Solution
Seven of Eight Recovered
Show the problem again
In a small trial, 8 patients receive a treatment and 7 of them recover; 9 patients receive a control and 2 of them recover. Apply Fisher's exact test for independence of treatment and recovery, conditioning on all margins (8 treated, 9 control, 9 recoveries in total). Define the two-sided p-value as the sum of the null probabilities of all tables whose probability is at most that of the observed table. Give the p-value as a reduced fraction.
Worked solution
Conditioned on the margins, the number $a$ of recovered treated patients is hypergeometric: $P(a)=\binom8a\binom9{9-a}/\binom{17}9$ with $\binom{17}9=24310$. The weights $\binom8a\binom9a$ for $a=0,\dots,8$ are 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9 (sum 24310). The observed table ($a=7$) has weight 288. Tables at most as likely: $a=8$ (9), $a=1$ (72) and $a=0$ (1), while $a=2,\dots$ up to 6 are all more likely. Total $288+9+72+1=370$, so $p=370/24310=37/2431\approx0.0152$. The one-sided p-value would be $297/24310=27/2210$.
Source: Fisher exact test, from R. A. Fisher, The Design of Experiments (1935). Numbers and framing written for AxiomIQ. Statement written for AxiomIQ.