To determine how many more adults the researcher needs to sample for the distribution of the sample proportion $\hat{p}$ to be approximately normal, we can use the guideline based on the Central Limit Theorem (CLT) for proportions. The CLT states that the distribution of $\hat{p}$ will be approximately normal if both $np \geq 10$ and $n(1 - p) \geq 10$, where:

• $n$ is the sample size
• $p$ is the population proportion

The problem provides initial sample information (25 adults) and asks for additional sample size needed. Let's analyze each case:

Part (a): $p = 0.15$

1. We need $np \geq 10$ and $n(1 - p) \geq 10$.

2. Substitute $p = 0.15$:

   • $np = 0.15n$
   • $n(1 - p) = 0.85n$

3. Solve each inequality:

   • $0.15n \geq 10$ ⟹ $n \geq \frac{10}{0.15} \approx 66.67$
   • $0.85n \geq 10$ ⟹ $n \geq \frac{10}{0.85} \approx 11.76$

4. Since both conditions must be satisfied, we need $n \geq 66.67$. Rounding up, we require $n = 67$.

5. Since the researcher has already sampled 25 adults, the additional sample size required is: $67 - 25 = 42$

Thus, for $p = 0.15$, the researcher needs to sample 42 more adults.

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Part (b): $p = 0.20$

1. Again, we need $np \geq 10$ and $n(1 - p) \geq 10$.

2. Substitute $p = 0.20$:

   • $np = 0.20n$
   • $n(1 - p) = 0.80n$

3. Solve each inequality:

   • $0.20n \geq 10$ ⟹ $n \geq \frac{10}{0.20} = 50$
   • $0.80n \geq 10$ ⟹ $n \geq \frac{10}{0.80} = 12.5$

4. Since both conditions must be satisfied, we need $n \geq 50$.

5. Since the researcher has already sampled 25 adults, the additional sample size required is: $50 - 25 = 25$

Therefore, for $p = 0.20$, the researcher needs to sample 25 more adults.

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Summary of Results

• (a) The researcher needs to sample 42 more adults if 15% support the changes.
• (b) The researcher needs to sample 25 more adults if 20% support the changes.

Do you have any questions or need further explanation?

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Here are some related questions that might help deepen understanding:

1. How would the required sample size change if the support percentage were higher than 20%?
2. Why is the normal approximation useful in sampling distributions of proportions?
3. How does increasing the sample size affect the margin of error in estimating population proportions?
4. What other factors besides sample size might influence the shape of the sampling distribution?
5. How would the results differ if we used a 90% confidence interval instead of the normal approximation?

Tip: When dealing with proportions, always ensure both $np$ and $n(1 - p)$ are sufficiently large to apply the normal approximation.