Question

a. The average rate of emission radioactive particles from a source was measured over a long period, and found to be 10 particles per unit time. After an experimental treatment had been applied to the source, a further sample was examined and emitted 17 particles in unit time. Test at 5% the null hypothesis that the rate of emissions is unchanged.

Answer 1

The number of emitted particles has Poisson distribution.

$P\{\xi=k\}=\frac{\lambda^k}{k!}e^{-\lambda}, k=0,1,\ldots$

We have $\lambda_1=10, \lambda_2=17.$

We will take samples of sizes $n_1=20, n_2=20$ from the first and the second populations.

$P\{\xi=20\}=\frac{10^{20}}{20!}e^{-10}\approx 0.0019.\\ P\{\xi=20\}=\frac{17^{20}}{20!}e^{-17}\approx 0.0692.$

We get relative frequencies:

$\omega_1=0.0019\\ \omega_2=0.0692$

$H_0: p_1=p_2; H_1: p_1\neq p_2\\ \text{where } p_i \text{ is probability that event A (the particle emits)}\\ \text{will happen in the i-th population}.$

We will use the following random variable:

$u=\frac{\omega_1-\omega_2}{\sqrt{(\omega_1+\omega_2)(1-(\omega_1+\omega_2))(\frac{1}{n_1}+\frac{1}{n_2}})}\\ u_{obs}\approx -0.8281\\ \Phi(u_{cr})=\frac{1-\alpha}{2}=0.475\\ u_{cr}=1.96\\ (-\infty,-1.96)\cup (1.96,\infty)\text{ --- critical region}.\\ u_{obs}\text{ does not fall into the critical region. So we accept } H_0.$

The rate of emission did not change after an experimental treatment.