Supplement to Letter from R. 0. Gilbert to T. McCraw dated September 22, 1976
Concerning Sampling Plans for Enewetak Cleanup Survey.
I.
Confidence Limits on True Average (Median) Concentration.
x
Pu concentration
y
log, x
- ue
If x is distributed lognormally, then
Prob[p < y +48 = |-a
(since the y; are normal),
where s = standard deviation of the y's.
y = mean of logs of the sample data,
u = true (unknown) mean of logs
t = "t" value for specified a and n-] degrees of freedom.
Then exp(y + ts/vn) is an approximate (1-a)% upper limit on the median
of the lognormal distribution (original data).
The median is that con-
centration above which and below which half the observations lie.
For Janet (data taken from Fig. B.8.1.i in NVO-140) we have
n = 139, y = 2.180, and s = 1.152
For
a = 0.01, 0.05, and 0.10 we find:
100 (1-a)% Upper
_o
O01
ti38
2.35
05
1. 66
10
-10
1.29
10
Interpretation:
Limit on Median
11 pCi/g
For a = .01 we state:
We are 99% sure that the true
(unknown) median Pu concentration on Janet is less than
or equal to 11 pCi/g (if the data are lognormal).
Discussion:
An alternative approach would be to assume the mean x of the
Pu concentrations is approximately normally distributed.
Then
an upper confidence limit on the true (inknown) mean would be
computed as x
+ ‘8 » Where S now refers to the standard devia-
tion of the original untransformed observations.
Since for
Janet we have n = 139, x = 15.9 pCi/g, s = 20.9 pCi/g we find
the approximate limits: