Question

A pathologist has been studying the frequency of bacterial colonies within the field of a microscope using samples of throat cultures from healthy adults. Long-term history indicates that there is an average of 2.94 bacteria colonies per field. Let r be a random variable that represents the number of bacteria colonies per field. Let O represent the number of observed bacteria colonies per field for throat cultures from healthy adults. A random sample of 100 healthy adults gave the following information.

r	0	1	2	3	4	5 or more
O	13	16	28	18	18	7

(a) The pathologist wants to use a Poisson distribution to represent the probability of r, the number of bacteria colonies per field. The Poisson distribution is given below.P(r) =

e^−λλ^r

r!

Here λ = 2.94 is the average number of bacteria colonies per field. Compute P(r) for r = 0, 1, 2, 3, 4, and 5 or more. (Round your answers to three decimal places.)

P(0)	=
P(1)	=
P(2)	=
P(3)	=
P(4)	=
P(5 or more)	=

(b) Compute the expected number of colonies E = 100P(r) for r = 0, 1, 2, 3, 4, and 5 or more. (Round your answers to one decimal place.)

E(0)	=
E(1)	=
E(2)	=
E(3)	=
E(4)	=
E(5 or more)	=

(c) Compute the sample statistic χ² =

	(O − E)2 E

and the degrees of freedom. (Round your test statistic to three decimal places.)

d.f.	=
χ2	=

Answer 1

from above formula:

P(0)=	0.053
P(1)=	0.155
P(2)=	0.228
P(3)=	0.224
P(4)=	0.165
P(5 or more)=	0.175

b)

E(0)=	5.3
E(1)=	15.5
E(2)=	22.8
E(3)=	22.4
E(4)=	16.5
E(5 or more)=	17.5

c)

r	p(r )	observed	expected	chi square
		O	E	(O-E)2/E
0	0.053	13	5.3	11.19
1	0.155	16	15.5	0.02
2	0.228	28	22.8	1.19
3	0.224	18	22.4	0.86
4	0.165	18	16.5	0.14
5 or more	0.175	7	17.5	6.30
	total	100.00	100	19.690

df =categories-1 =4

X² =19.690

p value =

0.0006

reject Ho