- In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.
The results are summarised in the following table.
| Number of heads | 0 | 1 | 2 | 3 | 4 | 5 |
| Frequency | 3 | 10 | 45 | 62 | 38 | 12 |
Marcus believes that a \(\mathrm { B } ( 5,0.5 )\) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows
| Number of heads | 0 | 1 | 2 | 3 | 4 | 5 |
| Expected frequency | \(r\) | 26.56 | \(s\) | \(s\) | 26.56 | \(r\) |
- Find the value of \(r\) and the value of \(s\)
- Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the \(\mathrm { B } ( 5,0.5 )\) distribution is a good model for these data.
You should state clearly your hypotheses, the test statistic and the critical value used.
Nima believes that a better model for these data would be \(\mathrm { B } ( 5 , p )\) - Find a suitable estimate for \(p\)
To test her model, Nima uses this value of \(p\), to calculate expected frequencies as follows
| Number of heads | 0 | 1 | 2 | 3 | 4 | 5 |
| Expected frequency | 2.07 | 14.65 | 41.44 | 58.63 | 41.47 | 11.74 |
- State,
- giving your reasons, the degrees of freedom
- the critical value
that Nima should use for a test at the 5\% significance level.
- With reference to Marcus' and Nima's test results, comment on
- the probability of the coin landing on heads,
- the independence of the spins of the coin.
Give reasons for your answers.