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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?

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    $\begingroup$ Please see stats.stackexchange.com/questions/71127/… The concept is the same. I suppose this could even be marked as a duplicate. $\endgroup$ – Cristian Dima Sep 30 '13 at 17:17
  • $\begingroup$ @Merovingian Not quite. That Q. is for continuous variables with a specified standard deviation, this is for a count (i.e. binomial) proportion. $\endgroup$ – Glen_b -Reinstate Monica Aug 9 '14 at 9:18
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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).

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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.

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    $\begingroup$ (1) One crucial issue in any good analysis of this question is whether to use a two-sided or one-sided interval (or test). What justifies your two-sided approach? (2) By not exploiting the asserted $80\%$ rate, your solution lacks power. $\endgroup$ – whuber Nov 29 '13 at 21:16

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