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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).
Interval arithmetic is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Values are intervals, which can be represented in various ways, such as: [ 6 ]
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality ...
The Šidák correction is derived by assuming that the individual tests are independent. Let the significance threshold for each test be α 1 {\displaystyle \alpha _{1}} ; then the probability that at least one of the tests is significant under this threshold is (1 - the probability that none of them are significant).
The procedures of Bonferroni and Holm control the FWER under any dependence structure of the p-values (or equivalently the individual test statistics).Essentially, this is achieved by accommodating a `worst-case' dependence structure (which is close to independence for most practical purposes).
This application employs the model-view-controller design, which includes a data structure to represent the exam questions, a graphical user interface (GUI) for inputting student answers, and a set of algorithms written in JavaScript to process input and output. However, this application is a work in progress, as it cannot handle rounding errors.