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Statistics or sadistics?



Questions generated by today's lab.

A long-T radiactive sample was "counted" for one munute. The outcome
was A=145 -->st.dev.=12. Then the background was measured for one minute
yielding B=41 --> st.dev=6.4.

1) The estimated net A-B is 104. What error bar (st.dev.) should be
used for this result. I know it would be 18.4 if the result were
obtained from a multiplication, or from a division of A and B. But
what is the expected standard deviation of a differnce, or sum, in
terms of known sig_A and sig_B?

2) Subsequently the background, measured for 60 minutes, were 2110.
Thus B=35 --> st.dev.=sqrt(2110)/60=0.76. This is much better that
41 +/-6.4 . But the sample is no longer available for long counting.
Can we really benefit from the long counting of the background? My
answer is NO. We might know the long-term mean value of B very well
but we have no way of knowing what fraction of the net count of
(145 +/-12) was actually due to the background. A very long
counting of B does not help us to reduce the error bar around the
value of 104. Do you agree?