Problem

The manager of a car park claims that the number of cars entering the car park follows a Poisson distribution with mean $2.8$. The numbers of cars entering the car park are recorded on a working day during successive 5-minute periods. The following table contains the observed frequencies, together with most of the expected frequencies and their contributions to the $\chi^2$-test statistic.

Number of cars	0	1	2	3	4	5	$\ge 6$
Observed frequency	2	15	31	29	13	3	7
Expected frequency	6.081	17.03	23.84	$p$	15.57	8.721	6.511
$\chi^2$-test statistic	2.739	0.241	2.152	$q$	0.425	3.753	0.037

(a) Find the value of $p$ and the value of $q$.

(b) Carry out a goodness of fit test at the 5% significance level to investigate the manager’s claim.