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[1] [2] The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should ...
In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory.
Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals. [15] [16] [page needed] In particular, For every α in (0, 1), let (−∞, ξ n (α)] be a 100α% lower-side confidence interval for θ, where ξ n (α) = ξ n (X n,α) is continuous and increasing in α for each sample X n.
Thus the sampling distribution of the quantile of the sample maximum is the graph x 1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p 1/k N, q 1/k N]. Inverting this yields the corresponding confidence interval for the population maximum of [m/q 1/k, m/p 1/k].
For example, a pain-relief drug is tested on 1500 human subjects, and no adverse event is recorded. From the rule of three, it can be concluded with 95% confidence that fewer than 1 person in 500 (or 3/1500) will experience an adverse event. By symmetry, for only successes, the 95% confidence interval is [1−3/ n,1].
For example, f(x) might be the proportion of people of a particular age x who support a given candidate in an election. If x is measured at the precision of a single year, we can construct a separate 95% confidence interval for each age. Each of these confidence intervals covers the corresponding true value f(x) with confidence 0.
A stated confidence level generally applies only to each test considered individually, but often it is desirable to have a confidence level for the whole family of simultaneous tests. [4] Failure to compensate for multiple comparisons can have important real-world consequences, as illustrated by the following examples:
In the social sciences, a result may be considered statistically significant if its confidence level is of the order of a two-sigma effect (95%), while in particle physics and astrophysics, there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a discovery.