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 \\( \square \\) more american adults. (round up to the nearest integer.)

Explanation:

Step1: Recall the normal - approximation condition for the sample proportion

The distribution of \(\hat{p}\) is approximately normal if \(np\geqslant10\) and \(n(1 - p)\geqslant10\).

Step2: For part (a) where \(p = 0.2\)

Let \(n\) be the sample size. We know \(np\geqslant10\) and \(n(1 - p)\geqslant10\). Substituting \(p = 0.2\) into \(np\geqslant10\), we get \(n\times0.2\geqslant10\), so \(n\geqslant\frac{10}{0.2}=50\). Substituting \(p = 0.2\) into \(n(1 - p)\geqslant10\), we have \(n\times0.8\geqslant10\), so \(n\geqslant\frac{10}{0.8}=12.5\). The more restrictive condition is \(n\geqslant50\). Since the current sample size \(n_0 = 30\), the number of additional samples \(n - n_0=50 - 30=20\).

Answer:

20