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Question 4, 8.2.23
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A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 25 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of 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.)
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Step 1 ：To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, we need to satisfy the following conditions: \(np \geq 10\) and \(n(1-p) \geq 10\), where n is the sample size and p is the proportion of the population that supports the changes.

Step 2 ：(a) If 20% of all adult Americans support the changes, then \(p = 0.20\). We need to find the minimum n such that both conditions are satisfied.

Step 3 ：We already have a sample of 25 adults, so we need to find how many more adults we need to sample. Let's denote the additional number of adults by x. So, \(n = 25 + x\).

Step 4 ：Substituting n and p into the conditions, we get: \((25 + x) * 0.20 \geq 10\) and \((25 + x) * (1 - 0.20) \geq 10\).

Step 5 ：Solving these inequalities, we get: \(5 + 0.20x \geq 10\) and \(20 + 0.80x \geq 10\).

Step 6 ：Subtracting 5 and 20 from both sides of the inequalities, we get: \(0.20x \geq 5\) and \(0.80x \geq -10\).

Step 7 ：Dividing both sides of the inequalities by 0.20 and 0.80, we get: \(x \geq 25\) and \(x \geq -12.5\).

Step 8 ：Since x cannot be negative, we take the maximum of the two solutions. So, the researcher must ask at least \(\boxed{25}\) more American adults.

Step 9 ：(b) If 25% of all adult Americans support the changes, then \(p = 0.25\). We need to find the minimum n such that both conditions are satisfied.

Step 10 ：Substituting n and p into the conditions, we get: \((25 + x) * 0.25 \geq 10\) and \((25 + x) * (1 - 0.25) \geq 10\).

Step 11 ：Solving these inequalities, we get: \(6.25 + 0.25x \geq 10\) and \(18.75 + 0.75x \geq 10\).

Step 12 ：Subtracting 6.25 and 18.75 from both sides of the inequalities, we get: \(0.25x \geq 3.75\) and \(0.75x \geq -8.75\).

Step 13 ：Dividing both sides of the inequalities by 0.25 and 0.75, we get: \(x \geq 15\) and \(x \geq -11.67\).

Step 14 ：Since x cannot be negative, we take the maximum of the two solutions. So, the researcher must ask at least \(\boxed{15}\) more American adults.