A vaccine for Covid-19 is known to be $90\%$ effective, i.e. $90\%$ of vaccine recipients are successfully immunised against Covid-19. A new (different) vaccine is tested on $100$ patients and found to successfully immunise $96$ of the $100$ patients. Is the new vaccine better?
Hint: Assume the new vaccine is equally effective as the original vaccine and consider using an appropriate distribution.
I am not sure how to tackle this problem, but my answer is:
Not necessarily, since the sample is $100$ and it is unknown what is the sample of the first vaccine, hence we cannot know whether the second vaccine is better.
What's the correct way to answer this problem?
This is an Hypothesis Test exercise.
Consider as the null hypothesis of 90% success a binomial distribution. The extreme probability
$$\mathbb{P}[X\geq 96|p=0.9]\approx 2.40\%$$
Thus you can reject the hypothesis that the old vaccine is better than the new one with a p-value equal or less than 2.4%
this means that the test is significant but not higly significant.
Usually the test is significant if $\text{p-value} < 5%$ and highly significant if $\text{p-value}<1\%$