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.

The arithmetic is one subtraction over one standard error: 124.6 minus 128, divided by 1.84, gives t equal to −1.85 at 24 degrees of freedom, and the p-value is the tail area past it — 0.0385, integrated on this page rather than looked up. Everything hard about this topic happens after that number exists. Read it off the normal table instead of the t distribution and you get 0.0323, sixteen percent smaller and sixteen percent more convincing than the data earned. Announce "reject H0" without ever setting 0.0385 beside 0.05 and a reader has nothing to check. And run the procedure at n = 8 on data with one wild value, and the p-value stops meaning anything at all: this page moves a single measurement from 42 to 72, watches the sample mean travel further from the null hypothesis, and watches the evidence collapse from p = 0.023 to p = 0.142 at the same time.

8 STEPS · 6 QUICK CHECKS · THE t STATISTIC · p-VALUES AT df · ROBUSTNESS · v1

t = (x-bar − mu0)/(s/√n) · df = n − 1 · state the comparison to alpha · robustness ends at n = 8

Before you start

What you're looking at

The t distribution the test reads its p-value from, with the observed statistic marked and the tail area beyond it shaded — every area on this page is integrated, not quoted.

The question

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

Watch for

Step 7. One measurement moves from 42 to 72, the sample mean travels away from the null hypothesis, and the evidence gets weaker, not stronger.

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