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Audit sample size

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A ledger has $21$ entries with the sizes $1$ to $21$ , each size appearing once. An auditor draws $n$ entries without replacement, with all subsets equally likely, and uses the mean size of the drawn entries. What is the smallest $n$ for which the variance of this sample mean is at most $1$ ?

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Hint

Write the variance of the mean as a function of n and solve the inequality.

Solution

The population variance is $\frac{2 1 ^{2} - 1}{12} = \frac{110}{3}$ .
So $Var (mean) = \frac{\frac{110}{3}}{n} \times \frac{21 - n}{20} = \frac{11 ( 21 - n )}{6 n}$ .
The condition $11 (21 - n) \leq 6 n$ gives $n \geq \frac{231}{17}$ , about $13.6$ .
At $n = 13$ the variance is above $1$ and at $n = 14$ it is $\frac{77}{84}$ , so the smallest $n$ is $14$ .

Answer: $14$

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