Question

In an investigation into a patient’s red corpuscle count, the number of such corpuscles
appearing in each of 400 cells of a haemocytometer was counted. The results were as
follows:

No. of red blood corpuscles	0	1	2	3	4	5	6	7	8
No. of cells	40	66	93	94	62	25	14	5	1

It is thought that a Poisson distribution with mean .. provides an appropriate model for this
situation.
(a) Estimate mu, the Poisson parameter.
(b) Test the fit of the Poisson model.

Answer 1

(a) Estimate of $\mu$

$\bar{X}=\sum f.x / \sum f$

= 2.585

(b)

The null hypothesis

H0: The data fits the Poisson model.

Test statistic

(Oi - Ei)2

where Oi : observed frequency

Ei : expected frequency

x	Oi	Ei	(Oi-Ei)^2/Ei
0	40	30.15843	3.2115882
1	66	77.95955	1.83467907
2	93	100.7627	0.598035923
3	94	86.82387	0.593118212
4	62	56.10993	0.618303425
5	25	29.00883	0.553994587
6	14	12.49797	0.180516362
7	5	4.615322	0.032062071
8	1	1.491326	0.161870242
		399.4279	7.784168093

Therefore,

$\chi ^{2}=7.784$

Note :Expected frequencies are calculated as , Ei = 400* P(X=x) = 400*$\frac{e^{-\mu }\mu^{x}}{x!}$

df = 8

P value = 0.4548

Since P value > 0.05

, result is not significant

Fail to reject H0.

There is not enough evidence to conclude that the data do not fit Poisson model.