7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug.
A random sample of 10 patients with depression is taken from a doctor's records.
- Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug.
Given that the claim is correct,
- find the probability that the treatment will be successful for exactly 6 patients.
The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
- Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
- From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.