Note that a single confidence interval with 95 % coverage probability level will likely contain the population parameter it is meant to contain, i . e . in the long run 95 % of confidence intervals built in that way will contain the true population parameter.
32.
Note that a single confidence interval with 95 % coverage probability level will likely contain the population parameter it is meant to contain, i . e . in the long run 95 % of confidence intervals built in that way will contain the true population parameter.
33.
The tolerance interval differs from a confidence interval in that the confidence interval bounds a single-valued population parameter ( the mean or the variance, for example ) with some confidence, while the tolerance interval bounds the range of data values that includes a specific proportion of the population.
34.
Thus for inference purposes " t " is a useful " pivotal quantity " in the case when the mean and variance ( ?, ? 2 ) are unknown population parameters, in the sense that the " t "-value has then a probability distribution that depends on neither ? nor ? 2.
35.
However, it is a minimum variance unbiased estimate of the expected value of the median of three values and in this application of the theory it is the population parameter defined as " the expected value of the median of three values " which is being estimated, not the median of the population.
36.
It is not simply a matter of the population parameters ( mean and standard deviation ) being unknown it is that " regressions " yield " different residual distributions " at " different data points, " unlike " point estimators " of univariate distributions, which share a " common distribution " for residuals.
37.
Whereas a confidence interval's size is entirely due to sampling error, and will approach a zero-width interval at the true population parameter as sample size increases, a tolerance interval's size is due partly to sampling error and partly to actual variance in the population, and will approach the population's probability interval as sample size increases.
