Confidence Intervals – AP Statistics

Learning Objectives

Point Estimate: Our single sample $\hat{p}$ (or $\bar{x}$) is our absolute best guess for the unknown true Population parameter $\mu$.

Margin of Error ($ME$): We add and subtract $Z^* \times SE$ to form a 'net'. A higher $Z^*$ (more confidence) makes a wider net. A bigger sample $n$ shrinks the Standard Error ($SE$), tightening the net.

The Meaning of "95%": It does NOT mean there's a 95% chance the parameter is inside. It means if we repeat this sampling process 100 times, exactly 95 of our "nets" will successfully capture the true stationary parameter!

Tags

InferenceConfidenceZ-ScoreMargin of Error

       CONFIDENCE INTERVAL BOUNDS

       $\text{Estimate } \pm Z^* \times \frac{\sigma}{\sqrt{n}}$

True Pop Mean $\mu$ (Hidden!)

Captured $\mu$

Missed $\mu$!

Network Parameters

Confidence Level ($Z^*$) = 95%

Sample Size ($n$) = 30

MARGIN OF ERROR BREAKDOWN

Critical Val $Z^*$: 1.960

Standard Error: 2.74

Margin $ME$: ±5.37

LIVE SIMULATION TRACKER

Capture Rate: 96.0%

The Trade-off: To increase confidence to 99%, we must physically cast a WIDER net (larger margin of error). This makes our estimate 'safer' but less precise. You can fight this only by paying for a larger sample size $n$!