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
b. 20% of all adult Americans support the changes?

Verified step by step guidance

1. Identify the given population proportion \(p = 0.20\) (20% 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: \(np = 50 \times 0.20 = 10\) and \(n(1-p) = 50 \times 0.80 = 40\). Since \(np = 10\), the condition is just met, but to be safely above 10, the researcher may want a larger sample.
4. Set up the inequality \(n \times 0.20 \geq 10\) to find the minimum \(n\) needed for the normal approximation to hold comfortably. Solve for \(n\):
5. \[ n \geq \frac{10}{0.20} \]
6. Calculate the difference between this minimum \(n\) and the current sample size of 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 how p̂ varies from sample to sample. When the sample size is large enough, this distribution is approximately normal, centered at the true population proportion (p), with variability measured by the standard error. This concept allows us to make probability statements about p̂.

Conditions for Normal Approximation

The normal approximation to the sampling distribution of p̂ is valid when both np and n(1-p) are at least 10. This ensures enough expected successes and failures in the sample for the Central Limit Theorem to apply, making the distribution of p̂ approximately normal and suitable for inference.

Calculating Required Sample Size

To determine how many more individuals to sample, we use the condition np ≥ 10 and n(1-p) ≥ 10. Given p and the current sample size, we solve for n to meet these criteria. This calculation ensures the sample is large enough for the normal approximation to hold.