# Confidence interval

**confidence intervalsconfidence levelconfidence95% CICIconfidence beltconfidence limitconfidence limits95% confidence95% 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 unknown population parameter.wikipedia

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### Jerzy Neyman

**NeymanJerzy Spława-NeymanNeyman, Jerzy**

Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.

Neyman first introduced the modern concept of a confidence interval into statistical hypothesis testing and co-revised Ronald Fisher's null hypothesis testing (in collaboration with Egon Pearson).

### Frequentist inference

**frequentistfrequentist statisticsclassical**

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 unknown population parameter.

This is the inference framework in which the well-established methodologies of statistical hypothesis testing and confidence intervals are based.

### Coverage probability

**coveragecoverage probabilitiesprobability**

In statistics, the coverage probability of a technique for calculating a confidence interval is the proportion of the time that the interval contains the true value of interest.

### Point estimation

**point estimatepointpoint estimator**

Interval estimation can be contrasted with point estimation.

Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in the case of Bayesian inference.

### Statistical inference

**inferential statisticsinferenceinferences**

The principle behind confidence intervals was formulated to provide an answer to the question raised in statistical inference of how to deal with the uncertainty inherent in results derived from data that are themselves only a randomly selected subset of a population.

However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments.

### Sample size determination

**sample sizeSampling sizessample**

A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example, the number of people taking part in a survey.

For example, if a proportion is being estimated, one may wish to have the 95% confidence interval be less than 0.06 units wide.

### Interval estimation

**interval estimateintervalInterval (statistics)**

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 unknown population parameter. Interval estimation can be contrasted with point estimation.

### Confidence region

**3-dimensional uncertainty ellipsoiderror ellipseline of variation**

Confidence regions generalize the confidence interval concept to deal with multiple quantities.

In statistics, a confidence region is a multi-dimensional generalization of a confidence interval.

### Neyman construction

Neyman construction is a frequentist method to construct an interval at a confidence level C, \, such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction C\, of the time.

### Confidence and prediction bands

**confidence bandconfidence beltPrediction band**

A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data.

Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value.

### Student's t-distribution

**Student's ''t''-distributiont-distributiont''-distribution**

In the theoretical example below, the parameter σ is also unknown, which calls for using the Student's t-distribution.

The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.

### Statistical significance

**statistically significantsignificantsignificance level**

If the estimates of two parameters (for example, the mean values of a variable in two independent groups) have confidence intervals that do not overlap, then the difference between the two values is more significant than that indicated by the individual values of α.

Sometimes researchers talk about the confidence level

### Normal distribution

**normally distributedGaussian distributionnormal**

This example assumes that the samples are drawn from a Normal distribution. As the machine cannot fill every cup with exactly 250.0 g, the content added to individual cups shows some variation, and is considered a random variable X. This variation is assumed to be normally distributed around the desired average of 250 g, with a standard deviation, σ, of 2.5 g. To determine if the machine is adequately calibrated, a sample of n = 25 cups of liquid is chosen at random and the cups are weighed.

These values are used in hypothesis testing, construction of confidence intervals and Q-Q plots.

### Bootstrapping (statistics)

**bootstrapbootstrappingbootstrap support**

Bootstrapping allows assigning measures of accuracy (defined in terms of bias, variance, confidence intervals, prediction error or some other such measure) to sample estimates.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

Confidence intervals are closely related to statistical significance testing.

Hypothesis tests based on statistical significance are another way of expressing confidence intervals (more precisely, confidence sets).

### Statistics

**statisticalstatistical analysisstatistician**

A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data.

They introduced the concepts of "Type II" error, power of a test and confidence intervals.

### Null hypothesis

**nullnull hypotheseshypothesis**

For example, if for some estimated parameter θ one wants to test the null hypothesis that θ = 0 against the alternative that θ ≠ 0, then this test can be performed by determining whether the confidence interval for θ contains 0.

A statistical significance test shares much mathematics with a confidence interval.

### Standard deviation

**standard deviationssample standard deviationSD**

As the machine cannot fill every cup with exactly 250.0 g, the content added to individual cups shows some variation, and is considered a random variable X. This variation is assumed to be normally distributed around the desired average of 250 g, with a standard deviation, σ, of 2.5 g. To determine if the machine is adequately calibrated, a sample of n = 25 cups of liquid is chosen at random and the cups are weighed.

The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval.

### Regression analysis

**regressionmultiple regressionregression model**

Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis.

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

### Standard error

**SEstandard errorsstandard error of the mean**

In our case we may determine the endpoints by considering that the sample mean from a normally distributed sample is also normally distributed, with the same expectation μ, but with a standard error of:

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, e.g., confidence intervals).

### Prediction interval

**interval forecastsPIpredictive performance**

An analogous concept in Bayesian statistics is credible intervals, while an alternative frequentist method is that of prediction intervals which, rather than estimating parameters, estimate the outcome of future samples.

Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.

### Binomial proportion confidence interval

**Agresti–Coull intervalarcsine square root transformationbinomial confidence intervals**

See "Binomial proportion confidence interval" for better methods which are specific to this case.

In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials).

### Nuisance parameter

**parameter of interest**

The quantities φ in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them.

These provide both significance tests and confidence intervals for the parameters of interest which are approximately valid for moderate to large sample sizes and which take account of the presence of nuisance parameters.

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

This is often used in determining likelihood-based approximate confidence intervals and confidence regions, which are generally more accurate than those using the asymptotic normality discussed above.

### Credible interval

**credible intervalscredible regionBayesian interval**

An analogous concept in Bayesian statistics is credible intervals, while an alternative frequentist method is that of prediction intervals which, rather than estimating parameters, estimate the outcome of future samples. There are other answers, notably that provided by Bayesian inference in the form of credible intervals.

Credible intervals are analogous to confidence intervals in frequentist statistics, although they differ on a philosophical basis: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value.