A local council is proposing to ban dog-walking on the beach - HSC - SSCE Mathematics Extension 1 - Question 8 - 2024 - Paper 1

To find the smallest sample size ( n ) such that the standard deviation is less than 0.06, we set up the following inequality:

$\sqrt{\frac{p(1 - p)}{n}} < 0.06$

Substituting ( p = \frac{7}{12} ) into the inequality gives:

$\sqrt{\frac{\frac{7}{12} \left(1 - \frac{7}{12}\right)}{n}} < 0.06$

This simplifies to:

$\sqrt{\frac{\frac{7}{12} \cdot \frac{5}{12}}{n}} < 0.06$

Squaring both sides leads to:

$\frac{\frac{7}{12} \cdot \frac{5}{12}}{n} < 0.0036$

Rearranging gives:

$n > \frac{\frac{7}{12} \cdot \frac{5}{12}}{0.0036}$

Calculating the left side results in:

$n > \frac{\frac{35}{144}}{0.0036} = \frac{35}{144 \cdot 0.0036} = \frac{35}{0.5184} \approx 67.5$

Since ( n ) must be an integer, we round up:

Thus, the smallest sample size ( n ) is 68.