Applying hypothesis test to 1 data point Can you do a hypothesis test for a sample size of 1? For example let's say you have a big population and a sample of 1 and you want to tell if it is significant or not given that the sample of 1 had a big effect. Can you? How?
I.e. A drug trial where 100000 do not receive the drug and 1 person does. Can you use hypothesis testing to tell you anything given that the 1 person who did take the drug had a very very big effect?
 A: Purely mathematically, yes, as long as you make enough assumptions. E.g. if you assume a linear model with the same residual SD for both drug and control group, then you will get a hypothesis test that - if the assumptions are true - can give arbitrarily small p-values (etc.) even with just one patient on treatment.
Here's an example using R:
library(tidyverse)

set.seed(123)

lmfit1 = lm(data=tibble(treatment=c(1,rep(0,10000)), y=c(rnorm(1,mean=1000,sd=1), rnorm(10000,mean=0,sd=1))),
            y ~ factor(treatment))

summary(lmfit1)

which produces:
Call:
lm(formula = y ~ factor(treatment), data = tibble(treatment = c(1,
    rep(0, 10000)), y = c(rnorm(1, mean = 1000, sd = 1), rnorm(10000,
    mean = 0, sd = 1))))

Residuals:
    Min      1Q  Median      3Q     Max
-3.8432 -0.6658 -0.0089  0.6757  3.8498

Coefficients:
                     Estimate Std. Error  t value Pr(>|t|)   
(Intercept)         -0.002079   0.009989   -0.208    0.835   
factor(treatment)1 999.441603   0.998953 1000.489   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9989 on 9999 degrees of freedom
Multiple R-squared:  0.9901,   Adjusted R-squared:  0.9901
F-statistic: 1.001e+06 on 1 and 9999 DF,  p-value: < 2.2e-16

