6.5 Hypothesis tests

Set H₀ and H₁, choose a significance level, calculate a test statistic under H₀ and reject only when the result lies in the specified critical region or has p≤α.

Match one- or two-tailed wording, retain the parameter context and report the decision rather than claiming certainty.

A two-sided p-value of 0.04 rejects H₀ at 5% but not at 1%.

A p-value is not the probability that H₀ is true, and failure to reject is not proof of equality.

Approximate a binomial count by N(np,np(1−p)) or a Poisson count by N(λ,λ) when the distribution is sufficiently spread. Apply continuity correction to integer boundaries.

Translate “at most”, “at least” and exact counts into half-unit intervals before standardising, then compare with the exact model when the approximation is marginal.

P(X≥12) becomes P(Y>11.5) in the continuous approximation.

Using the uncorrected boundary can produce a visibly different tail probability.

Under H₀, the statistic has a known or approximated distribution. Critical values or p-values are calculated from that distribution and the direction of H₁.

Use the stated variance or standard error, preserve tails and degrees of freedom, and conclude in the original units.

A one-sided upper test uses P(T≥t_obs), whereas a two-sided test counts both tails of comparable extremity.

The same statistic value can lead to different decisions at different α or under different alternative hypotheses.

For testing μ=μ₀ with known σ, use Z=(X̄−μ₀)/(σ/√n). If σ is estimated and the syllabus model requires it, use the corresponding t statistic.

State H₀, calculate the standard error, choose the tail and compare with the correct critical value. The observed mean is evidence, not the null value itself.

With x̄=52, μ₀=50, σ=10, n=100, z=2; the decision depends on α and whether the test is one- or two-sided.

Using σ instead of σ/√n understates the evidence by ignoring the sample size.

For X~N(μ,σ²), standardise observations or sample means with z=(x−μ)/σ and use the standard normal tail. A two-sided alternative uses both tails.

Write the null parameter and variance clearly, draw the rejection region and convert back to the original variable when reporting a critical boundary.

Testing μ=100 with σ=15 and n=25 uses standard error 3, so x̄=106 gives z=2.

The test statistic’s denominator is the standard error for a mean, not the individual standard deviation σ.