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.
e. Would it be unusual that, in a survey of 200 Americans, 80 or fewer Americans can order a meal in a foreign language? Why?

Verified step by step guidance

1. Identify the parameter of interest: the population proportion \(p = 0.47\) represents the proportion of Americans who can order a meal in a foreign language.
2. Define the sample size \(n = 200\) and the observed number of successes \(X = 80\) (Americans who can order a meal in a foreign language).
3. Calculate the sample proportion \(\hat{p} = \frac{X}{n} = \frac{80}{200} = 0.4\).
4. Check if the sampling distribution of \(\hat{p}\) can be approximated by a normal distribution by verifying that \(np \geq 10\) and \(n(1-p) \geq 10\).
5. Calculate the mean and standard deviation of the sampling distribution of \(\hat{p}\): the mean is \(p = 0.47\) and the standard deviation is \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.47 \times 0.53}{200}}\); then compute the z-score for \(\hat{p} = 0.4\) using \(z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\) to determine how unusual the observed proportion is.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, 'success' is an American who can order a meal in a foreign language, with probability 0.47, and the number of trials is 200. This distribution helps calculate probabilities of observing a certain number of successes.

Normal Approximation to the Binomial

When the number of trials is large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies calculations, especially for probabilities of ranges of outcomes, such as 80 or fewer successes. The approximation requires checking conditions like np and n(1-p) being greater than 10.

Unusual Events and Probability Thresholds

An event is considered unusual if its probability is very low, often less than 5%. To determine if 80 or fewer Americans can order a meal in a foreign language is unusual, we calculate the probability of 80 or fewer successes. If this probability is below the threshold, the event is unusual; otherwise, it is not.