# Counting bacteria : confidence intervals from one measurement

Here are two biologically-inspired statistics problem, with very simple solutions. They show that, when you have informations about the way the observations are collected, one observation may be enough to give a confidence interval.

## Question 1

From a bacterial solution I sampled a small volume v in which I found no bacteria. Give an upper bound (with confidence 95%) for the concentration of bacteria in the solution.

## Question 2

From a bacterial solution I sampled a small volume v in which I found n bacteria (n > 20). Give a 95% confidence interval for the concentration of bacteria in the solution.

## Solutions

Short answers : for the first question $c < (3/v)$ and for the second $c \in [\frac{ n \pm 2\sqrt{n}}{v}]$.

Yep, it’s that simple ! See the solutions.