Question

a researcher studying public opinion of proposed social security changes obtains a simple random sample of 30 adult americans and asks them whether or not they support the proposed changes. to say that the distribution of \\( \hat { p } \\), the sample proportion of adults who respond yes, is approximately normal, how many more adult americans does the researcher need to sample in the following cases?
(a) 20% of all adult americans support the changes
(b) 25% of all adult americans support the changes
(a) the researcher must ask 33 more american adults. (round up to the nearest integer.)
(b) the researcher must ask \\( \square \\) more american adults. (round up to the nearest integer.)

Explanation:

Step1: Recall the condition for normal approximation of sample proportion

For the sampling distribution of \(\hat{p}\) to be approximately normal, we need \(np\geq5\) and \(n(1 - p)\geq5\). Let the total sample size be \(n\). We already have \(n_0 = 30\) samples.

Step2: Solve for \(n\) when \(p = 0.25\)

For \(p=0.25\), from \(np\geq5\) and \(n(1 - p)\geq5\). Substitute \(p = 0.25\) into \(np\geq5\), we get \(n\times0.25\geq5\Rightarrow n\geq\frac{5}{0.25}=20\). Substitute \(p = 0.25\) into \(n(1 - p)\geq5\), we have \(n\times(1 - 0.25)\geq5\Rightarrow n\times0.75\geq5\Rightarrow n\geq\frac{5}{0.75}=\frac{20}{3}\approx6.67\). The more restrictive condition is \(n\geq20\).

Since we already have \(n_0 = 30\) (which satisfies \(n = 30\geq20\)), the number of additional samples is \(0\)

Answer:

\(0\)