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

To determine the smallest sample size ( n ) such that the standard deviation of ( \hat{p} ) is less than 0.06, we first need to calculate the standard deviation of a proportion:

$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Where:

• ( p ) is the true proportion of households that have a dog, which is ( \frac{7}{12} )
• ( \sigma_{\hat{p}} ) is the standard deviation of ( \hat{p} )

Given that we want ( \sigma_{\hat{p}} < 0.06 ), we can set up the inequality:

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

Calculating ( p(1-p) ): ( 1 - \frac{7}{12} = \frac{5}{12} )\nSo, ( p(1-p) = \frac{7}{12} \times \frac{5}{12} = \frac{35}{144} )

Substituting into the inequality:

$\sqrt{\frac{\frac{35}{144}}{n}} < 0.06$

Squaring both sides:

$\frac{35}{144n} < 0.0036$

Rearranging gives:

$35 < 0.0036 \cdot 144n$ $35 < 0.5184n$

Now, solving for ( n ):

$n > \frac{35}{0.5184} \approx 67.5$

Therefore, the smallest integer value for ( n ) is 68. Thus, the answer is 68.