# Radioactive Decay: Analyzing 1000 Events at 5% Risk Level

• skrat
In summary, the experiment shows that out of 1000 events, 30% have a time between two consecutive events longer than 1 second and 5% have a time longer than 2 seconds. Using the theory of radioactive decay, the predicted numbers for these intervals are 63.21%, 23.2%, and 13.5% respectively. The calculated chi-squared value of 73.9 exceeds the expected value of 5.9915, indicating that the hypothesis of a 1-second decay time cannot be denied at a 5% risk level.

## Homework Statement

Analyzing 1000 events (each event is one radioactive decay of an unknown sample), we notice that the time between two consecutive events is larger than 1 second in 30% of the cases while in 5% it is longer than 2 seconds. Can we, at 5% risk level deny the hypothesis, that the characteristic decay time is 1 second long?

## The Attempt at a Solution

I really don't know how to begin here. I really don't. I do assume that I will have to use either Student distribution or Chi-squared distribution. But I have no idea how to start and what to do :/

Which distribution does a radioactive decay have?
Which fraction of events do you expect to be larger than 1 second, and how likely is it to have 300 out of 1000 then?

Events should be normally distributed.
Ammm If I understood your question correctly: 300 events should take longer than 1 second. How likely? Am, I would guess with 95% probability.

skrat said:
Events should be normally distributed.
The times of the events? Are you sure?
Ammm If I understood your question correctly: 300 events should take longer than 1 second. How likely? Am, I would guess with 95% probability.
300 is your observed number, I am asking for the prediction. And the 95% is something you will have to compare your result with later.

Well radioactive decay is a completely stochastic process, knowing this I would say it is normally distributed around characteristic time. But I guess it isn't. My next option, which is a result of guessing, would be that the number of decays observed over a given time interval obeys Poission statistics (distribution). If that is the case, than I would highly appreciate a two sentenced explanation, if not than I am lost. :/

Again, I thought that since all events are stochastic and since measurements show that in 30% of the consecutive decays the time is longer than 1 second, it is also my prediction that 30% of number of decays over a given time interval will always be longer than 1 second.

Do you know the concept of half-life? What does it suggest if there is a fixed time during which half of the remaining particles decay?

Ok, I think I understand what I am supposed to do. Please check if this seems to be good.

##N=1000## is the number of decays. Experiment tells us that ##N(1 s<t<2 s)=300## and ##N(t>2 s)=50## therefore ##N(t<1 s)=650##.
Of course $$\frac{dP}{dt}=\frac 1 \tau e^{-\frac t \tau}$$ So the theory for first interval (##t<1s##) predicts $$F_1=\int _0^1\frac 1 \tau e^{-\frac t \tau}=1-e^{-1}=0.6321$$ and similary ##F_2=\int _1^2\frac 2 \tau e^{-\frac t \tau}=e^{-1}-e^{-2}=0.232## and ##F_3=\int _2^\infty\frac 2 \tau e^{-\frac t \tau}=e^{-2}=0.135##.

This now leaves me with $$\chi ^2 =\frac{(650-632)^2}{632}+\frac{(300-232)^2}{232}+\frac{(50-135)^2}{135}=73.9$$ while the data from the tables say that $$\chi ^2(3-1) ^{ 5 \text{%}}=5.9915$$

So I guess the answer is no?

I think the 30% "longer than one second" include the 5% "longer than 2 seconds".

The three values are not uncorrelated (because each event has to be in one of the classes) so you cannot add their ##\chi^2##-contributions like that, but the last term alone is sufficient to draw the same conclusion.

## 1. What is radioactive decay and why is it important to study?

Radioactive decay is the process by which unstable atoms release energy and particles to become more stable. It is important to study because it plays a crucial role in various fields such as nuclear energy, radiometric dating, and medical imaging.

## 2. What does it mean to analyze 1000 events at a 5% risk level?

Analyzing 1000 events at a 5% risk level means that out of 1000 trials, there is a 5% chance of obtaining a result that is due to chance rather than the actual phenomenon being studied. This level of risk is typically used in scientific studies to ensure the reliability of the results.

## 3. How is radioactive decay measured and recorded?

Radioactive decay is measured and recorded using a variety of methods such as counting the number of decay events over a period of time, measuring the decay products, or using specialized instruments like a Geiger counter. The data is then analyzed and recorded in a graph or table for further study.

## 4. What factors can affect the rate of radioactive decay?

The rate of radioactive decay can be affected by various factors such as the type of radioactive material, the temperature, and the presence of other particles. Additionally, external factors such as pressure and electric fields can also influence the rate of decay.

## 5. What are the potential risks of studying radioactive decay?

The potential risks of studying radioactive decay depend on the materials and methods used. In general, there is a risk of exposure to radiation which can have harmful effects on the body. It is important for scientists to follow proper safety protocols and use protective equipment to minimize these risks.