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A confidence interval for the parameter , with confidence level or coefficient , is an interval determined by random variables and with the property: The number , whose typical value is close to but not greater than 1, is sometimes given in the form (or as a percentage ), where is a small positive number, often 0.05.
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).
It is usually determined on the basis of the cost, time or convenience of data collection and the need for sufficient statistical power. For example, if a proportion is being estimated, one may wish to have the 95% confidence interval be less than 0.06 units wide. Alternatively, sample size may be assessed based on the power of a hypothesis ...
when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and; as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.
In contrast, it is worth noting that other confidence interval may have coverage levels that are lower than the nominal , i.e., the normal approximation (or "standard") interval, Wilson interval, [8] Agresti–Coull interval, [13] etc., with a nominal coverage of 95% may in fact cover less than 95%, [4] even for large sample sizes.
The dependence of the confidence intervals on sample size is further illustrated below. For N = 10, the 95% confidence interval is approximately ±13.5789 standard deviations. For N = 100 the 95% confidence interval is approximately ±4.9595 standard deviations; the 99% confidence interval is approximately ±140.0 standard deviations.
Calculating the confidence interval. Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sided t value from the table is 1.372 . Then with confidence interval calculated from
A tolerance interval ( TI) is a statistical interval within which, with some confidence level, a specified sampled proportion of a population falls. "More specifically, a 100×p%/100× (1−α) tolerance interval provides limits within which at least a certain proportion ( p) of the population falls with a given level of confidence (1−α ...