2 A drug for treating a particular minor illness cures, on average, 78\% of patients. Twenty people with this minor illness are selected at random and treated with the drug.

1. (A) Find the probability that exactly 19 patients are cured.
   (B) Find the probability that at most 18 patients are cured.
   (C) Find the expected number of patients who are cured.
2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.