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.

Theoretical $p=1/3$ for a fair die, so $S.E.=\sqrt{pq/n}=\sqrt{(1/3)(2/3)/9000}\approx0.00497$.

Given. $n=9000$ throws. For an unbiased die the probability of getting a $3$ or a $4$ is

$p=\frac{2}{6}=\frac13,\qquad q=1-p=\frac23.$

(The observed $3240$ successes are not used when computing the S.E. under the unbiased assumption.)

Formula: standard error of proportion

$S.E.=\sqrt{\frac{pq}{n}}.$

Substitute: …