Question 8

A local council is proposing to ban dog-walking on the beach. It is known that the proportion of households that have a dog is 712 \displaystyle \frac7{12}.

The local council wishes to poll nn households about this proposal.

Let p^\hat{p} be the random variable representing the proportion of households polled that have a dog.

What is the smallest sample size, nn, for which the standard deviation of p^\hat{p} is less than 0.060.06?

A. 67

B. 68

C. 94

D. 95

Solution

In this question, p^\hat{p} is the sample proportion taken from nn households. We can model this random variable using the binomial distribution, which means its variance is given by

Var⁡(p^)=p(1−p)n\operatorname{Var}(\hat{p}) = \frac{p(1-p)}{n}

and the standard deviation is given by

σ2=p(1−p)n\sigma^2 = \frac{p(1-p)}{n}

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