9.6. Confidence Tests 553
7.Compareχ^2 /νwithχ^2 ν,α/ν.Let us now see what we can infer from this comparison.Case-1,χ^2 /νχ^2 ν,α/ν:We are up toα×100% confident that our hypothesis
was correct.Case-2,χ^2 /ν > χ^2 ν,α/ν:This may mean one of the following.1.The model we have chosen to represent the system is not adequate.
2.The model is adequate but there are some bad data points in the sample.
It takes only a few large excursions in the data that are far away from the
mean to yield a large value of chi-square. Care should therefore be taken
to ensure that proper filtration of the data is performed to eliminate such
data points.
3.The data values are not uniformly distributed about their means. The is
the most troubling scenario, since it would mean that this goodness-of-fit
method is not really applicable and we should either resort to some other
method or look closely at the data to find out if just a few values are
causing this deviation from the normal distribution. Generally, discarding
a few data points does the trick.Case-3,χ^2 /ν < χ^2 ν,α/ν: This means that the squares of the random normal
deviates are less than expected, a situation that demands as much attention as
the previous one. The following possibilities exist for this case.1.The expected means were overestimated. This does not mean that the
model was wrong.
2.There are a few data points that have caused the chi-square value to
become too small.9.6.B Student’stTest..........................
Student’sttest is the most commonly used method of comparing the means of
two low statistics data samples. To perform the test, first the following quantity is
evaluated.
t=|x ̄ 1 −x ̄ 2 |
σ 12(9.6.3)
Here ̄x 1 and ̄x 2 represent the means of first and second datasets andσ 12 is the
standard deviation of the difference between the two means. It can be computed
from
σ 12 =[
σ^21
N 1+
σ^22
N 2] 1 / 2
, (9.6.4)
whereσ 1 andσ 2 are the standard deviations of the two datasets havingN 1 andN 2
number of data points. Note that here what we have done is to simply taken the
square root of the sum of the standard errors associated with each dataset.
The next step is to compare the calculatedt-value with the tabulated one. The
tabulated values, derived from the Student’stdistribution we presented earlier, are