Question

The following counts of vehicles arriving at a toll station, over 1-minute intervals, were made by an engineering student: 2,0,4,1,2,4,2,1,6,2,5,4,4,3,3,2,2,1,2,3,2,5,2,0,1,2,0,3,1,1, 2,3,5,6,3,4,3,4,0,6,3,2,1,4,4,0,1,4,3,7,0,0,2,3,2,4,3,2,4,5 (a) Assuming the vehicle arrivals were generated by a Poisson process, compute the maximum likelihood estimate of the arrival rate. (b) Test the hypothesis that the arrival counts follow a Poisson distribution, at the α=0.05 significance level.

Answer 1

a)

sample mean is unbiased estimator of lambda

sample mean = sum of all terms / number of terms

b)

this is chi-square goodness of fit test

	p	Oi	Ei	(Oi-Ei)^2/Ei
0	0.0695	7	4.1690	1.9224
1	0.1853	8	11.1174	0.8741
2	0.2471	15	14.8231	0.0021
3	0.2196	11	13.1761	0.3594
4	0.1464	11	8.7841	0.5590
5	0.0781	4	4.6848	0.1001
>5	0.0541	4	3.2455	0.1754
	1	60	60.0000	3.9926
		TS	3.9926
		critical value	12.5916
		p-value	0.6777

TS = 3.9926

critical value = 12.5916

since TS < critical value

we fail to reject the null hypothesis

we conclude that there is not sufficient evidence that the arrival counts does not follow a Poisson distribution

Formulas

	p	Oi	Ei	(Oi-Ei)^2/Ei
0	=POISSON.DIST(A2,8/3,0)	7	=$C$10*B2	=(C2-D2)^2/D2
1	=POISSON.DIST(A3,8/3,0)	8	=$C$10*B3	=(C3-D3)^2/D3
2	=POISSON.DIST(A4,8/3,0)	15	=$C$10*B4	=(C4-D4)^2/D4
3	=POISSON.DIST(A5,8/3,0)	11	=$C$10*B5	=(C5-D5)^2/D5
4	=POISSON.DIST(A6,8/3,0)	11	=$C$10*B6	=(C6-D6)^2/D6
5	=POISSON.DIST(A7,8/3,0)	4	=$C$10*B7	=(C7-D7)^2/D7
>5	=1-SUM(B2:B7)	4	=$C$10*B8	=(C8-D8)^2/D8
	=SUM(B2:B8)	=SUM(C2:C8)	=SUM(D2:D8)	=SUM(E2:E8)
		TS	=SUM(E2:E8)
		critical value	=CHISQ.INV(0.95,6)
		p-value	=1-CHISQ.DIST(E10,6,1)