Worked exercise — Dog or cat: the class survey

Problem

In a class of 24 students the question is asked: “Do you prefer dogs or cats?”. The answers are: 13 dogs, 11 cats. What is the percentage of those who prefer dogs, and with what uncertainty?

Solution

Step 1 — Proportion and percentage. The fraction of those who prefer dogs is

$p = \frac{13}{24} = 0{,}542 \quad\text{i.e.}\quad 54{,}2\%$

Step 2 — Uncertainty on the proportion. A finite sample of $N$ individuals always carries a statistical uncertainty, given by $\text{err}(p) = \sqrt{p(1-p)/N}$:

$\text{err}(p) = \sqrt{\frac{0{,}542\cdot 0{,}458}{24}} \approx \sqrt{\frac{0{,}248}{24}} \approx 0{,}102$

that is, about $10{,}2\%$.

Step 3 — Result. Dogs win with

$\ev{(54{,}2 \pm 10{,}2)\%}$

that is, an interval $[\,44{,}1\%\;;\;64{,}3\%\,]$. For such a small sample the uncertainty is enormous: with only 24 students it is not statistically clear whether the majority really prefers dogs or cats!

Linked atoms

Argomenti: Physical quantities and measurement Concetti: Significant figures and uncertainty