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) 10% of all adult americans support the changes
(b) 15% of all adult americans support the changes
(a) the researcher must ask 82 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: Check the normal - approximation condition for sample proportions

The normal approximation to the sampling distribution of \(\hat{p}\) is valid when \(np\geq5\) and \(n(1 - p)\geq5\).

Step2: For part (a)

Given \(p = 0.1\). Let the sample size be \(n\). We need \(np\geq5\) and \(n(1 - p)\geq5\). Substituting \(p = 0.1\) into \(np\geq5\), we get \(n\times0.1\geq5\Rightarrow n\geq50\). Since the initial sample size \(n_0=30\), the number of additional samples \(n - n_0=50 - 30=20\). But wait, we also check \(n(1 - p)\): when \(n = 50\), \(n(1 - p)=50\times(1 - 0.1)=45\geq5\)

Step3: For part (b)

Given \(p = 0.15\). Using the condition \(np\geq5\), we substitute \(p = 0.15\) into \(np\geq5\), so \(n\times0.15\geq5\Rightarrow n\geq\frac{5}{0.15}=\frac{100}{3}\approx33.33\). Since \(n\) must be an integer, \(n = 34\). Also check \(n(1 - p)=34\times(1 - 0.15)=34\times0.85 = 28.9\geq5\). The number of additional samples is \(n - n_0=34 - 30 = 4\)

Answer:

(a) \(20\)
(b) \(4\)