Unit 6 | Confidence Intervals

https://fiveable.me/ap-stats/unit-6 for a general overview of unit 6, https://schraderhsmath.weebly.com/uploads/6/0/6/6/60662375/ap_stats_cheat_sheet.pdf for a good cheat sheet

What is z*?

critical value from the standard normal distribution

What is \hat p ?

The estimated proportion of successes in a statistical sample. (sample proportion)

What is n?

sample size

Formula for confidence interval for a proportion

Confidence Interval = Sample Proportion ± Margin of Error

\hat{p}\pm z^{*}\sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}

What is the z* for a 95% confidence level, and what is the z* for a 99% confidence level?

1.96, 2.58

Higher confidence (e.g. 95% to 99%) means a (wider/narrower) interval and a (wider/narrower) margin of error.

Explain why this is the case.

wider, wider

To be more certain of capturing the true parameter, you must include a broader range of values.

Lower confidence (e.g. 99% to 95%) means a (wider/narrower) interval and a (wider/narrower) margin of error.

Explain why this is the case.

narrower, narrower

To have a lower confidence level, you can accept a smaller range of values for the parameter, resulting in a more precise interval and reduced margin of error.

z* (increases/decreases) with higher confidence levels.

increases

As z* grows, the margin of error grows by this proportion… (equation)

ME=z^{*}\times\frac{\sigma}{\sqrt{n}}

Interpret a confidence level (e.g. what should you write to interpret a confidence level?)

In repeated procedures of sample size n from the same population, we would expect to capture the true mean (or proportion) CONTEXT in approximately the specified confidence level (e.g., 95%) of the intervals constructed.

Interpret a confidence interval (e.g. what should you write to interpret a confidence level?)

We are 95% confident that the true mean (or proportion) CONTEXT is between (#, #).