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Download a pdf copy of this help file here |
This set of results, rearranged for use in a simulation model, compare
the means and standard deviations between two populations X and Y that
are assumed Normally distributed (Normal(mX, sX) and Normal(mY,
sY)), and from which we have nX
and nY random observations {xi}
and {yi}.
The sample mean is given by:
and the sample standard deviation is given by:
denotes a Chisq
distribution with n degrees of freedom,
i.e. VoseChisq(n)
denotes a Student-t
distribution with n degrees of freedom,
i.e. VoseStudent(n)
Example 1: X, Y have the same standard deviation s, but it is unknown
Example 2: sX, sY are known
Example 3: sX, sY may be different
Example 1: X, Y have unknown means mX, mY
Example 2: X, Y have known means mX, mY
Known as the Behrens-Fisher problem, there is no exact solution is available for this example and it is a place where statisticians have a lot of fun fighting over the methods to use (Bayesian inference has an exact solution). Classical statistics does offer an approximate (known as the Welch) solution which is a good compromise between opinions:
where 
is the nearest
integer to
where 
is an F
distribution. We can also generate values from this uncertainty distribution
by using the ratio of Chisq
distributions identity:
or, for simulation purposes:
