Foreign Language According to a study done by Wakefield Research, the proportion of Americans who can order a meal in a foreign language is 0.47.
c. Describe the sampling distribution of p̂, the proportion of Americans who can order a meal in a foreign language. Be sure to verify the model requirements.

Verified step by step guidance

1. Identify the population proportion \(p = 0.47\), which represents the true proportion of Americans who can order a meal in a foreign language.
2. Define the sample proportion \(\hat{p}\) as the statistic representing the proportion of Americans in a sample who can order a meal in a foreign language.
3. Check the conditions for the sampling distribution of \(\hat{p}\) to be approximately normal: the sample should be a simple random sample, and the sample size \(n\) should be large enough so that both \(np \geq 10\) and \(n(1-p) \geq 10\).
4. If the conditions are met, describe the sampling distribution of \(\hat{p}\) as approximately normal with mean \(\mu_{\hat{p}} = p\) and standard deviation (standard error) \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\).
5. Summarize that the sampling distribution of \(\hat{p}\) models the variability of the sample proportion around the true population proportion \(p\) when taking repeated samples of size \(n\).

Key Concepts

Sampling Distribution of a Sample Proportion

The sampling distribution of a sample proportion (p̂) describes the probability distribution of p̂ over many random samples of the same size from a population. It is approximately normal if certain conditions are met, centered at the true population proportion (p), with variability measured by the standard error.

Model Requirements for Normal Approximation

To use the normal model for the sampling distribution of p̂, the sample size must be large enough so that both np and n(1-p) are at least 10. This ensures the distribution is approximately normal, allowing for valid probability calculations and confidence intervals.

Standard Error of the Sample Proportion

The standard error (SE) measures the typical variability of the sample proportion p̂ from the true proportion p. It is calculated as SE = sqrt[p(1-p)/n], where n is the sample size. This quantifies the expected sampling variability in p̂.