An Interval of Gaps, Not of Scores

You'll learnhow the two-sample t interval is assembled from its three pieces, why the critical value comes from the t family and what the z habit silently costs, which conditions run once and which run twice, what the four-part interpretation must contain, and what actually narrows an interval.

Method A scored 78.4 from 30 students; method B scored 71.2 from 28. The difference is 7.2 points — and 7.2 from one pair of samples is not the answer, it is the center of one. Reach t* standard errors both ways: 7.2 ± 2.052 × 2.341 = 7.2 ± 4.80, giving (2.40, 12.00). Every number in that range is a plausible GAP between the two population means — not a score — and zero is not among them, which is what lets the study say method A is plausibly ahead. Use z's 1.96 instead of t's 2.052 and the interval claims 95% while covering about 94%: this page integrates the t distribution and prints the shortfall, rather than asking you to take the t table on faith.

8 STEPS · 6 QUICK CHECKS · THE T INTERVAL · CONDITIONS TWICE · INTERPRETATION · v1

7.2 ± 2.052 × 2.341 = (2.40, 12.00) · z 1.96 claims 95% and covers 94.0% · zero outside

Before you start

What you're looking at

A number line of GAPS: differences in mean score, method A minus method B. The dot is 7.2, the difference one pair of samples produced.

The question

One pair of samples gives 7.2. What range of true differences could plausibly have produced it — and is zero, meaning no difference at all, among them?

Watch for

The arms reach t* × SE each way, and every choice in the topic lives inside those two factors: t against z, df 27 against 53, and a standard error that adds the two variances.

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