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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 intuition behind the CDF-based approach is that bounds on the CDF of a distribution can be translated into bounds on statistical functionals of that distribution. Given an upper and lower bound on the CDF, the approach involves finding the CDFs within the bounds that maximize and minimize the statistical functional of interest.
The lower bound is weaker than that obtained by stopping the series after the term. A looser version of this bound is that ! + (,] for all . Stirling's formula for ...
Solving for Q(x) provides the lower bound. The geometric mean of the upper and lower bound gives a suitable approximation for (): () +, Tighter bounds ...
The Exploratory Software for Confidence Intervals tutorial programs that run under Excel; Confidence interval calculators for R-Squares, Regression Coefficients, and Regression Intercepts; Weisstein, Eric W. "Confidence Interval". MathWorld. CAUSEweb.org Many resources for teaching statistics including Confidence Intervals.
For example, to calculate the 95% prediction interval for a normal distribution with a mean (μ) of 5 and a standard deviation (σ) of 1, then z is approximately 2. Therefore, the lower limit of the prediction interval is approximately 5 ‒ (2⋅1) = 3, and the upper limit is approximately 5 + (2⋅1) = 7, thus giving a prediction interval of ...
In the graph at right the top line y = n − 1 is an upper bound valid for all n other than one, and attained if and only if n is a prime number. A simple lower bound is φ ( n ) ≥ n / 2 {\displaystyle \varphi (n)\geq {\sqrt {n/2}}} , which is rather loose: in fact, the lower limit of the graph is proportional to n / log log n .
A tolerance interval bounds this variance by introducing a confidence level , which is the confidence with which this interval actually includes the specified proportion of the population. For a normally distributed population, a z-score can be transformed into a " k factor" or tolerance factor [ 6 ] for a given γ {\displaystyle \gamma } via ...