical estimates, thereby supporting decision-making in scenarios involving uncertainty, such as scientific research, quality control, or policy analysis. By providing lower and upper bounds for parameters like population means, proportions, or variances, it allows users to evaluate the potential range of the true value and assess risks associated with sampling error. This method enhances the interpretability of data beyond point estimates, offering a more complete picture of inferential conclusions.^[2]^[5] The concept of interval estimation emerged in the early 20th century within the frequentist paradigm, with Jerzy Neyman providing its formal foundation in 1937 through his development of confidence intervals as a systematic tool for estimation.^[6] In standard notation, an interval estimator yields [L, U], where L is the lower bound and U is the upper bound, both functions of the sample data, designed such that, in repeated sampling, the proportion of intervals covering the true parameter \theta is equal to the predetermined confidence level.^[4]

Relation to point estimation

Point estimation involves deriving a single value from sample data to approximate an unknown population parameter \theta, serving as the best guess based on the observed information. A classic example is the use of the sample mean \bar{X} as a point estimate for the population mean \mu.^[7] Point estimates, however, do not incorporate the inherent variability from the sampling process, which can foster overconfidence by presenting the parameter as known with exact precision rather than as an approximation subject to error.^[8] Interval estimation extends this approach by constructing a range of plausible values around the point estimate, thereby explicitly accounting for sampling uncertainty and providing a more complete assessment of reliability.^[2] In practice, point and interval estimates complement each other, with the point estimate frequently acting as the center or midpoint of the interval to merge the specificity of a targeted value with the contextual breadth of uncertainty quantification.^[9] For estimating the population mean, this relationship is illustrated by expanding the point estimate \bar{X} into an interval of the form \bar{X} \pm z \cdot SE, where SE represents the standard error of the mean and z is a critical value from the standard normal distribution, without altering the central role of \bar{X}.^[8]

Frequentist Methods

Confidence intervals

A confidence interval is a range of plausible values for an unknown population parameter, derived from sample data in the frequentist paradigm. Formally, a (1-α)100% confidence interval for a parameter θ is a random interval [L, U], where L and U are functions of the observed data, such that the probability that the interval contains the true θ equals 1-α when considering repeated sampling from the population. This long-run frequency interpretation was introduced by Jerzy Neyman as a method to quantify estimation uncertainty without assuming a prior distribution on θ.^[10] Confidence intervals are typically constructed using pivotal quantities, which are functions of the data and the parameter whose probability distributions are free of unknown parameters. For the mean μ of a normal distribution N(μ, σ²) with known σ, the pivotal quantity is the standardized sample mean Z = √n (¯X - μ)/σ, which follows a standard normal distribution N(0,1). Inverting this pivot yields the confidence interval ¯X ± z_{α/2} σ / √n, where z_{α/2} is the (1-α/2) quantile of the standard normal distribution. When σ is unknown, the Student's t-distribution replaces the normal: the pivot T = √n (¯X - μ)/s follows a t-distribution with n-1 degrees of freedom, producing the interval ¯X ± t_{α/2, n-1} s / √n, where s is the sample standard deviation. For estimating the variance σ² of a normal distribution, the pivotal quantity (n-1)s² / σ² follows a chi-squared distribution with n-1 degrees of freedom, leading to the interval [(n-1)s² / χ²_{1-α/2, n-1}, (n-1)s² / χ²_{α/2, n-1}], where χ² denotes the chi-squared quantiles. For large sample sizes, the central limit theorem justifies asymptotic approximations to construct confidence intervals, even when exact distributions are unavailable. The sample mean ¯X is approximately normal with mean μ and variance σ²/n, allowing the normal-based interval ¯X ± z_{α/2} σ / √n (or using s in place of σ). Similarly, for a binomial proportion p based on n trials with k successes, ˆp = k/n is approximately normal with mean p and variance p(1-p)/n, yielding the Wald interval ˆp ± z_{α/2} √[ˆp(1-ˆp)/n]. These approximations perform well when np and n(1-p) are both at least 5–10.^[11] Example: Normal mean. Consider a random sample of size n=25 from N(μ, σ²=1) with observed ¯X=10. The 95% confidence interval (α=0.05, z_{0.025}=1.96) is 10 ± 1.96 × 1 / 5, or [9.608, 10.392], capturing the true μ in 95% of repeated samples from this population. Example: Binomial proportion. In a poll of n=1000 voters, 520 support a candidate, so ˆp=0.52. The 95% interval is 0.52 ± 1.96 √[0.52×0.48/1000] ≈ [0.489, 0.551], indicating the population support p lies in this range with 95% confidence over repeated polls.^[11] In modern computational statistics, the bootstrap method provides an empirical alternative for approximating confidence intervals when analytical pivots are complex or unavailable. This nonparametric approach involves resampling the original data with replacement to generate B bootstrap samples, computing the statistic (e.g., mean or proportion) for each, and using the bootstrap distribution's quantiles or percentiles to form the interval, such as the basic bootstrap interval [2¯X - ¯X^_{(1-α/2)}, 2¯X - ¯X^{(α/2)}], where ¯X^*{(q)} is the q-th quantile of the bootstrap means. The percentile method simply takes the (α/2) to (1-α/2) quantiles of the bootstrap statistics directly. Bootstrap intervals achieve coverage probabilities close to the nominal 1-α for large B (typically 1000+), offering flexibility for non-standard distributions.^[12]

