# how to analyze the results for this binomial test?

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?

• 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. – A. Webb Mar 27 '15 at 21:24

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 $\alpha = 0.05$).