A one-sided sign test
For the two examiners, the second gave the higher mark on 8 of 12 exams. Find the p-value of the sign test against the one-sided alternative that the second examiner is more generous.
Worked answer (click to show)
Under H_0, N\sim\operatorname{Bin}(12,1/2), and large N is evidence that the second examiner is more generous. The p-value is
\mathbb{P}(N\ge8)=\frac1{2^{12}}\Bigl[\binom{12}8+\binom{12}9+\binom{12}{10}+\binom{12}{11}+\binom{12}{12}\Bigr]=\frac{495+220+66+12+1}{4096}=\frac{794}{4096}=0.194.
There is still no evidence of a difference.