One-tailed tests (the probability method)

Carrying out a one-tailed binomial hypothesis test by the probability (p-value) method: assume H0, model X ~ B(n, p0), find the probability of a result at least as extreme as the one observed in the direction of H1 — P(X ≥ x) for an upper test or P(X ≤ x) for a lower test — compare it with the significance level, and write a full conclusion in context.

Statistics · Hypothesis testing

One-tailed tests (the probability method)

Why it works

To test a directional claim, we assume the null hypothesis is true and ask how likely a result as extreme as the observed one would be. If that is below the significance level, the data are too surprising for

H_0

to stand.

The probability (p-value) method, step by step.

Define the variable and, assuming $H_0 : p = p_0$ , state the model $X \sim B(n, p_0)$ .
State $H_0$ and $H_1$ (one-tailed).
With observed value $x$ , find the probability of $x$ **or more extreme in the

direction of

H_1

**:

upper test ( $H_1 : p > p_0$ ): $P(X \ge x) = 1 - P(X \le x-1)$ ;
lower test ( $H_1 : p < p_0$ ): $P(X \le x)$ .

Compare with $\alpha$ : if the probability is $< \alpha$ , the result is in the

critical region — reject $H_0$ ; otherwise do not reject $H_0$ .

Write the conclusion in context.

Why "or more extreme". A test asks whether the result is unusually high (or low), so we measure the whole tail from the observed value outward, not just the single value

P(X = x)

. That tail probability is the p-value.

Getting the tail right. For an upper test the evidence for

H_1

is a large

x

, so use the upper tail

P(X \ge x)

— and remember

P(X \ge x) = 1 - P(X \le x-1)

(mind the

x-1

). For a lower test use

P(X \le x)

directly.

Conclusions come in two halves. Always state both: the statistical decision ("there is/ is not sufficient evidence to reject

H_0

") and what it means in context ("…so the proportion of … has increased").