Uncertainty in measurement (GRE question)

In summary, a student needs to count for a certain amount of time in order to establish the rate of decay for a long-lived radioactive isotope with an uncertainty of 1 percent. This can be determined using the Poisson distribution and the standard deviation of the measured decay events. The time required can be found by dividing the standard deviation by the average number of decays per second.
  • #1
0ddbio
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This is actually a GRE problem, I'm just trying to go through and understand them all for studying purposes, but I don't understand how to do this one at all.

Homework Statement


A student makes 10 one-second measurements of the disintegration of a sample of long-lived radioactive isotope and obtains the following values.
3, 0, 2, 1, 2, 4, 0, 1, 2, 5
How long should the student count to establish the rate to an uncertainty of 1 percent?

Homework Equations


I have no idea. But I would guess that standard deviation might be involved..
[tex]\sigma_{x}^{2}=\langle x^{2}\rangle-\langle x\rangle^{2}[/tex]

The Attempt at a Solution


But I am unsure how to get time involved in this. But the standard deviation is 2.4
Any help is appreciated.
 
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  • #2
Here's the Wikipedia article on the Poisson distribution: http://en.wikipedia.org/wiki/Poisson_distribution

The Poisson distribution is the probability density function for the number of events that will occur during some interval if the average number of events that occur per interval is a known constant (it will not always be exactly the average because the process is random; the Poisson distribution is what describes the spread). For a radioactive isotope, an "event" is a decay. Obviously as the sample decays the expected number of decays per second decreases, but I think we are meant to ignore that since the isotope is "long-lived," and presumably the sample is very large. The Poisson distribution has the interesting property that the (ideal, not sample) standard deviation is the square root of the average,

The average of the ten numbers you listed is 2, so we can take that to be the average number of decays per second. If you take longer time intervals, the average and standard deviation both increase.

This problem is asking you to find the time required for the standard deviation divided by the average to reach 0.01.
 
  • #3

FAQ: Uncertainty in measurement (GRE question)

What is uncertainty in measurement?

Uncertainty in measurement refers to the inherent imprecision or error that exists when making any kind of measurement. It is the range of values within which the true value of a measurement is expected to lie.

Why is uncertainty in measurement important?

Uncertainty in measurement is important because it provides a measure of the reliability and accuracy of a measurement. It allows us to understand the limitations and potential errors in our data and results.

How is uncertainty in measurement calculated?

Uncertainty in measurement is calculated by analyzing various sources of error and combining them using statistical methods. This can include factors such as instrument precision, human error, and environmental conditions.

How can uncertainty in measurement be reduced?

Uncertainty in measurement can be reduced by using more precise instruments, improving measurement techniques, and controlling environmental factors that may affect the measurement. It is also important to repeat measurements and use statistical analysis to determine the most accurate value.

What is the difference between uncertainty and accuracy?

Uncertainty refers to the range of values within which the true value of a measurement is expected to lie, while accuracy refers to how close a measurement is to its true value. In other words, uncertainty is a measure of imprecision, while accuracy is a measure of correctness.

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