A prediction interval for some population is an interval on the real line
constructed so that it will contain \(k\) future observations or averages
from that population with some specified probability \((1-\alpha)100\%\),
where \(0 < \alpha < 1\) and \(k\) is some pre-specified positive integer.
The quantity \((1-\alpha)100\%\) is call the confidence coefficient or
confidence level associated with the prediction interval. The function
predIntNorm
computes a standard prediction interval based on a
sample from a normal distribution.
The function predIntNormTestPower
computes the probability that at
least one out of \(k\) future observations or averages will not be contained in
a prediction interval based on the assumption of normally distributed observations,
where the population mean for the future observations is allowed to differ from
the population mean for the observations used to construct the prediction interval.
The function predIntLnormAltTestPower
assumes all observations are
from a lognormal distribution. The observations used to
construct the prediction interval are assumed to come from a lognormal distribution
with mean \(\theta_2\) and coefficient of variation \(\tau\). The future
observations are assumed to come from a lognormal distribution with mean
\(\theta_1\) and coefficient of variation \(\tau\); that is, the means are
allowed to differ between the two populations, but not the coefficient of variation.
The function predIntLnormAltTestPower
calls the function
predIntNormTestPower
, with the argument delta.over.sigma
given by:
$$\frac{\delta}{\sigma} = \frac{log(R)}{\sqrt{log(\tau^2 + 1)}} \;\;\;\;\;\; (1)$$
where \(R\) is given by:
$$R = \frac{\theta_1}{\theta_2} \;\;\;\;\;\; (2)$$
and corresponds to the argument ratio.of.means
for the function
predIntLnormAltTestPower
, and \(\tau\) corresponds to the argument
cv
.