The Null Names One Curve, and x-bar Is Not a Hypothesis

You'll learnhow to state hypotheses about a population mean before any computation, why the null must carry equality and the alternative must take its direction from the research question, how a paired design changes the parameter to a mean difference, and why the procedure is t.

A manufacturer claims its batteries last 128 hours on average; a consumer group suspects less and tests 25 of them, finding a mean of 124.6. The whole setup happens before that 124.6 is allowed to influence anything: the hypotheses are about mu, the mean of all batteries — never about x-bar, which is a known number the data simply report — the null pins one value because a p-value is an area under one specific curve, and the alternative points where the suspicion pointed, not where the sample leaned. This page draws the curve the null names, then breaks each rule and computes what it costs: a null at a different center changes the p-value tenfold on screen, and choosing the tail after peeking at the data turns a 5 percent test into a 10 percent test with both shaded tails visible. Then the paired version — subtract first, hypotheses about the mean difference, degrees of freedom counting pairs — and the reason every one of these tests is t: sigma had to be estimated.

8 STEPS · 6 QUICK CHECKS · HYPOTHESES ABOUT mu · ONE NULL CURVE · PAIRED SETUP · v1

H0: mu = 128 · Ha: mu < 128 · direction from the question · paired: mu-d, df = pairs − 1

Before you start

What you're looking at

The curve a null hypothesis names: the sampling distribution of x-bar if mu really were 128, with the observed 124.6 marked in its tail.

The question

Why must hypotheses be about mu rather than x-bar, why must the null carry equality, and where does the alternative's direction come from?

Watch for

Step 3: the same observed mean under two candidate nulls. The tail area changes tenfold, which is why a p-value needs the null to name one curve.

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