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

Explanation:

Step1: Check the normality condition

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 know the initial sample size \(n_0=30\).

Step2: For \(p = 0.25\)

Set up the inequalities:

• \(n\times0.25\geq5\) and \(n\times(1 - 0.25)=n\times0.75\geq5\). The more restrictive condition is from \(n\times0.25\geq5\) (since \(0.25<0.75\)). Solving \(n\times0.25\geq5\) gives \(n\geq\frac{5}{0.25}=20\). But we also know from the general rule for a good approximation (some textbooks use \(np\geq10\) and \(n(1 - p)\geq10\)). Using \(np\geq10\) and \(n(1 - p)\geq10\)
• From \(n\times0.25\geq10\), we get \(n=\frac{10}{0.25}=40\)
• From \(n\times0.75\geq10\), we get \(n=\frac{10}{0.75}\approx13.33\)

Since \(n = 40\) satisfies both \(n\times0.25=40\times0.25 = 10\) and \(n\times0.75=40\times0.75=30\)

Step3: Calculate the additional number of samples

The initial sample size \(n_0 = 30\). The required sample size \(n=40\). The number of additional samples \(=n - n_0=40 - 30=10\)

Answer:

10