4.2 Naming the two hypotheses
We have the reasoning. What are the two positions we are reasoning between?
The previous section did the whole of a hypothesis test without naming anything. We assumed the administration’s claim, asked how often a sample would stray as far from it as ours did, and found the answer was about once in forty.
Let us give the two positions in that argument their names.
We start from a sample \(x_1, x_2, \ldots, x_n\) drawn from a population with unknown mean \(\mu\) and standard deviation \(\sigma\). The claim under scrutiny is the null hypothesis:
\[H_0 : \mu = 165\]
The position we would move to if the claim collapses is the alternative hypothesis:
\[H_A : \mu \neq 165\]
Why begin by assuming the claim is true?
At first this looks backwards. If we suspect the claim is wrong, why start by supposing it is right?
Because the assumption is what gives us something to measure against. Notice what we actually did in the previous section: we could only simulate those fifty thousand samples because we had a specific value of \(\mu\) to simulate from. Had we begun with “the mean is not 165,” there would have been nothing to draw samples from — not 165 covers 164, and 170, and 3,000, and each would give a different answer.
So we grant the claim, temporarily, and see where it leads. If our sample would be quite ordinary under it, we have no reason to doubt it. If our sample would be extraordinarily unlikely under it, then the claim itself becomes hard to believe.
The null hypothesis is always the boring option — nothing is going on, the claim stands, the difference is zero. We never prove it. We only ever fail to knock it down.
This is why we say “fail to reject \(H_0\)” and never “accept \(H_0\).” A court that acquits has not proven innocence; it has failed to prove guilt.
4.2.1 Which hypothesis does the claim go in?
A claim arrives. Does it become \(H_0\) or \(H_A\)?
This is one of the most common places for a beginner to go wrong, and the way out is not to memorise a rule but to ask a question first.
Before writing anything down, ask yourself:
What result would convince me?
Suppose a report claims that the mean CAT score of applicants is more than 499. What would persuade you? Almost everyone answers the same way: a sample mean comfortably above 499. A sample averaging 470 would not support the report, and neither would one averaging exactly 499.
That answer has already determined the test. The thing you would find convincing — a mean above 499 — is the thing you are gathering evidence for, and a test only ever produces evidence against \(H_0\). So the claim has to sit in the alternative:
\[H_0 : \mu \leq 499 \qquad H_A : \mu > 499\]
The direction follows too. You would be convinced by large values, so it is large values that must count against the null, and the test looks in the upper tail.
Two questions settle every case.
What am I trying to gather evidence for? That goes in \(H_A\). A test can only accumulate evidence against the null, so whatever you hope to establish must be what survives when the null falls.
Which direction would count as evidence? That fixes the inequality in \(H_A\), and with it the tail the test looks in.
Examination questions usually signal this in their phrasing, and the signal is worth recognising.
| The question asks | The claim becomes | Because |
|---|---|---|
| “Is there evidence to support the claim?” | \(H_A\) | The claim is what you are gathering evidence for. |
| “Can you reject the claim?” | \(H_0\) | The claim is what you are challenging, and you are asking whether the data can knock it down. |
Where the equality goes
One structural rule holds in every case, and it will save you from a good deal of trouble:
The equality always belongs in the null hypothesis.
\(H_0\) takes \(=\), \(\leq\) or \(\geq\). \(H_A\) takes \(\neq\), \(<\) or \(>\), and never equality.
The reason is the one from earlier in this section. Every test begins by assuming one specific value of \(\mu\) and computing how likely our sample would be under it. Only a hypothesis containing an equality supplies such a value — \(\mu \leq 499\) can be tested at its boundary, 499, whereas \(\mu > 499\) names no particular number to compute with.
A claim phrased with a strict inequality — “the mean is more than 499” — therefore cannot go into \(H_0\) as stated. If you are asked for evidence supporting it, the pair is
\[H_0 : \mu \leq 499 \qquad H_A : \mu > 499\]
Writing \(H_0: \mu \geq 499\) is the common error. It puts the claim in the null and points the test in the wrong direction, so a sample mean above 499 — evidence for the claim — ends up producing a \(p\)-value near 1 and a “fail to reject.” The arithmetic is fine; the conclusion is backwards.
The CAT report, worked through
A report claims the mean CAT score of applicants is more than 499. A sample of 100 applicants has mean 502, with \(\sigma = 10.6\). Is there enough evidence at \(\alpha = 0.01\) to support the report?
We settled the hypotheses above by asking what would convince us:
\[H_0 : \mu \leq 499 \qquad H_A : \mu > 499\]
Right-tailed, so the critical value is \(Z_{0.01} = 2.33\) — positive, not negative.
Before computing anything, check the direction. The sample mean is 502 and the claim is “more than 499.” The sample points the same way as the claim, so whatever the arithmetic returns, it must come out favourable to the report. If it does not, the tail has been pointed the wrong way.
xbar_c <- 502; mu_c <- 499; sigma_c <- 10.6; n_c <- 100
z_c <- (xbar_c - mu_c) / (sigma_c / sqrt(n_c))
c(z_calc = z_c,
z_crit = qnorm(0.99),
p_value = pnorm(z_c, lower.tail = FALSE)) # upper tail: H_A is ">"#> z_calc z_crit p_value
#> 2.830189 2.326348 0.002326\(Z_{calc} = 2.83 > 2.33\), and \(p = 0.0023 < 0.01\). We reject \(H_0\), which means the data support the report’s claim — and the answer points the way the sanity check said it would.
Make that check a habit, and make it before you trust any \(p\)-value rather than after.
Ask which way the sample points relative to the claim, and decide what answer you should therefore expect. If the arithmetic hands back something else — a \(p\)-value of 0.998 and “no support” when the sample sits above the claimed value — the mistake is almost always in \(H_A\), not in the calculation.