Tolerance intervals

Tolerance intervals are statistical intervals designed to contain at least a specified proportion β of the population distribution with a given confidence level 1-α.^[13] Unlike confidence intervals, which focus on capturing unknown population parameters such as the mean, tolerance intervals target the coverage of the population itself, ensuring that a proportion β (e.g., 95%) of future observations or items from the population fall within the interval [L, U] with confidence 1-α (e.g., 95%).^[13] This dual specification of coverage proportion and confidence distinguishes tolerance intervals as tools for bounding population variability rather than parameter estimation.^[14] Tolerance intervals come in two main types: two-sided and one-sided. Two-sided tolerance intervals are symmetric around the sample mean and aim to capture the central β proportion of the population, providing bounds for both lower and upper tails.^[13] One-sided tolerance intervals, in contrast, provide either a lower bound L (capturing at least β of the population above L) or an upper bound U (capturing at least β below U), which are useful for scenarios involving minimum or maximum specifications, such as ensuring product strength exceeds a threshold.^[13] For normally distributed data, tolerance intervals are commonly constructed using the sample mean \bar{X} and standard deviation s. The two-sided tolerance interval takes the form \bar{X} \pm k \cdot s, where the tolerance factor k is determined based on the sample size n, coverage proportion β, and confidence level 1-α, often involving factors from the non-central chi-squared or Student's t distributions to account for sampling variability.^[13] For example, the factor k can be computed as k = z_{(1 + \beta)/2} \sqrt{\frac{n-1}{ \chi^2_{\gamma; n-1} }} \cdot \sqrt{1 + \frac{1}{n}}, where z is the standard normal quantile and \chi^2 is the chi-squared critical value, though exact methods adjust for the non-centrality to ensure the coverage probability.^[13] One-sided versions modify this by using appropriate quantiles for the lower or upper tail. In applications, tolerance intervals are particularly valuable in quality control to establish specification limits that bound manufacturing variability, ensuring that a high proportion of produced items meet standards without focusing solely on parameter estimates like those in confidence intervals.^[14] For instance, in engineering and pharmaceutical production, they help verify that at least 99% of items fall within acceptable tolerances with 95% confidence, aiding decisions on process acceptability.^[15] Tolerance intervals were developed in the 1940s amid the rise of statistical process control and engineering standards, with seminal contributions from statisticians like Abraham Wald and Samuel Wilks.^[16]

Prediction intervals

A prediction interval provides a range within which a future observation from the same underlying distribution is likely to fall, with a specified probability, based on an existing sample. Formally, a (1-α)100% prediction interval for a future observation X_{n+1} is an interval that contains X_{n+1} with probability 1-α, conditional on the observed data.^[17] For data assumed to follow a normal distribution with unknown mean and variance, the standard construction of a two-sided (1-α)100% prediction interval for a single future observation is

\bar{X} \pm t_{\alpha/2, n-1} \, s \sqrt{1 + \frac{1}{n}},

where \bar{X} is the sample mean, s is the sample standard deviation, n is the sample size, and t_{\alpha/2, n-1} is the upper \alpha/2 quantile of the Student's t-distribution with n-1 degrees of freedom.^[18] This formula arises from the fact that the future observation follows a t-distribution centered at \bar{X}, scaled by the estimated standard error that includes both parameter estimation uncertainty and the new observation's variability.^[18] Prediction intervals are inherently wider than confidence intervals for the population mean, as they account for not only the sampling variability in estimating the mean but also the additional randomness inherent in the future observation itself.^[19] The term \sqrt{1 + 1/n} in the formula reflects this dual source of uncertainty: the "1" captures the variance of the new observation, while $1/n addresses the estimation error, which diminishes as n increases.^[19] In practice, prediction intervals are applied in scenarios such as linear regression to estimate the range for a new response value at a given predictor level, or in time series forecasting to bound individual future data points beyond the mean forecast.^[20] For instance, in regression, the interval widens farther from the center of the data due to increased extrapolation uncertainty combined with observational noise.^[21] When the normality assumption does not hold, nonparametric methods construct prediction intervals without relying on specific distributional forms. One approach uses order statistics from the sample to define intervals; for example, the (r)th and (s)th order statistics can form a distribution-free interval that covers a future observation with probability at least 1-α, selected such that the coverage is optimal based on the sample ranks.^[22] Alternatively, the bootstrap method generates nonparametric prediction intervals by resampling the original data with replacement to simulate the empirical distribution of possible future observations, then taking percentiles of the bootstrapped predictions to form the interval endpoints.^[12] This resampling technique approximates the sampling distribution of the predictor and accounts for both estimation and observational variability in a model-free manner.^[12]

