4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 4.1}
| Number of vehicles | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | \(\geqslant 11\) |
| Frequency | 36 | 33 | 14 | 10 | 4 | 1 | 0 | 0 | 1 | 0 | 1 | 0 |
\end{table}
- Use the results to find an estimate for \(\mu\).
The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Fig. 4.2}
| \multirow[b]{2}{*}{1} | A | B | C | D | E |
| Number of vehicles | Observed frequency | Poisson probability | Expected frequency | Chi-squared contribution |
| 2 | 0 | 36 | 0.2725 | 27.2532 | 2.8073 |
| 3 | 1 | 33 | 0.3543 | 35.4291 | |
| 4 | 2 | 14 | | | 3.5400 |
| 5 | \(\geqslant 3\) | 17 | | | 0.5145 |
| 6 | | | | | |
- Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
- C4
- D5
- E3
- In this question you must show detailed reasoning.
Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level. - Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged.
Make two comments about this revised test.