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Probability of a value of x, given a mean value of x bar and standard deviation s 
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#1
Jul110, 08:46 PM

P: 300

Let's say I monitors my local aquifer (Edwards) using well J17 over the past 8 summers (not counting this summer). I found that the mean water level/day is 680 ft above sea level, with a standard deviation of 30. Assuming a normal distribution, how do I find the probability that the aquifer level will drop to between 660 and 641 ft sometime this summer, at which time Stage 1 drought restrictions are in place?



#2
Jul110, 08:50 PM

P: 300

Also, how do I find the probability that the level will be exactly 641 ft?



#3
Jul110, 09:03 PM

P: 587

Assuming a normal distribution won't do you any good. The water levels in the aquifer at different days are certainly dependent so you need to model them as a time series. One you have estimated a suitable time series model from your observations, the question of how likely the water level is to drop to below a certain threshold can be answered in different ways.
One thing which usually comes up in this context is that the target region for which you want to estimate the probability is well outside the typical range of the time series under consideration and reaching it is therefore considered an extreme event, which then leads to the discipline called extreme value theory for time series. 


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