Solved – how to analyze the results for this binomial test

hypothesis testing

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?

Best Answer

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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