Judging a Claim with a Mean Interval

A confidence interval is a range of plausible values for the true mean, built from your sample. If someone claims the mean equals some number, you check whether that number sits inside the interval. If it does, your data are consistent with the claim, nothing to object to. If it falls outside, your sample mean and spread say that value is an unlikely place for the true mean to live. That is evidence against the claim, built without ever running a full hypothesis test.

This works because the interval already encodes how far values can reasonably stray from your sample mean, given the sample size and variability, measured in standard errors using a t-distribution since the population standard deviation is unknown. A claimed value just outside the interval is a near miss, barely incompatible. One far outside is strongly contradicted. The distance from the claim to the nearer boundary is a rough gauge of how implausible that claim is, the same logic a two-sided test uses, just read visually off an interval instead of computed as a p-value.

This connects directly to significance tests for a mean: a claim outside a 95% interval would also be rejected by a two-sided test at alpha = 0.05, and a claim inside would fail to be rejected. The interval just shows the whole range of acceptable claims at once, instead of testing one value at a time.