Bayesian Methods

Credible intervals

In Bayesian statistics, a credible interval provides a range of plausible values for an unknown parameter based on the posterior distribution derived from the observed data and prior beliefs. Specifically, a (1-α)100% credible interval is defined as an interval [a, b] such that the posterior probability P(θ ∈ [a, b] | data) = 1 - α, where θ represents the parameter of interest.^[23]^[24] Credible intervals can be constructed in different ways depending on the desired properties. The equal-tailed credible interval is obtained by taking the central portion of the posterior distribution, specifically the interval between the α/2 and 1 - α/2 quantiles of the posterior: [q_{α/2}, q_{1-α/2}], where q_p denotes the p-th quantile.^[23]^[24] An alternative is the highest posterior density (HPD) interval, which selects the shortest interval that contains 1 - α of the posterior probability mass, ensuring that every point inside the interval has a higher posterior density than any point outside it; this approach is particularly useful for skewed posteriors as it minimizes interval width.^[24] Unlike frequentist confidence intervals, credible intervals explicitly incorporate prior beliefs about the parameter, which influence the shape and location of the posterior distribution. For instance, when using conjugate priors, the prior seamlessly updates with the likelihood to form the posterior; a common example is the beta prior for a binomial proportion parameter θ, where if the prior is Beta(α, β) and the data consist of s successes in n trials, the posterior is Beta(α + s, β + n - s).^[23]^[25] To illustrate, suppose a beta prior Beta(7, 3) reflects prior data equivalent to 6 successes and 2 failures, and new data show 11 successes in 12 trials; the posterior becomes Beta(18, 4), yielding an 80% equal-tailed credible interval approximately [0.709, 0.914] via the inverse cumulative distribution function of the beta distribution.^[23] This interval shrinks and shifts based on the data while retaining the influence of the prior, especially when sample sizes are small. A key advantage of credible intervals is their direct probabilistic interpretation: given the data and prior, there is a 1 - α probability that the true parameter lies within the interval, providing a straightforward measure of uncertainty about the parameter itself post-data.^[24]

Posterior predictive intervals

Posterior predictive intervals provide a Bayesian approach to estimating intervals for future observations, incorporating uncertainty from both the parameter estimates and the stochastic nature of new data. These intervals are derived from the posterior predictive distribution, which represents the distribution of a new data point conditional on the observed data. Specifically, a (1-α)100% posterior predictive interval contains the values within which a future observation X_{n+1} is expected to fall with probability $1 - \alpha, given the data. The construction of posterior predictive intervals begins with the posterior predictive density, defined as

p(X_{n+1} \mid \text{data}) = \int p(X_{n+1} \mid \theta) \, p(\theta \mid \text{data}) \, d\theta,

where the integral marginalizes over the posterior distribution of the parameters \theta. In practice, for conjugate models like the normal distribution with unknown mean and unknown variance, the posterior predictive distribution follows a Student's t-distribution, allowing closed-form intervals. For instance, if the data are normally distributed with unknown mean \mu and unknown variance \sigma^2, and a normal-inverse-gamma prior is used, the predictive interval for a new observation is centered at the posterior mean of \mu and scaled by the t-distribution with degrees of freedom equal to the sample size minus one, reflecting both epistemic and aleatoric uncertainty. In non-conjugate or complex models, intervals are typically obtained via Monte Carlo integration: samples are drawn from the posterior p(\theta \mid \text{data}) using methods like Markov chain Monte Carlo (MCMC), and for each \theta^{(s)}, a predictive sample X_{n+1}^{(s)} is generated from the likelihood p(X_{n+1} \mid \theta^{(s)}); the empirical quantiles of these X_{n+1}^{(s)} then form the interval. Unlike credible intervals, which quantify uncertainty in the parameters \theta alone by integrating solely over the posterior, posterior predictive intervals extend this to future data by also accounting for the sampling variability inherent in the observation process, thus providing a more complete prediction for unseen values. This distinction ensures that posterior predictive intervals are generally wider than corresponding credible intervals, as they capture the full predictive uncertainty. In modern Bayesian practice, MCMC algorithms such as those implemented in Stan or JAGS are widely used for computing these intervals, enabling their application to hierarchical and high-dimensional models where analytical solutions are infeasible.

