Mistake Master · AP Statistics · Unit 4 · Step-Through Animation
The p-Value Is Where the Interval Touches Zero
You'll learnhow the two-sample t statistic is assembled, how to read its p-value at either accepted df, what a conclusion must contain to be checkable, why pooling has no licence here even when it barely changes the answer, and exactly how the test and the interval are the same statement.
One subtraction over one standard error: t = 7.2 / 2.341 = 3.08, and at the conservative 27 degrees of freedom the one-sided p-value is 0.0024. Then the part usually taught as a rule to memorise. The test and the interval do not merely "agree" — build the interval at confidence 1 − p and its critical value is the t whose tail area is p, which is 3.08 itself, so the half-width is 3.08 × 2.341 = 7.2 and the interval is exactly (0.00, 14.40). The p-value is the confidence level at which the lower endpoint lands on zero, and this page asserts it to the last bit rather than asking you to take it on faith.
8 STEPS · 6 QUICK CHECKS · THE STATISTIC · THE p-VALUE · TEST AND INTERVAL · v1
t = 7.2 / 2.341 = 3.08 · one-sided p = 0.0024 at df 27 · the interval at 1 − p starts exactly at 0.00
Before you start
What you're looking at
The teaching-methods study one last time: a difference of 7.2 points, a standard error of 2.341, and the t distribution that says how surprising that is if the two population means are equal.
The question
The test and the interval are always taught as two procedures that happen to agree. Why do they agree — and is "agree" even the right word?
Watch for
Step 6, where the interval is rebuilt at confidence 1 − p and its lower endpoint lands exactly on zero. That is not a coincidence; it is one line of algebra.
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