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I have a question regarding binom.test() in R (3.3.3 on Windows 10):

I have a null hypothesis that the p(success)=0.0033 and I want to know whether I can reject this null hypothesis at a signficance level of 0.10 (two-tailed).

I observed 1 success in 1497 trials so I use:

> binom.test(1,1497,0.0033,conf.level=0.9)
        Exact binomial test
data:  1 and 1497
number of successes = 1, number of trials = 1497, p-value = 0.1062
alternative hypothesis: true probability of success is not equal to 0.0033
90 percent confidence interval:
 3.426347e-05 3.164954e-03
sample estimates:
probability of success 
          0.0006680027 

My null hypothesis p=0.0033 is outside the confidence interval (> 0.00316) so I am tempted to reject it, but then when I look at the p-value, it is 0.1062 (> 0.10) which would say I cannot reject it.

Is this an inconsistency or am I missing something?

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@Tim provided a link to a more general question regarding t-tests.

The answer for your question would be that the 90% confidence interval around the estimated value is not identical to the acceptance region of the hypothesis test, but there is only a link or similarity between the 2. Your p-value of 0.1062 is close enough to the threshold of 10% to make this seemingly paradox occur. But it is not an actual paradox.

The confidence interval is based on the estimated value of the 'probability of success' and so are the standard deviations which are responsible for the length of the confidence interval.

The acceptance region of the test which is performed here would be based on your assumed p0=0.0033 and so are the standard deviations which make this acceptance region symmetric interval around 0.0033 wider than the confidence interval. This is just enough for the acceptance region around 0.0033 to still include your estimated probability of success 0.0006680027, whereas the slightly less wide confidence interval does not include 0.0033 leading to the effect you see here and perceive as a paradox.

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