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 $normalcdf(.8, e99, .15, 0.0319)$?
Best Answer
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).