Porisma
Back to reading

Hypothesis tests for a mean · Use a p-value

Follow the alternative, then compare the p-value

Choose the alternative and significance level before inspecting the sample. Use this map only after the test statistic has a justified standard-normal null model, exact or approximate as stated.

  1. Count the tail or tails named by the alternative

    For a population-mean test, start from H₀: μ = μ₀. Use the pre-specified alternative, even if the observed statistic points the other way.

    Let Φ\Phi be the standard normal CDF and zz the observed statistic.

    Decrease · H₁: μ < μ₀Use the left tail

    p=Φ(z)p=\Phi(z). An observation far in the negative direction gives a small p-value; a positive observation does not justify switching to a right-sided test.

    Increase · H₁: μ > μ₀Use the right tail

    p=1−Φ(z)p=1-\Phi(z). Count values at least as large as the observation under the null model.

    Either direction · H₁: μ ≠ μ₀Include both equally extreme tails

    p=2[1−Φ(∣z∣)]p=2[1-\Phi(|z|)] for this symmetric standard-normal model. Use the magnitude of zz; do not double a probability greater than one half.

    Check the reference model before using these formulas. Normal-population mean tests with known variance are exact; large-sample normal tests require the stated sampling and approximation assumptions.

  2. Compare unrounded probabilities with the fixed level

    Write the significance level as a probability: 5% means α = 0.05. Compare the unrounded p-value using the declared rejection rule.

    Reject H0H_0 when p≤αp\le\alpha.

    p ≤ αReject the null hypothesis

    There is sufficient evidence, at the stated level and under the model assumptions, in favour of the pre-specified alternative. State the conclusion about the population mean in context.

    p > αDo not reject the null hypothesis

    There is insufficient evidence against the null at this level. This does not prove or accept the null hypothesis, nor assign it a probability of being true.

    A p-value is conditional on the null model, not a probability that the hypothesis is true. Preserve full precision near the boundary; do not repeat sampling to obtain a preferred decision.

Try a worked example

Read a case, make a prediction, then reveal the reasoning. The examples keep their stated values.

Same z, different pre-specified alternatives

Illustration: separately pre-specified right-sided and two-sided standard-normal tests each observe z=2z=2. Compare their p-values at 5%.

Show reasoning

  1. 1. Use the upper tail for the right-sided test

    Count values at least as large as the observed statistic under the null model.

    p=P(Z≥2)≈0.0228\begin{aligned}p&=P(Z\ge2)\\&\approx0.0228\end{aligned}

  2. 2. Include both extreme tails for the two-sided test

    The standard normal distribution is symmetric. Count values at least as far from zero as the observation, in either direction.

    p=P(∣Z∣≥2)≈0.0455\begin{aligned}p&=P(|Z|\ge2)\\&\approx0.0455\end{aligned}

  3. 3. Apply the declared level

    Use full calculator precision for the comparison; the displayed values are rounded. Both p-values are below 0.05, so both tests reject at 5%. Their different p-values reflect their different alternatives. Do not choose or switch the alternative after seeing the observed sign.

What does p = 0.032 mean?

Illustration from the unit: a right-sided test gives p = 0.032. Interpret this probability and compare decisions under pre-specified 5% and 1% rules.

Show reasoning

  1. 1. Condition on the null model

    If the null model were true, results at least this extreme in the stated direction would have probability 0.032. This is not a 3.2% probability that the null hypothesis is true.

  2. 2. Compare with each fixed rule

    Reject at 5%, but do not reject at 1%. The data and p-value stay the same; the smaller significance level requires stronger evidence.

    0.01<0.032<0.050.01<0.032<0.05

  3. 3. Keep the conclusion limited

    Non-rejection does not establish the null hypothesis. Statistical significance also does not measure the size or practical importance of an effect.