Statistics, Module 4: Confidence Intervals

What is a confidence interval?

Is a method that uses sample data to produce a range of plausible values for an unknown population parameter.

What is the confidence level?

(e.g. 90%, 95%, 99%) Is the long-run success rate of the method. (after many repetitions, about that fraction of the intervals would contain the true parameter.)

What is a point estimate?

The center of the interval

What is the margin of error (ME)?

How far we extend to each side of the point estimate.

What is the standard error (SE)?

Measures sampling variability of the estimator.

What is a critical value?

A multiplier from a reference distribution (\(\z\) or \(t\)) that depends on the chosen confidence level. 

When do we use \(z\)?

When population SD (\(\sigma\)) is known (rare in practice) and conditions for normality are met. 

When do we use \(t\)?

When population SD is unknown and you estimate with sample SD (\(s\))

Does \(t\) distribution has thicker tails to account for extra uncertainty?

True.

What are the conditions for a CI (confidence interval) for a population mean?

Data come from a random and independent sample and/or population is approximately normal or sample size is large.

What is a correct way to interpret a CI?

We are 95% confident the true mean is between A and B.

Is this a correct way to interpret a CI: There is a 95% probability the true mean is between A and B. 

False

Comparing CIs are based on mostly intuition…

Narrower intervals indicate more precision (often due to larger \(n\) or lower variability).

Overlapping vs. nonoverlapping CIs can…

Hint at differences, but formal hypothesis testing is a separate procedure.

\(\,\mu\,\)

Population mean

\(\,\bar{x}\,\)

Sample mean

\(\,\sigma\,\)

Population SD

\(\,s\,\)

Sample SD

\(\,p\,\)

Population proportion

\(\,z^*\,\)

Critical value from standard normal (depends on confidence)

\(\,t^*\,\)

Critical value from \(t\) distribution (depends on confidence and df)