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I have established the following hypothesis:

The probability of successfully passing the course in this trial is less than 50% (alternative hypothesis)

The probability of failing the course in this trial is greater than or equal 50% (null hypothesis)

and perform a binomial test in R like the following:

binom.test(10,30,p=0.5)

which means that from 30 students only 10 pass the course, so I am making the hypothesis than 50% of the students will pass the course. I got the following results:

Exact binomial test

data:  10 and 30
number of successes = 10, number of trials = 30, p-value = 0.09874
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
 0.1728742 0.5281200
sample estimates:
probability of success 
             0.3333333 

How can I interpret these results? does it mean that I fail to reject the null hypothesis so that I do not have enough evidence that more of half of the class would fail the course?

Update: So should I apply?

binom.test(20,30,.5,alternative="less")

        Exact binomial test

data:  20 and 30 
number of successes = 20, number of trials = 30, p-value = 0.9786
alternative hypothesis: true probability of success is less than 0.5 
95 percent confidence interval:
 0.0000000 0.8066916 
sample estimates:
probability of success 
             0.6666667 

So I failed to reject the null hypothesis, that means that there is not enough evidence that more of half of the class will fail the course?

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  • $\begingroup$ Your current wording for null and alternative hypotheses appear to say the same thing. Your null is $50\% \le P(Fail) = 1 - P(\neg Fail) = 1 - P(Pass)$ or $P(Pass) \le 1 - 50\% = 50\%$. So it is difficult to know what your formulation is. Take Greg Snow's advice and attempt to formulate everything in the affirmative -- passing. $\endgroup$ – A. Webb Mar 27 '15 at 21:24
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A couple of points:

You don't have equality in either hypothesis, it should go in the null: probability of failing is greater than or equal to 50%.

Your hypotheses are in terms of failing, but your data is in terms of passing, this can cause confusion, it would probably be clearer to keep these consistent.

Your hypotheses are one-sided but you ran the two-sided test (note the 'not equal' in the output). Rerunning this with the proper one-sided test (look at the arguments on the help page for binom.test) will make a big difference in your final conclusion (especially if you are using the traditional $\alpha = 0.05$).

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  • $\begingroup$ thanks @GregSnow I updated the question, is fine the conclussion I arrived? $\endgroup$ – Layla Mar 27 '15 at 21:00
  • $\begingroup$ @Nadine, your 2 hypotheses are still the same (you changed both pass to fail and greater to less), so it is still not clear. If 20 out of 30 passed and you want to show the pass rate is less than 50% then your code is correct, but if 10 out of 30 passed then it is backwards and your conclusion will be very different. $\endgroup$ – Greg Snow Mar 27 '15 at 21:23
  • $\begingroup$ what I want to hypothesize is that more of half of the class will fail the course, and according to my data only 10 people out of 30 pass it, the other 20 failed the course $\endgroup$ – Layla Mar 27 '15 at 22:53
  • $\begingroup$ @Layla, look at your last set of output. It says "number of successes = 20" and also "probability of success is less than 0.5", those 2 are not consistent with what you say you are testing. You need to change either the 20 to 10, or the "less" to "greater" (but not both). $\endgroup$ – Greg Snow Mar 28 '15 at 16:49

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