Q.A die is thrown $9000$ times and a throw of $3$ or $4$ is observed $3240$ times. Find the standard error of the proportion for an unbiased die.

With $p = \tfrac13,\ q = \tfrac23,\ n = 9000$, the standard error of the proportion is $\sqrt{pq/n} \approx 0.00497$.

Step 1 — theoretical proportion for an unbiased die. A throw of $3$ or $4$ has

$p = \frac{2}{6} = \frac{1}{3}, \qquad q = 1 - p = \frac{2}{3}.$

(The observed count $3240$ describes the sample; the S.E. for an unbiased die uses the theoretical $p = \tfrac13$.)

Step 2 — standard error of the proportion. With $n = 9000$, …