9231 P41 - Nov 2025 - Q2 - 6 marks
6602
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 | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(\geq 6\) |
|---|---|---|---|---|---|---|---|
| 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.
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