Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

496 CHAPTER 9
This result shows that the measured difference is 1.62 standard deviations from zero.
From a table of areas under a normal curve, we find that 1.62 standard deviations
includes 44.74% of the area on each side of the mean, or 89.48% of the area between
±1.62 standard deviations. We interpret this result as saying that we would expect
a difference this great 10 or 11 times out of 100 (100.00 − 89.48 = 10.52) if the
two samples came from the same population. Therefore, the difference between
the two samples is not statistically significant. To be statistically significant at the
95% confidence level, the difference between the two means would have to be great
enough to be observed only 5 times or less out of 100. When we are using the 95%
confidence level, there is a 5% chance of incorrectly concluding that there is a
difference between the two means when, in fact, the two means are members of the
same population, and the observed difference is due to errors of random sampling.
Minimum Detectable Activity
The minimum detectable activity (MDA) is important in low-level counting, when the
count rate of a sample is almost the same as the count rate of the background, Under
these conditions, the background is measured with a blank—that is, with a sample
holder, such as a planchet—and everything else that may be counted with the actual
sample, except that there is no activity in the blank. The MDA is defined as the
smallest quantity of radioactivity that could be distinguished from the blank under
specified conditions. The MDA depends on the lower limit of detection and on the
counting efficiency of a counting system.
In Example 9.14, we wanted to know whether the two means differed. In determining
the MDA, we wish to know whether the sample activity, M1, is greater than
the blank activity, M2, not merely different from M2. To answer this question we use
a “one-tailed” t test. In the one-tailed test, we resort to the cumulative area under
the normal curve. To include 95% of the area under a normal curve, we have 50%
from −∞ to the mean, and from the mean to the next 45% of the area is 1.645σ .
Therefore, to be significant at the 95% confidence level when the one-tailed test is
appropriate, the difference between the means must be ≥1.645σ .
To decide whether or not a given sample has activity in it, we arbitrarily choose
some count rate greater than that of a zero-activity blank that must be exceeded by
the sample. If this decision level LC is chosen so that it will be exceeded only 5 times
out of 100 by a zero-activity blank, then the net count rate of the sample must be
greater than that of the blank by 1.645 standard deviations of the net count rate.
This decision level considers only an alpha or type 1 error—that is, saying that there
is activity in the sample when in fact there is none. Such an error is often called a
false positive. The 95% confidence level means that we will be wrong, or have false
positives 5 times out of 100. A beta, or type 2, error is one where we say that there is
no activity when there really is activity. That is, a type 2 error is a false negative. Itis
possible to have a very large type 2 error even if the type 1 error is only 5%. Figure
9-41 illustrates the situation where the sample really contains radioactivity and its
mean count rate is equal to the decision level of the background. In this case, the
type 2 error can be 50%. The power of a statistical test, which is defined as
Power = 1 − β, (9.52)