Question

2. A fire station logs the number of callouts occuring each day for a year and tabulates the results: O < c(c0-8, c1-12, c2-36, c3-54 , c4-67 , c5-66, с7-37, c8-23, с9-10 , с10-11) c6-41,

Answer 1

$(a)~Let~X=random~variable~denoting~no.~of~call outs~in~a~day~and\\ X~takes~any~value~between~\{0,1,2,3,.....\}~and~since~the~observed~frequency\\ ~of~X\geq 11~is~zero~so~we~can~assume~X\sim Poisson(\lambda).\\ (b)$

R code:

f=c(8,12,36,54,67,66,41,37,23,10,11)
sum(f)

Output:

> sum(f)
[1] 365

$(c)~Since~\lambda~is~unknown~so~we~can~estimate~\lambda~by~using~method~of~moment~estimation~and\\ \hat{\lambda}=\bar{x}=\frac{1}{365}\sum_{i=0}^{10}if_i$

R code:

x=0:10
f=c(8,12,36,54,67,66,41,37,23,10,11)
m=sum(x*f)/sum(f)
m
prob=0:10*0
for(i in 1:10)
{
prob[i]=dpois(i-1,m)
}
prob[11]=1-sum(prob)
prob #estimated probability
e=365*prob
e#Expected frequency

Output

> e#Expected frequency
[1] 3.164364 15.024228 35.667105 56.448487 67.003581 63.625866 50.348688
[8] 34.150401 20.268029 10.692388 8.606863

(e)

R code:

x=0:10
f=c(8,12,36,54,67,66,41,37,23,10,11)
m=sum(x*f)/sum(f)
m
prob=0:10*0
for(i in 1:10)
{
prob[i]=dpois(i-1,m)
}
prob[11]=1-sum(prob)
prob #estimated probability
e=365*prob
e#Expected frequency
chisq=sum((f-e)^2/e)
chisq

Output:

> chisq
[1] 11.24837

(f)

p-value:

R code:

1-pchisq(chisq,11-2)

Output:

0.2590679