Finding Time Given Number Of Trials And Precision

[lalover]

1. The problem statement, all variables and given/known data

We want to measure the specific activity (number of decays per second) of a radioactive source so that we can use it to calibrate the equipment of the gamma-ray experiment. We use an electronic counter and a timer to measure the number of decays in a given time interval. In round numbers we measure 1000 decays in 5 minutes of observation. For how long should we measure in order to obtain a specific activity with a precision of 1%? Explain.


2. Relevant equations



3. The attempt at a solution

I'm not sure where to begin. I thought about using the formula 1/sqrt(N), but wasn't sure if I could set that equal to the percent of precision (1%). I also hope I put this question in the right category. Thanks in advance!!

[davo789]

You're on the right lines by using 1/sqrt(N) = error. You've just got to work out how to say '1%' in a different way.

[lalover]

Yeah I was thinking of using '.01' and setting it equal to 1/sqrt(N). If I did that, I got an answer of 50 minutes. Does that thinking seem reasonable? Thanks.

[lalover]

Well don't you have to convert the relative error by the amount measure to get absolute error? So would it be 0.01 times 1000? I'm not sure what the amount measured would be here.

[uart]

Yeah I was thinking of using '.01' and setting it equal to 1/sqrt(N). If I did that, I got an answer of 50 minutes. Does that thinking seem reasonable? Thanks.

Yes that's what I had in mind when I read the OP. Make \sqrt{N}/N = 0.01, which corresponds to 10000 counts or approximately 50 min.

BTW. I'm assuming that your 1/sqrt(N) comes from sqrt(N)/N, right?

[lalover]

Well our professor just gave us 1/sqrt(N), but I can see how it's equivalent to sqrt(N)/N. So you disagree with what davo789 said?

[davo789]

Sorry chaps, I've just realised that I've got myself into a terrible muddle!

sqrt(N) is the absolute error on N. So therefore, as you say, sqrt(N)/N is the relative error, and uart's answer is correct, of course! I've edited my OP just to make things clear; apologies for any confusion!

[lalover]

So the original answer of 50 minutes is correct? And "precision of 1%" is just another way of saying the relative error is equal to .01? Thanks again.

[davo789]

Yes, 50mins is correct. Yes, those two statements are equivalent. What you have found is how many measurements you need to take before you have the accuracy/error/precision of 1%.

[lalover]

Ok, great. Thanks a lot.