Question

During the World War II, rockets were fired to attack the city of London. To manage the crisis, London was subdivided into many identical regions. Ambulances and fire fighters were assigned in each region to observe the number of rocket hits while providing rescue. At the same time, statisticians predicted the number of rocket hits follows Poisson distribution. Using goodness of fit test, test whether or not the number of rocket hits follows a Poisson distribution at α=5%.

Number of rocket hits	Poisson Probability	Observed number of regions
0	0.39	229
1	0.37	211
2	0.17	93
3	0.05	35
4 or more	0.02	8

Answer 1

The total frequency here is computed as:
= 229 + 211 + 93 + 35 + 8 = 576

Using the given poisson probabilities, the expected frequencies are computed here as:
E_i = Total Frequency * Poisson Probability

Number of rocket hits	Poisson Probability	Observed number of regions	Expected Observations	(O_i - E_i)^2/E_i
0	0.39	229	224.64	0.0846
1	0.37	211	213.12	0.0211
2	0.17	93	97.92	0.2472
3	0.05	35	28.8	1.3347
4 or more	0.02	8	11.52	1.0756
	1	576	576	2.7632

Using the given observed and expected frequencies as in the above table, we obtain the chi square test statistic for the goodness of fit test here as:

For n - 1 = 4 degrees of freedom, the p-value here is computed from the chi square distribution tables here as:

As the p-value here is 0.6 > 0.05 which is the level of significance, therefore the test is not significant here and we cannot reject the null hypothesis here. Therefore we don't have sufficient evidence here that the given frequencies dont fit the poisson distribution.