6 Cosmic rays passing through the upper atmosphere cause muons, and other types of particle, to be formed. Muons can be detected when they reach the surface of the earth. It is known that the mean number of muons reaching a particular detector is 1.7 per second. The numbers of muons reaching this detector in 200 randomly selected periods of 1 second are shown in Fig. 6.1.
\begin{table}[h]
| Number of muons | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Frequency | 34 | 65 | 55 | 24 | 14 | 6 | 2 | 0 |
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\end{table}
- Use the values of the sample mean and sample variance to discuss the suitability of a Poisson distribution as a model.
The screenshot in Fig. 6.2 shows part of a spreadsheet to assess the goodness of fit of the distribution Po(1.7).
\begin{table}[h]
| A | B | C | D | E |
| 1 | Number of muons | Observed frequency | Poisson probability | Expected frequency | Chi-squared contribution |
| 2 | 0 | 34 | 0.1827 | 36.5367 | 0.1761 |
| 3 | 1 | 65 | | | |
| 4 | 2 | 55 | 0.2640 | 52.7955 | 0.0920 |
| 5 | 3 | 24 | 0.1496 | 29.9175 | 1.1704 |
| 6 | 4 | 14 | | | 0.1299 |
| 7 | \(\geqslant 5\) | 8 | 0.0296 | 5.9230 | 0.7284 |
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
- Calculate the missing values in each of the following cells.
- C3
- D3
- E3
- Explain why the numbers for 5, 6 and at least 7 muons have been combined into the single category of at least 5 muons, as shown in Fig. 6.2.
- In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.