Question

a researcher studying public opinion of proposed social security changes obtains a simple random sample of 35 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.) (b) the researcher must ask \\( \square \\) more american adults (round up to the nearest integer.)

Explanation:

Step1: Check the condition for normal approximation

For the sampling distribution of \(\hat{p}\) to be approximately normal, we need \(np\geq5\) and \(n(1 - p)\geq5\).

Step2: Case (a) when \(p = 0.2\)

Let the sample size be \(n\). We know \(np\geq5\) and \(n(1 - p)\geq5\). Substituting \(p = 0.2\), from \(np\geq5\), we have \(n\times0.2\geq5\Rightarrow n\geq\frac{5}{0.2}=25\). From \(n(1 - 0.2)\geq5\), we have \(n\times0.8\geq5\Rightarrow n\geq\frac{5}{0.8}=6.25\). The more restrictive condition is \(n\geq25\). Since the current sample size \(n_0 = 35\), we already satisfy the condition. Wait, no, there is a mistake. The formula for the normal approximation of the sampling distribution of \(\hat{p}\) is that \(np(1 - p)\geq10\) (more accurate rule).

$$np(1 - p)\geq10$$

Substitute \(p = 0.2\):

$$n\times0.2\times(1 - 0.2)\geq10$$

$$n\times0.2\times0.8\geq10$$

$$n\geq\frac{10}{0.2\times0.8}=\frac{10}{0.16}=62.5$$

Since the current sample size \(n_0=35\), the number of additional samples \(n - n_0=63 - 35 = 28\)

Step3: Case (b) when \(p = 0.25\)

Use the formula \(np(1 - p)\geq10\)
Substitute \(p = 0.25\):

$$n\times0.25\times(1 - 0.25)\geq10$$

$$n\times0.25\times0.75\geq10$$

$$n\geq\frac{10}{0.25\times0.75}=\frac{10}{0.1875}\approx53.33$$

Since the current sample size \(n_0 = 35\), the number of additional samples \(n - n_0=54 - 35=19\)

Answer:

(a) \(28\)
(b) \(19\)