Alternative Approaches

Fiducial intervals

Fiducial intervals represent a historical frequentist approach to interval estimation, introduced by Ronald A. Fisher in 1930, which derives a probability distribution for an unknown parameter by inverting a pivotal quantity while treating the observed data as fixed. Unlike traditional probability statements about data given parameters, fiducial inference assigns probabilities directly to parameter values based on the observed sample, aiming to provide objective bounds without invoking prior distributions.^[26] The construction of fiducial intervals relies on a pivotal quantity whose distribution is known and free of unknown parameters. For instance, when estimating the mean \mu of a normal distribution with known standard deviation \sigma from a sample of size n, the pivotal Z = (\bar{X} - \mu) \sqrt{n} / \sigma \sim N(0,1) is inverted to obtain the fiducial interval \bar{X} - z_{1 - \alpha/2} \sigma / \sqrt{n} < \mu < \bar{X} + z_{1 - \alpha/2} \sigma / \sqrt{n}, where z_{1 - \alpha/2} is the (1 - \alpha/2)-th quantile of the standard normal distribution; this yields symmetric bounds for equal tails but can be asymmetric if unequal tail probabilities are chosen.^[27] Fiducial methods encountered substantial controversies, stemming from ambiguities in their precise definition and challenges in extending them beyond single-parameter cases, where multiple valid pivots could produce conflicting fiducial distributions.^[28] Critics, including Jerzy Neyman, pointed out logical inconsistencies, such as violations of conditioning principles and paradoxes in multiparameter settings, which undermined their reliability.^[27] Despite these issues, fiducial inference exerted influence on early statistical thought, particularly in shaping discussions around inductive reasoning in estimation.^[29] A notable example involves the variance parameter \sigma^2 in a normal model with n i.i.d. observations, where the pivotal quantity v = (n-1)s^2 / \sigma^2 \sim \chi^2_{n-1} leads to a fiducial distribution for \sigma^2 via inversion, resulting in bounds that assign fiducial probabilities analogous to a scaled inverse chi-square; in simple scenarios, this yields intervals resembling those from uniform assumptions on transformed parameters.^[30] From a modern perspective, fiducial intervals are recognized as precursors to confidence intervals, coinciding with them in basic one-parameter problems, but they are criticized for failing to consistently ensure coverage probabilities and for lacking a robust general framework. Largely superseded by more rigorous methods in mainstream statistics, fiducial ideas persist in niche applications and generalized forms—such as generalized fiducial inference developed since the 2000s, which addresses foundational issues and applies to non-standard models including big data scenarios—valued for their intuitive appeal in uncertainty quantification despite foundational flaws.^[28]^[31]

Likelihood-based intervals

Likelihood-based intervals, also known as likelihood ratio or profile likelihood intervals, are constructed by identifying parameter values where the likelihood function, or its profile version in multiparameter models, falls to a specified fraction of its maximum value. For a nominal 95% interval in the case of a single parameter of interest, this corresponds to values of the parameter θ where the likelihood L(θ | data) equals L(θ̂ | data) × exp(-χ²_{1,0.95}/2), with χ²_{1,0.95} ≈ 3.841, so exp(-1.9205) ≈ 0.146; an approximation sometimes used is L(θ | data) = L(θ̂ | data) / 8.^[32]^[33] In models with nuisance parameters, the profile likelihood is employed: for a parameter of interest ψ, the profile log-likelihood l_p(ψ) is obtained by maximizing the full log-likelihood over the nuisance parameters φ for each fixed ψ, yielding l_p(ψ) = max_φ l(ψ, φ | data). The interval consists of ψ values satisfying 2 [l_p(ψ̂) - l_p(ψ)] ≤ χ²_{1,0.95} ≈ 3.841, solved numerically via grid search or optimization.^[34]^[32] Under standard regularity conditions, these intervals possess desirable asymptotic properties, with the likelihood ratio statistic -2 log [L_p(ψ)/L_p(ψ̂)] following a χ² distribution with degrees of freedom equal to the number of parameters of interest (Wilks' theorem). This yields consistent coverage probabilities approaching the nominal level as sample size increases, often outperforming Wald intervals in small samples or skewed distributions due to better approximation of the true sampling distribution.^[34]^[35] For the rate parameter λ of an exponential distribution with n independent observations x_1, ..., x_n, the maximum likelihood estimate is λ̂ = n / ∑x_i. The likelihood ratio interval inverts the test statistic 2n [ln(λ̂/λ) + (λ/λ̂) - 1] ≤ 3.841 to find the bounds, providing an asymmetric interval that accounts for the positive support and potential skewness in small samples.^[36]^[32] These intervals offer key advantages, including invariance under reparameterization—unlike Wald intervals, which depend on the choice of parameterization—and applicability in complex models such as generalized linear models (GLMs), where they handle nonlinearity and multiple nuisance parameters effectively without relying on asymptotic normality assumptions.^[34]^[35]

