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I'm currently in the planning stages of an experiment that I hope will detect a 0.155% signal variation (relative magnitude) within a reasonable time frame (less than 6 months ideally). I've calculated the rate of (usable) data will be around 68 events per day, though it should be stressed this is a random variable. Now I'm trying to work out how many days will I need to run the detector for to see the variation with a confidence level of $3\sigma$?

Some other details that may (or may not) be relevant include: the variation in the signal is expected to be sinusoidal with a period of 0.5 days. For this reason I reduced my useful event rate to 34 (Ie half) as clearly there is no variation to see when the sinusoidal signal is at or close to the mean value.

I've been googling for a method to predict the size of a data set necessary to see such a small signal variation but have come up with nothing. I would be extremely grateful for any hints / tips anyone could offer.

EDIT: With the greatest thanks to @shabbychef I feel this question has now been satisfactorally answered on physics.SE here.

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  • $\begingroup$ I cannot figure out what you are trying to detect: the amplitude? a DC offset? $\endgroup$ – shabbychef Jul 11 '11 at 18:26
  • $\begingroup$ @shabbychef: the only data I'm interested in is the detection rate, whilst the detector can record (something related to) the energy it is merely the variation of the rate that I need to 'see'. Apologies if I'm being a bit vague, I've been asked to keep certain aspects of the experiment confidential for the time being. $\endgroup$ – qftme Jul 11 '11 at 18:40
  • $\begingroup$ so perhaps the random variable of interest is 'detects per day' and you are trying to estimate the variance in that RV? $\endgroup$ – shabbychef Jul 11 '11 at 18:51
  • $\begingroup$ @shabbychef: yes the random variable is 'detects per day' but what I'm really trying to estimate is how long I need to run the detector to get a definite sign of its variance. The difficulty being the relative magnitude of the variance is only 0.155% and there are only approximately 68 events per day. An analogous experiment would be like trying to 'see' the ebb-and-flow of the tide (0.5 day period) by locating a photon detector at the bottom of the ocean (though of course this wouldn't work and is silly, but it is quite similar.) Hope that clarifies it a bit, let me know if not. $\endgroup$ – qftme Jul 11 '11 at 20:42
  • $\begingroup$ the sign of the variance is positive! But seriously, what exactly are you trying to estimate? If the RV is the number of detects per day, are you trying to estimate the mean, the variance, the coefficient of variation, autocorrelation in time, what? $\endgroup$ – shabbychef Jul 11 '11 at 21:05
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If you can put your question in the form of a t-test, then Lehr's rule can be used to estimate the required sample size. For a 2-sided, one sample t-test at the 0.05 level, one can achieve power of 0.80 by using $n = 8 / \Delta^2$, where $\Delta$ is the 'signal to noise' (mean divided by standard deviation). I cannot figure out if you mean this to be 0.155%, but if so, you're looking at roughly 100K years!

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