4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of \(\alpha\)-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds.
| Number of particles emitted, \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | \(\geqslant 11\) |
| Frequency | 57 | 203 | 383 | 525 | 532 | 408 | 273 | 139 | 45 | 27 | 10 | 6 |
It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.
- Find the contribution to the \(\chi ^ { 2 }\) value of the frequency of 273 corresponding to \(x = 6\) in a goodness of fit test for a Poisson distribution.
- Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
- Given also that the calculated value of \(\chi ^ { 2 }\) is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.