Key Properties and Considerations

One-sided versus two-sided intervals

In interval estimation, two-sided intervals provide bounds on both sides of a point estimate, capturing the range within which the parameter is likely to lie with a specified confidence level. These intervals are typically constructed as [L, U], where the lower limit L and upper limit U are determined such that the probability of each tail is \alpha/2 for a total significance level \alpha, often using critical values like z_{\alpha/2} from the standard normal distribution for large samples.^[37] They can be symmetric around the point estimate if the distribution is symmetric, or asymmetric in cases like proportions or skewed data.^[38] One-sided intervals, in contrast, focus on a single direction of uncertainty, either providing a lower bound [L, \infty) where the entire \alpha probability is in the upper tail, or an upper bound (-\infty, U] with \alpha in the lower tail. Construction involves using the critical value z_{\alpha} (or t_{\alpha} for small samples), which is larger than z_{\alpha/2}, resulting in a bound that extends farther from the point estimate compared to the corresponding side of a two-sided interval.^[37]^[38] The choice between one-sided and two-sided intervals depends on the research question and directional nature of the inference. Two-sided intervals are preferred for general uncertainty quantification, where the parameter could plausibly deviate in either direction, providing a complete range without assuming a specific hypothesis direction.^[39] One-sided intervals are appropriate for directional hypotheses, such as verifying that a process meets a minimum threshold, as in quality control where only exceeding an upper limit for defects is concerning.^[38]^[40] For the same confidence level, a one-sided interval offers tighter control on the bounded side by allocating the full \alpha to that tail, but the finite bound is wider than the corresponding bound in a two-sided interval due to the larger critical value. This trade-off allows more precise statements about one direction at the expense of information on the other, which is unbounded. Coverage probability remains $1 - \alpha in the specified direction, similar to two-sided intervals.^[37]^[38] A practical example is in manufacturing, where a one-sided upper confidence interval might be used to assess the proportion of defective components, ensuring with 95% confidence that the true proportion defective does not exceed a quality threshold (e.g., 5%) based on sample data, without concern for a lower limit. This application is common in statistical quality control to accept or reject lots efficiently.^[38]^[40]

Coverage and width

In frequentist interval estimation, the coverage probability refers to the long-run proportion of intervals that contain the true parameter value across repeated random samples from the population, often set at a nominal level such as 95%.^[41] This probability is guaranteed to be at least the nominal level for valid procedures, though it can exceed it in conservative methods.^[42] In contrast, Bayesian credible intervals interpret coverage as the posterior probability that the true parameter lies within the interval, directly quantifying uncertainty given the data and prior.^[24] Coverage probabilities can be exact, matching the nominal level precisely under the assumed model, or approximate, relying on large-sample asymptotics that may lead to deviations, particularly for small samples or skewed distributions.^[42] Key factors influencing coverage include sample size, which improves the reliability of approximations as it increases, and distributional assumptions, where mismatches can cause undercoverage (below nominal) or overcoverage.^[43]^[44] Conservative intervals, such as the exact Clopper-Pearson method for binomial proportions, deliberately exceed the nominal coverage to ensure the probability never falls below the target, though this often results in wider intervals.^[42] The width of an interval, defined as the difference between its upper bound U and lower bound L, serves as a direct measure of estimation precision, with narrower widths indicating greater certainty about the parameter.^[45] The expected width E[U - L] quantifies average precision across repeated samples. In Bayesian estimation, highest posterior density (HPD) intervals minimize this width for a fixed coverage probability by selecting the shortest region containing the desired posterior mass, offering optimal precision among equal-tailed alternatives.^[46] To optimize interval width, sample size planning is essential; for a confidence interval on a normal population mean, the required sample size n to achieve a desired full width W is approximately n \approx \left( \frac{z_{\alpha/2} \sigma}{W/2} \right)^2, where z_{\alpha/2} is the critical value from the standard normal distribution and \sigma is the population standard deviation.^[47] For non-normal cases, such as binomial or Poisson distributions, coverage probabilities and expected widths are evaluated through Monte Carlo simulations or exact computations to assess performance beyond asymptotic guarantees.^[42]^[48]

Interpretation and Applications

Common pitfalls in interpretation

One of the most persistent misconceptions in interpreting frequentist confidence intervals is the belief that a 95% confidence level means there is a 95% probability that the true parameter lies within the specific interval calculated from the data.^[49] In reality, the confidence level refers to the long-run frequency property: if the same procedure were repeated many times, 95% of such intervals would contain the true parameter, but for any single realized interval, the probability statement cannot be made post-data because the parameter is fixed and unknown. This error stems from conflating the interval's coverage probability, defined before data collection, with a posterior probability about the parameter's location.^[49] In contrast, Bayesian credible intervals do allow for the direct probabilistic interpretation that there is a 95% posterior probability the parameter falls within the interval, given the data and prior, which avoids this frequentist pitfall but introduces reliance on prior specifications.^[24] Jerzy Neyman himself warned against such misinterpretations in his foundational 1937 paper, emphasizing that confidence intervals describe a method's performance across repeated applications rather than a probability for a fixed interval, yet these cautions have often been overlooked in practice. A related common error is the misinterpretation of overlapping or non-overlapping confidence intervals when comparing two groups. While overlapping 95% confidence intervals do not rule out a statistically significant difference (as a direct test of the difference may still reject the null), non-overlapping intervals indicate a significant difference at a level stricter than α=0.05, assuming similar standard errors; however, a direct test of the difference is always recommended for precise assessment.^[50] Additionally, practitioners frequently ignore underlying assumptions, such as normality of the sampling distribution, leading to invalid intervals when data violate these conditions (e.g., in small samples or skewed distributions), which can distort coverage and inflate error rates.^[51] Frequentist approaches are particularly prone to overemphasizing long-run frequency guarantees in one-off analyses, where users may dismiss practical relevance for a single study by fixating on hypothetical repetitions, whereas Bayesian methods risk underemphasizing model sensitivity to priors in isolated contexts.^[24] To mitigate these pitfalls, intervals should always be reported alongside point estimates, sample sizes, and explicit statements of assumptions and confidence levels to provide full context.^[52] Visualizations, such as forest plots or error bars, aid comprehension by illustrating uncertainty without implying probabilistic containment, and adjustments for multiple comparisons (e.g., via Bonferroni correction) are essential to control family-wise error rates when testing several intervals.^[52]

