Confidence intervals (HL only)

A range of believable values for the true population mean, built as x̄ ± margin of error. A 95% interval is produced by a method that captures the true mean about 95% of the time.

x̄ ± t*·s_{n-1}/√n, where t* comes from the t-distribution with df = n − 1.

df = n − 1 (one less than the sample size). The GDC's t-interval applies this automatically.

The unbiased estimate s_{n-1} (the GDC's 'Sx', dividing by n − 1), because the true population σ is unknown and must be estimated from the sample.

Use the t-interval menu: enter x̄, s_{n-1} and n (or the raw data) and the confidence level (C-Level), then read off the interval (a, b).

It makes the interval NARROWER: a larger n increases √n in the denominator, so the margin t*·s/√n shrinks — a more precise estimate.

It makes the interval WIDER: a higher confidence level uses a larger t*, so the margin grows — you cast a wider net to be more sure of trapping μ.

If the claimed value lies INSIDE the interval it is plausible (consistent with the data); if it lies OUTSIDE, the data give evidence against the claim.