One Subtraction, One Tail Area, and a Sentence That Shows Its Work

You'll learnhow to compute the one-sample t statistic and read its p-value at the right degrees of freedom, why the conclusion must state the comparison to alpha in context, why a test and a confidence interval always agree, and where the robustness of t procedures runs out.

Twenty-five batteries came in at a mean of 124.6 hours against a claimed 128. The arithmetic that turns that into a verdict is one subtraction over one standard error, and it gives t = −1.85 at 24 degrees of freedom; the p-value is the tail area past it, 0.0385, integrated on this page rather than looked up. Everything hard about the topic happens after that number exists. Read it off the normal table and you get 0.0323 — sixteen percent smaller, and sixteen percent more convincing than the data earned. Announce "reject H0" without setting 0.0385 beside 0.05 and a reader has nothing to check. And at n = 8 with one wild value, the p-value stops meaning anything: this page slides a single measurement from 42 to 72 and you watch the sample mean travel further from the null while the evidence collapses. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE t STATISTIC · p-VALUES AT df · ROBUSTNESS · v2

One subtraction, one tail area, one sentence — and a p-value that needs its conditions.

Before you start

What you're looking at

The t distribution the test reads its p-value from. Every area on this page is integrated here, not quoted from a table.

The question

Once the statistic exists, what makes the conclusion complete — and when does the p-value stop being worth reporting?

Step 1 / 12