Practical uses across fields

In statistics and scientific research, interval estimation is widely applied to bound effect sizes in experimental settings, such as clinical trials evaluating drug efficacy. For instance, confidence intervals around the difference in treatment outcomes help determine whether a new drug demonstrates statistically significant benefits over a placebo, with the 95% confidence interval often used to assess the range of plausible treatment effects.^[53] In non-inferiority trials, the upper bound of the confidence interval for the risk difference is examined to confirm that a test drug is not worse than an active control by more than a predefined margin.^[52] In engineering, particularly reliability testing, interval estimation quantifies uncertainty in failure rates and mean time between failures (MTBF) for components and systems. Confidence intervals for failure rates, derived from sparse or censored data, enable engineers to predict system unavailability and set tolerance limits, ensuring designs meet safety standards in applications like aerospace or manufacturing.^[54] For example, in exponential failure models, two-sided confidence intervals around the MTBF estimate provide bounds for operational reliability, with widths reflecting data precision and informing maintenance schedules.^[55] Economists employ interval estimation to convey uncertainty in macroeconomic forecasts, such as confidence intervals around projected GDP growth rates. These intervals, often at 70% or 90% levels, illustrate the range of potential outcomes based on econometric models, aiding policymakers in assessing fiscal risks and economic stability.^[56] In regression analyses of economic indicators, confidence intervals for coefficients quantify the precision of relationships, like those between interest rates and growth, supporting decisions in monetary policy.^[57] In machine learning, interval estimation addresses prediction uncertainty through methods like conformal prediction, which generates prediction sets with guaranteed coverage probabilities regardless of the underlying model distribution. This approach is particularly useful in high-stakes applications, such as medical diagnostics or autonomous systems, where bounding prediction errors enhances decision reliability by quantifying both aleatoric and epistemic uncertainty.^[58] Conformal prediction extends traditional intervals by providing distribution-free guarantees, making it a modern tool for uncertainty-aware AI deployments.^[59] Software tools facilitate interval estimation in practice; in R, the confint() function computes confidence intervals for model parameters, integrating seamlessly with hypothesis testing frameworks like lm(). Similarly, Python's SciPy library offers functions in scipy.stats, such as t.interval(), for calculating intervals from t-distributions, enabling efficient computation in data analysis pipelines.^[60] A notable case study involves election polling, where confidence intervals around vote margins estimate the uncertainty in candidate leads. For the 2020 U.S. presidential election, polls reported 95% confidence intervals that captured the true margin in only about 60% of cases, highlighting the impact of sampling variability and non-response bias on predictive accuracy.^[61] Such intervals guide interpretations of race closeness, informing campaign strategies while underscoring the need to avoid over-reliance on point estimates alone.^[62]

