4.2 Inference using normal and t-distributions

Set a null hypothesis H₀ and an alternative H₁ before calculating. The test statistic or p-value measures how surprising the sample would be if H₀ were true.

Choose a significance level α, identify the rejection region or compare p with α, and state the decision in context. A one- or two-tailed alternative changes the region.

At α=0.05, p=0.03 leads to rejection of H₀; it does not prove H₁, only that the observed result is sufficiently inconsistent with the null model.

Failing to reject H₀ is not accepting it, and statistical significance does not measure the size or practical importance of an effect.

When samples estimate a common population mean, a pooled estimate weights each sample mean by its sample size: x̄=(n₁x̄₁+n₂x̄₂)/(n₁+n₂). Variance pooling requires its own assumptions.

Pool only when the samples target the same parameter and independence and comparable modelling assumptions are reasonable. Do not average sample means equally unless sample sizes match.

Means 12 from n=20 and 15 from n=10 give pooled mean (20·12+10·15)/30=13, not 13.5.

Pooling cannot repair biased sampling, and a pooled mean does not imply the two populations have identical distributions.

A hypothesis-test statistic compares the observed estimate with the null value after scaling by its standard error. Under H₀, its sampling distribution determines critical values or p-values.

Use the correct distribution and degrees of freedom, match the tail to H₁, and distinguish a statistic calculated from data from the random variable describing its repeated-sample behaviour.

A two-sided test with statistic z=2.1 has p=2P(Z≥2.1), not just one tail; whether this rejects H₀ depends on α.

A 2.1 standard-error difference is not automatically “significant”; the alternative, tail count and chosen α still matter.

A confidence interval has the form estimate ± critical value × standard error. It gives a range produced by a method that captures the fixed population parameter in a stated proportion of repeated samples.

Choose the correct normal or t-based model and use the sample size and variability assumptions. A 95% interval is about the method’s long-run coverage, not a 95% probability attached to the already fixed mean.

If x̄=20 and the margin is 1.5, report (18.5,21.5) in the measurement units and explain the confidence level.

A wider interval is not automatically worse: it may reflect higher confidence or a smaller sample. The interval does not contain 95% of individual observations.

For two independent means, an interval for μ₁−μ₂ is the sample difference plus or minus a critical value times its standard error. The order of subtraction fixes the sign.

Check independence, variance assumptions and whether a paired design should instead be analysed through within-pair differences. If zero lies in the interval, a two-sided 5% test would not reject equality.

An interval for μA−μB of (1.2,4.8) supports a positive difference; an interval (−0.6,2.1) does not establish a direction at that confidence level.

An interval crossing zero is not proof of no effect, and separate intervals for μA and μB are not equivalent to an interval for their difference.