- A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.
| Number of successes | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Number of practices | 4 | 6 | 3 | 12 | 10 | 7 | 4 | 2 | 2 |
The researcher decides to model this data using a binomial distribution.
- State two necessary assumptions that the researcher made in order to use this model.
- Show that the mean number of successes per sample is 3.54
He decides to use this mean to calculate expected frequencies. The results are shown in the table below.
| Number of successes | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Expected frequency | 0.47 | 2.96 | 8.23 | 13.07 | \(f\) | 8.23 | 3.27 | 0.74 | \(g\) |
- Calculate the value of \(f\) and the value of \(g\). Give your answers to 2 decimal places.
- Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.