Probability question about left-handed US presidents. There have been 7 presidents since Gerald Ford (including Ford).  5 have been left-handed, 2 have been right handed.  While 90% of the population are right handed and 10% left handed.
Let's neglect how many elections they won / terms they served and how good they were (Carter and GWBush were the righties).  Can someone tell me the left-handed bias that has been showed by this trend.  I.e. how over-represented are southpaws for their percent of population?
 A: Let's try to ignore (the crucial possibility of) selection bias. If we were to pick $n = 7$ Presidents at random and find $X = 5$ left-handed ones, then two ways come to mind (out of possibly many) to express the incongruity with 10% left handed in the population.
(a) Population Mean vs. observed number. Under random sampling we might model $X \sim Binom(7, .1).$ So the population mean is $\mu = E(X) = 7(.1) = 0.7,$ so we might expect to see 1 (or 0) left-handers. By contrast, the observed proportion is $\hat p = X/n = 5/7 = 0.7142,$
which estimates the population mean as $\hat \mu = 7(\hat p) = X = 5.$ Intuitively, this seems much greater than
"expected."
(b) P-value. Under the null hypothesis that $P(\text{Left-handed}) = p = .1$ and given the data that 5 of 7 are left-handed,
we have $P(X \ge 5|n=7, p=.1) = 0.00018$ which is much smaller than $0.05 = 5\%.$ So we might say that we strongly Reject $H_0: p = .1$ in favor of $H_a: p > .1$. (In R statistical software, the statement
1 - pbinom(4, 7, .1) returns 0.0001765.)
