Statistics interpretation of a real world application (Sampling Proportions) A report states that 15% of men in the US are left handed. If a random sample of 125 men contains only 10 men who are left handed, is it reasonable to assert that 15% of all males are left-handed is unlikely? Or is that particular random sample unlikely? 
We can see here that the sample proportion is $\hat{p} = \frac{x}{n} = \frac{10}{125} = 0.08$ which is less than the population proportion. We find that the probability of this occurring is 1.43%, does this mean that we reject the assertion that 15% of men in the US are left handed or do we say that the random sample is unlikely?
What about the flip side - we find that in a random sample of 125 men 100 men are left handed, is it reasonable to assert that 15% are left handed is unlikely? Or is that particular random sample unlikely? We find that the probability of this occurring is $100*normalcdf(-e99, .8, .15, 0.0319) = 100\%$, does this mean that we find this occurrence extremely likely? Or should I say $normalcdf(.8, e99, .15, 0.0319)$? 
 A: I think answering this question requires use of Bayes theorem. 
The extent to which you think the particular random sample is unlikely, or the 15% hypothesis is false, depends on how strong your 'prior' belief is that the 15% hypothesis is correct. Your prior beliefs would presumably depend on your knowledge of previous studies which produced the 15% estimate.
Denote the hypothesis that $X\%$ of men are left handed as $X\%LH$.
Bayes theorem then says:
$$ Pr(X\%LH|data) = Pr(data|X\%LH)*Pr(X\%LH)/Pr(data) $$
Here $Pr(data|X\%LH)$ is the probability of the data given that $X\%LH$ is true, $Pr(X\%LH)$ is your prior probability of  $X\%LH$, and Pr(data) is just a normalising constant which ensures that the quantity on the left hand side integrates to 1 with respect to $X\%LH$ (so it is a valid probability distribution). 
A: If evidence 1 states 15% and evidence 2 states 8%, the likelihood of the true population parameter being 9.5 is more likely, given no other information. However, if we take into account that each evidence has different statistical power, we could weigh the information differently. For example, consider if evidence 1 is based on a sample size of 1250. We would infer that the smaller sample is unlikely to reflect the actual population as accurate as the larger sample. However, we also need to consider to what extent the samples are representative of the population. Perhaps evidence 1 has a better coverage of factors that affect handedness that is considered to be a genetic trait.
If evidence 1 provides the actual population parameter, having asked all American men, then we would say that the random sample is not representative.
For evidence 3, 20%, the above applies again.
