Social Security Reform A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 50 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of p̂, the sample proportion of adults who respond yes, is approximately normal, how many more adult Americans does the researcher need to sample if
a. 10% of all adult Americans support the changes?

Verified step by step guidance

1. Identify the given population proportion \(p = 0.10\) (10% support the changes) and the current sample size \(n = 50\).
2. Recall the condition for the sampling distribution of the sample proportion \(\hat{p}\) to be approximately normal: both \(np \geq 10\) and \(n(1-p) \geq 10\) must hold.
3. Calculate the current values: \(np = 50 \times 0.10 = 5\) and \(n(1-p) = 50 \times 0.90 = 45\). Notice that \(np\) is less than 10, so the sample size is not large enough for normal approximation.
4. Set up the inequality \(np \geq 10\) to find the minimum sample size \(n\) needed: \(n \times 0.10 \geq 10\) which simplifies to \(n \geq \frac{10}{0.10}\).
5. Calculate the minimum \(n\) from the inequality, then subtract the current sample size (50) to find how many more adults need to be sampled.

Key Concepts

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion (p̂) describes the distribution of p̂ values over many samples from the population. It is approximately normal if the sample size is large enough, typically when both np and n(1-p) are at least 10, ensuring the Central Limit Theorem applies.

Normal Approximation Condition (np and n(1-p) Rule)

For the sample proportion to be approximately normally distributed, the expected number of successes (np) and failures (n(1-p)) must each be at least 10. This rule helps determine the minimum sample size needed to use normal approximation methods for inference.

Sample Size Determination

Sample size determination involves calculating the minimum number of observations needed to satisfy conditions like np ≥ 10 and n(1-p) ≥ 10. Given a population proportion p, this ensures the sampling distribution of p̂ is approximately normal, enabling valid statistical inference.