In a training center, a new strategy was implemented. After the implementation of this new strategy the training center claimed that on an average $80\%$ of the students passed. In order to find out whether this claim can be justified, we take a random sample of $30$ students and see that $60\%$ of the students passed. Do we have sufficient evidence to accept the $80\%$ claim?
It is generally easier to reject a claim than to "accept" it and if you consider failing to reject a claim to be the same as accepting it then that leads to poor science based on too small of sample sizes (http://dilbert.com/strips/comic/2004-04-13/).
Traditional hypothesis testing could look at testing "At least 80%" or "At Most 80%". There are also tests of equivalence, but you need to determine what you mean by "equivalent", e.g. you could establish equivalence if you were 95% confident that the true percentage was between 75% and 85% (or another meaningful region), but the given data will not support that.
If you really need proof that the value is exactly 80% then you will need an infinite amount of data (or at least the entire population).
Probably not. A 95% confidence interval for a binomial distribution with 18 out of 30 students passing is 0.4 to 0.77.
It's not an outlandish claim, but it also doesnt seem very likely. What I don't understand is why you don't have access to everyone's results...but that probably falls well outside the realm of statistics.