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Point estimation and interval estimation, and hypothesis testing are three main ways of learning about the population parameter from the sample statistic.
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[PDF] Confidence Intervals Point Estimation Vs - Stat@Duke
Point estimation gives us a particular value as an estimate of the population parameter. . Interval estimation gives us a range of values which is likely to ...
[9]
Some Statistical Basics - B. Gerstman - San Jose State University
Interval estimation provides a interval that has a calculated likelihood of capturing the parameter. For example, a 95% confidence interval for population p ...<|control11|><|separator|>
[10]
[PDF] Outline of a Theory of Statistical Estimation Based on the Classical ...
X-Outline of a Theory of Statistical Estimation Based on the. Classical Theory of Probability. By J. NEYMAN. Reader in Statistics, University College, London.
[11]
[PDF] Stat 5102 Notes: More on Confidence Intervals
Feb 24, 2003 · Pivotal quantities allow the construction of exact confidence intervals, mean- ing they have exactly the stated confidence level, as opposed to ...
[12]
8.1 - One Sample Proportion | STAT 200
The Central Limit Theorem states that if the sample size is sufficiently large then the sampling distribution will be approximately normally distributed.
[13]
Bootstrap Methods for Standard Errors, Confidence Intervals, and ...
This is a review of bootstrap methods, concentrating on basic ideas and applications rather than theoretical considerations.
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7.2.6.3. Tolerance intervals for a normal distribution
Definition of a tolerance interval, A confidence interval covers a population parameter with a stated confidence, that is, a certain proportion of the time.
[15]
[PDF] Normal Tolerance Interval Procedures in the tolerance Package
Abstract Statistical tolerance intervals are used for a broad range of applications, such as quality control, engineering design tests, ...
[16]
Using tolerance intervals for assessment of pharmaceutical quality
In quality control of drug products, tolerance intervals are commonly used methods to assure a certain proportion of the products covered within a pre ...
[17]
Full validation using β-content, γ-confidence tolerance interval
Tolerance intervals (TIs) were introduced in the late 1940s when control charts and statistical industrial process control were being developed. They have ...
[18]
[PDF] MIL-STD-414
MIL-STD-414. 11 blvs7. I. MILITARY STANDARD. SAMbLING. PROCEDURES AND. -. ~M. ORfkMOO$-10. TABLES. FOR INSPECTION BY VARIABLES. FOR PERCENT DEFECTIVE. 1. .—.
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8.2 - A Prediction Interval for a New Y | STAT 415
The prediction interval for a new observation $Y_{n+1}$ can be made to be narrower in the same ways that we can make the confidence interval for the mean \(\ ...
[20]
[PDF] Lecture 31 The prediction interval formulas for the next observation ...
In this lecture we will derive the formulas for the symmetric two-sided prediction interval for the n + 1-st observation and the upper-tailed prediction ...
[21]
The distinction between confidence intervals, prediction ... - GraphPad
Prediction intervals tell you where you can expect to see the next data point sampled. Assume that the data are randomly sampled from a Gaussian distribution.
[22]
5.5 Distributional forecasts and prediction intervals - OTexts
A prediction interval gives an interval within which we expect yt y t to lie with a specified probability. For example, assuming that distribution of future ...
[23]
3.3 - Prediction Interval for a New Response | STAT 501
By calculating the interval at the sample's mean of the predictor values ( x h = x ¯ ) and increasing the sample size n, the confidence interval's standard ...
[24]
Full article: Teaching Prediction Intervals - Taylor & Francis Online
Jul 7, 2022 · In this article I present a strategy for involving students in the development of a nonparametric prediction interval.
[25]
[PDF] Chapter 8. Statistical Inference 8.2: Credible Intervals
Definition 8.2.1: Credible Intervals. Suppose you have iid samples x = (x1, ..., xn) from some distribution with unknown parameter Θ. You are in the Bayesian ...
[26]
Understanding and interpreting confidence and credible intervals ...
Dec 31, 2018 · Interpretation of the Bayesian 95% confidence interval (which is known as credible interval): there is a 95% probability that the true (unknown) ...
[27]
STAT415 Handouts - 12 Beta-Binomial Model - Bookdown
For example, you could specify the endpoints of a prior 98% credible interval. ... So, the Beta distributions form a conjugate prior family for Binomial ...
[28]
[PDF] fisher-1930-inverse-probability.pdf - Error Statistics Philosophy
Feb 15, 2016 · Inverse Probability. By R. A. FISHER, SC.D., F.R.S., Gonville and. CaiusCollege; Statistical Dept., Rothamsted Experimental Station.
[29]
Can't Take the Fiducial Out of Fisher (if you want to understand the ...
Feb 22, 2022 · We take the simplest example,…the normal mean when the variance is known, but the considerations are fairly general. The lower limit, [with ...
[30]
[PDF] Fiducial inference then and now - arXiv
Dec 19, 2020 · We conduct a review of the fiducial approach to statistical inference, following its journey from its initiation by R. A. Fisher, through ...
[31]
Fiducial Inference - SpringerLink
The term “fiducial distribution” first appeared in Fisher's 1930 paper [CP 84], where the distribution of the correlation coefficient was considered.Missing: original | Show results with:original
[32]
None
### Example for Normal Variance Fiducial Distribution
[33]
Maximum Likelihood, Profile Likelihood, and Penalized Likelihood
The 95% LR confidence interval is 1.47, 61.1, as can be read off the profile log-likelihood plot in Figure 1 (or found numerically with the data used to ...
[34]
Profile Likelihood Confidence Intervals – Gary C. White
Profile likelihood confidence intervals are based on the log-likelihood function. For a single parameter, likelihood theory shows that the 2 points 1.92 units ...
[35]
Profile Likelihood for Estimation and Confidence Intervals
Profile-likelihood confidence intervals are particularly useful in nonlinear models. The command pllf computes and plots the maximum likelihood estimate and ...
[36]
A robust and efficient algorithm to find profile likelihood confidence ...
