Two Columns Are Not Two Samples, and 1.96 Is Not 2.064

You'll learnhow a one-sample t interval is assembled from its three parts, what using a z critical value actually costs in delivered coverage, how a paired design collapses into a single sample of differences, and which object the finished interval is a claim about.

Twenty-five batteries give a mean of 124.6 hours and a standard deviation of 9.2, and the 95 percent interval is 124.6 plus or minus 2.064 times 1.84 — about 120.8 to 128.4 hours. Every number in that sentence is built on this page, including the wrong ones: the 1.96 that produces an interval delivering only 93.8 percent (integrated here, not asserted), and the band that would actually hold 95 percent of individual batteries, which is nearly five times wider than the interval a student mistakes for it. Then the design question that decides everything downstream: six students measured before and after are not two samples of six. Subtract first and the interval on the differences is (0.4, 7.0), clear of zero; run the two columns as independent groups and the standard error triples, the interval spans zero, and the improvement the pairing plainly shows disappears into noise the design was built to cancel.

8 STEPS · 6 QUICK CHECKS · THE ONE-SAMPLE t INTERVAL · PAIRED DIFFERENCES · COVERAGE · v1

x̄ ± t*·s/√n · df = n − 1 · paired data: subtract first, one sample of differences

Before you start

What you're looking at

A number line of battery lifetimes. Every interval this topic argues about — right ones and wrong ones — is drawn as a bracket on it, built from computed parts.

The question

Three choices decide the interval: t or z, which degrees of freedom, and — when each individual was measured twice — whether the data is one sample of differences or two samples.

Watch for

Step 7. The same six students, analysed two ways: the paired interval clears zero and the two-sample misread of identical data does not.

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