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For a confidence level, there is a corresponding confidence interval about the mean , that is, the interval [, +] within which values of should fall with probability . Precise values of z γ {\displaystyle z_{\gamma }} are given by the quantile function of the normal distribution (which the 68–95–99.7 rule approximates).
The probability density function (PDF) for the Wilson score interval, plus PDF s at interval bounds. Tail areas are equal. Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ( , + ) has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.
The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not statistically significantly different from the point estimate at the .05 level." [20] Interpretation of the 95% confidence interval in terms of statistical significance.
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
Binomial proportion confidence interval; Binomial regression; ... Computational formula for the variance; ... Margin of error;
Given a sample from a normal distribution, whose parameters are unknown, it is possible to give prediction intervals in the frequentist sense, i.e., an interval [a, b] based on statistics of the sample such that on repeated experiments, X n+1 falls in the interval the desired percentage of the time; one may call these "predictive confidence intervals".
A confidence interval states there is a 100γ% confidence that the parameter of interest is within a lower and upper bound. A common misconception of confidence intervals is 100γ% of the data set fits within or above/below the bounds, this is referred to as a tolerance interval, which is discussed below.
when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and