One-Proportion Confidence Intervals for a School Survey

All flashcards

What do $n$ and $x$ represent in a school place-of-birth survey?

$n$ is the number of students surveyed, and $x$ is the number of surveyed students born in the state.

What is the difference between $p$ and $\hat{p}$ in a school survey?

$p$ is the unknown population proportion of students born in the state. $\hat{p}=x/n$ is the proportion observed in the sample.

What does the random variable $\hat{P}$ represent?

The sample proportion that would result from repeatedly taking samples of the same size; it varies across samples.

What are the tail areas for a 90% two-sided confidence interval?

The combined tail area is $0.10$; each tail has area $0.05$, and the central area is $0.90$.

Why does a higher confidence level produce a wider interval for the same sample?

It requires a larger critical value $z^*$, increasing the error bound and widening the interval.

Quiz questions

1. Which expression gives a one-proportion $z$ confidence interval for population proportion $p$?Answer: $\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
2. What does a 90% confidence level mean for intervals made by repeatedly using the same sampling method?Answer: About 90% of those intervals would contain the true population proportion.
3. A survey asks only one convenient class about birthplace. What is a key concern when estimating the proportion for the whole school?Answer: The class may not represent all students in the school.
4. If the sample proportion and confidence level stay fixed, what generally happens to interval width when sample size increases?Answer: The interval generally becomes narrower because the estimated standard error decreases.
5. Using the common success-failure check for a one-proportion interval, which sample condition supports an approximately normal sampling distribution?Answer: $n\hat{p}\geq 10$ and $n(1-\hat{p})\geq 10$