Question

2. For determining the half-lives of radioactive isotopes, it is important to know what the background radiation is for a given detector over a certain period. A γ -ray detection experiment is conducted over 300 one-second intervals (sample g1).

Generate the sample using the R codes below:

set.seed(12345678)

g1 = rpois(300,3)

State the sample (g1) that you obtained. Do these look like observations of a Poisson random variable with mean λ = 3? To answer this question, do the following:

(a) Calculate the sample mean and sample variance. Are they approximately equal?

(b) Find the frequencies of 0, 1, 2, . . . , 7, and 8 or more.

(c) Construct a Poisson probability histogram with λ = 3 and a relative frequency histogram of the sample on the same graph using part (b). Comment on this graph.

(d) Use α = 0.05 and a chi-square goodness-of-fit test to test whether the sample looks like observations of a Poisson random variable with mean λ = 3.

Answer 1

#The R-code is
#a)
set.seed(12345678)
g1=rpois(300,3)
mean(g1) #mean
var(g1) #variance
#Yes, mean and variance are approximately equal to 3
# Output of this code
> #The R-code is
> #a)
> set.seed(12345678)
> g1=rpois(300,3)
> mean(g1) #mean
[1] 2.893333
> var(g1) #variance
[1] 2.563835
> #Yes, mean and variance are approximately equal to 3

#b)
f=table(g1);f #Frequency table
#Output of the code is
> #b)
> f=table(g1);f #Frequency table
g1
0 1 2 3 4 5 6 7 8
17 38 79 64 54 32 10 4 2

#c)
Prob=c(dpois(0:7,3),1-ppois(7,3))
y=rep(0:8,300*Prob)
hist(g1,freq=FALSE,breaks=9,col=2)
hist(y,freq=FALSE,breaks=9,col=3,add=T)
legend("topright",legend=c("Random number from P(3)","Poisson distribution with lambda = 3 "),fill=2:3)
#The genertaed random sample is good fit for poisson distribution
#Output of code is
> #c)
> Prob=c(dpois(0:7,3),1-ppois(7,3))
> y=rep(0:8,300*Prob)
> hist(g1,freq=FALSE,breaks=9,col=2)
> hist(y,freq=FALSE,breaks=9,col=3,add=T)
> legend("topright",legend=c("Random number from P(3)","Poisson distribution with lambda = 3 "),fill=2:3)
> #The genertaed random sample is good fit for poisson distribution

Histogram of g1 80 Random number from P(3) Poisson distribution with lambda = 3 09 Frequency 40 20 0 0 4 6 8 N g1

#d)
#The Null and alternative hypothesis are
#H0:The sample is good fit for poisson distribution with lambda=3
#Ha:The sample is not good fit for poisson distribution with lambda=3
chisq.test(f,p=Prob)
#Here pvalue=0.5073>alpha=0.05 then we fail to reject the null hypothesis Ho and conclude that sample looks like observations of a Poisson random variable with mean λ = 3
#Output of the code is
> #d)
>#The Null and alternative hypothesis are
>#H0:The sample is good fit for poisson distribution with lambda=3
>#Ha:The sample is not good fit for poisson distribution with lambda=3
> chisq.test(f,p=Prob)

Chi-squared test for given probabilities

data: f
X-squared = 7.2746, df = 8, p-value = 0.5073

Warning message:
In chisq.test(f, p = Prob) : Chi-squared approximation may be incorrect
>#Here pvalue=0.5073>alpha=0.05 then we fail to reject the null hypothesis Ho and conclude that sample looks like observations of a Poisson random variable with mean λ = 3