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.