Question

"A state legislator wishes to survey residents of her district to see what proportion of the registered voters are aware of her position on using state funds to pay for a new Entertainment Center. What sample size is necessary if the 95% CI for population proportion to be within 5% of the true proportion?"

Answer 1

Answer:

The correct answer to the following question will be "384".

Step-by-step explanation:

We are having,

$P(-0.05$

On solving, we get

⇒  $P(\frac{-0.05}{\sqrt{\frac{pq}{n}}}$

⇒  $\frac{0.05}{\sqrt{\frac{p(1-p)}{n}}} =1.96$

On applying cross-multiplication, we get

⇒  $0.05^2=1.96^2\times \frac{P(1-p)}{n}$

⇒  $0.05^2n=1.96^2-1.96^2p^2$

⇒  $1.96^2p^2-1.96^2p+0.05^2n=0$...(equation 1)

When "p" seems to be the population proportion, (equation 1) approach, which is special

∴ $b^2-4ac=0$

On putting the values in the above expression, we get

⇒  $(1.96)^4-4\times 0.05^2\times 1.96^2n=0$

⇒  $1.98^2-4\times 0.05^2n=0$

⇒  $n=\frac{1.96^2}{4\times 0.05^2}$

⇒  $n=384.16=384$