Question

Consider the following frequency table of observations on the random variable X.

Values	0	1	2	3	4	5
Observed Frequency	8	25	22	21	16	8

(a) Based on these 100 observations, is a Poisson distribution with a mean of 2.4 an appropriate model? Perform a goodness-of-fit procedure with α=0.05. Which of the following is the correct conclusion?

(b) Which of the following are the correct bounds on the P-value for this test.

Answer 1

Ho: distribution is poisson distribtuion

H1: distribution do not follow poisson distribution

total observation = ΣO =100      

expected proportion = e-µ * µ^x / x!

expected frequency = expected proportion * total observation      
µ =    2.400

X	observed (O)		expected proportion = e-2.4* 2.4^x / x!	expected frequency ( E )	(O-E)²/E
0	8		0.091	9.072	0.127
1	25		0.218	21.772	0.478
2	22		0.261	26.127	0.652
3	21		0.209	20.901	0.000
4	16		0.125	12.541	0.954
5	8		0.096	9.587	0.263

chi square test statistic,X² = Σ(O-E)²/E =   2.47
  
level of significance, α=   0.05              
Degree of freedom=k-2 =    6   -   1   =   5
Critical value =    11.0705   [ Excel function: =chisq.inv.rt(α,df) ]          

since, test stat < critical value, fail to reject Ho

there is not enough evidence that distribution follow poisson distribution..

b)

P value =   0.7804   [ excel function: =chisq.dist.rt(test-stat,df) ]      

0.05< p value <0.10