May 7, 2021 · Profile likelihood confidence intervals are a robust alternative to Wald's method if the asymptotic properties of the maximum likelihood estimator are not met.
[37]
Exponential distribution - Maximum likelihood estimation - StatLect
In this lecture, we derive the maximum likelihood estimator of the parameter of an exponential distribution.
[38]
[PDF] Intro to confidence intervals - Stat@Duke
Two-sided confidence interval for parameter θ is given byˆθ ± zα. 2. SEˆ θ. One-sided confidence intervals for parameter θ are given by (−∞,ˆθ + zαSEˆ θ. ] or.
[39]
7.2.4.1. Confidence intervals - Information Technology Laboratory
One-sided confidence intervals, A one-sided confidence interval can also be constructed simply by replacing each z α / 2 by z α in the expression for the lower ...Missing: manufacturing | Show results with:manufacturing
[40]
What are the differences between one-tailed and two-tailed tests?
A two-tailed test will test both if the mean is significantly greater than x and if the mean significantly less than x.
[41]
[PDF] One-sided confidence intervals in discrete distributions
One-sided con- fidence intervals are useful in many applications. See Duncan (1986) and Montgomery. (2001) for applications in quality control. In the present ...
[42]
[PDF] Lecture 5: Confidence Intervals - Oxford statistics department
Two – sided confidence intervals are used when we are interested in inferring. IAUL – DEPARTMENT OF STATISTICS. PAGE 5. Page 6. DESCRIPTIVE STATISTICS FOR ...
[43]
[PDF] Approximate is Better than “Exact” for Interval Estimation of Binomial ...
Mar 22, 2012 · For interval estimation of a proportion, coverage probabil- ities tend to be too large for "exact" confidence intervals based on inverting the ...
[44]
[PDF] Coverage versus Confidence - Wolfram
Such a probability is called a coverage probability. Confidence is defined as the infimum of the coverage probabilities. The following definitions can be found ...Missing: frequentist | Show results with:frequentist
[45]
[PDF] Actual Coverage Probabilities for Confidence Intervals on the ... - SCB
It depends on a number of factors such as the type of estimator and design used, the underlying population, and the sample size. Consequently, there is a great ...
[46]
What do confidence intervals say about precision (if anything)?
Mar 30, 2016 · The width of a confidence interval indicates the precision of our knowledge about the parameter. Narrow confidence intervals show precise ...Narrow confidence interval -- higher accuracy? - Cross ValidatedWhat is the difference between confidence intervals and precision?More results from stats.stackexchange.com
[47]
Statistical Rethinking (2nd ed) - 3 Sampling the Imaginary - Bookdown
Right: 50% highest posterior density interval, HPDI. This interval finds the narrowest region with 50% of the posterior probability. Such a region always ...
[48]
7.5 Calculating the Sample Size for a Confidence Interval
To find the sample size, we need to find the z -score for the 95% confidence interval. This means that we need to find the z -score so that the entire area ...
[49]
[PDF] Coverage Probabilities of Poisson Interval Estimates
Among the intervals considered are frequentist confidence intervals based on normal approximation and Bayesian posterior probability intervals resulting from a ...
[50]
Continued misinterpretation of confidence intervals - NIH
There is widespread misunderstanding of confidence intervals, even among experts, and the data shows that researchers have robust misunderstandings of them.
[51]
Statistical tests, P values, confidence intervals, and power: a guide ...
If two confidence intervals overlap, the difference between two estimates or studies is not significant. No! The 95 % confidence intervals from two ...
[52]
Violating the normality assumption may be the lesser of two evils
We argue that violating the normality assumption bears risks that are limited and manageable, while several more sophisticated approaches are relatively error ...
[53]
Using the confidence interval confidently - PMC - NIH
The concept of the CI was introduced by Jerzy Neyman in a paper published in 1937 (3). It has now gained wide acceptance although many of us are not quite ...
[54]
[PDF] Non-Inferiority Clinical Trials to Establish Effectiveness - FDA
Nonetheless, the upper bound of the 95% confidence interval is typically used to judge the effectiveness of the test drug in the NI study, just as a two-sided ...
[55]
Interval estimates of average failure rate and unavailability per ...
A method of interval estimation of average failure rate and unavailability per demand when data are sparse, vague or bad is proposed.
[56]
Confidence Intervals for MTBF - Accendo Reliability
The key when calculating these confidence intervals is to know if the data is time or failure censored, then use the correct formula for degrees of freedom.
[57]
[PDF] Box E: Forecast Confidence Intervals - Reserve Bank of Australia
Graph E1 shows 70 per cent and 90 per cent confidence intervals around the forecasts for underlying inflation and GDP growth on a year-ended basis. The top ...
[58]
2. Theory | Treasury.gov.au
Nov 21, 2013 · Reporting the 70 and 90 per cent confidence intervals below gives a sense of the risks around the forecasts and follows the practice of the RBA.
[59]
Quantifying Deep Learning Model Uncertainty in Conformal Prediction
Jun 1, 2023 · We propose a probabilistic approach in quantifying the model uncertainty derived from the produced prediction sets in conformal prediction.
[60]
Uncertainty quantification for probabilistic machine learning in earth ...
Jul 13, 2024 · Conformal prediction is a mathematical framework that can quantify both epistemic and aleatoric uncertainty in combination. It is amenable to ...
[61]
Statistical functions (scipy.stats) — SciPy v1.16.2 Manual
This module contains a large number of probability distributions, summary and frequency statistics, correlation functions and statistical tests.Scipy.stats.mstats · Stats.norm · Stats.expon · Stats.binom
[62]
Election polls are 95% confident but only 60% accurate, Berkeley ...
Oct 26, 2020 · Not nearly as confident as the pollsters claim, according to a new Berkeley Haas study. Most election polls report a 95% confidence level. Yet ...
[63]
Election polls are 95% confident but only 60% accurate
In an analysis of polls for more than 6,000 contests, we have found that the actual vote total for a given candidate fell within the 95% confidence interval for ...