Hypothesis testing

What is a null hypothesis H0?

A statement about a population parameter that we assume to be true; usually a "no change" or "no effect" claim. Stated in terms of a parameter value, e.g. H0: p = 0.7.

What is an alternative hypothesis H1?

The statement we test against H0; it specifies the kind of departure from H0 we are looking for, e.g. H1: p ≠ 0.7, p > 0.7 or p < 0.7.

What is a test statistic?

The sample quantity we calculate and compare to a distribution to decide whether to reject H0 (e.g. the number of successes, or the sample mean).

What is the significance level?

The probability of rejecting H0 when it is actually true; the maximum risk of a false-positive we are willing to accept (e.g. 5%, 1%).

What is a 1-tail test?

A test where H1 is directional, e.g. H1: p > 0.7 or H1: p < 0.7.

What is a 2-tail test?

A test where H1 is non-directional, e.g. H1: p ≠ 0.7. The significance level is split between the two tails.

What is the critical value?

The boundary value of the test statistic; if the test statistic is beyond it, we reject H0.

What is the critical region (rejection region)?

The set of values of the test statistic for which H0 is rejected.

What is the acceptance region?

The set of values of the test statistic for which H0 is NOT rejected.

What is a p-value?

The probability, assuming H0 is true, of obtaining a test statistic at least as extreme as the one observed.

Decision rule using p-values

If p ≤ significance level, reject H0; otherwise do not reject H0.

What is the correct form of a conclusion when rejecting H0?

"There is evidence at the X% level to reject H0. It is likely that [statement about the population in context]." Conclusions should reflect that they are not certain.

What is the correct form of a conclusion when not rejecting H0?

"There is insufficient evidence at the X% level to reject H0. There is no reason to suppose that [statement about the population]." Avoid saying H0 is "accepted" or "true".

Why is "accept H0" the wrong conclusion?

Failing to reject H0 doesn't prove H0 is true — it just means the sample doesn't give enough evidence to overturn it.

How do you conduct a hypothesis test for a binomial proportion?

State H0 and H1 in terms of p; assume H0 and use X ~ B(n, p₀); calculate the probability of the observed result (or more extreme); compare to the significance level; conclude in context.

Why is the actual significance of a binomial test usually less than the stated level?

Because the binomial distribution is discrete, the critical region's exact probability is ≤ the nominal significance level, rarely equal to it.

On OCR A H240, is the normal approximation used for tests on a binomial proportion?

No — use the binomial distribution directly.

For a sample from N(μ, σ²) of size n, what is the distribution of the sample mean X̄?

X̄ ~ N(μ, σ²/n).

How do you test the mean of a normal distribution with known variance?

Standardise the sample mean: Z = (X̄ − μ₀)/(σ/√n); compare to critical values from the standard normal, or use a p-value.

What is Pearson's product–moment correlation coefficient r?

A measure (between −1 and 1) of how closely the points of a bivariate sample lie to a straight line. r = ±1 means perfect linear, r = 0 means no linear correlation.

How is Pearson's r used in a hypothesis test?

Test H0: ρ = 0 against H1: ρ ≠ 0 (or one-tailed). Compare the sample r to a critical value (from tables) for the given n and significance level, or to a p-value, assuming bivariate normal data.

What does it mean if the sample r is in the critical region?

There is evidence to reject H0, i.e. evidence of correlation in the population.

What assumption is made about the data when using Pearson's r in a hypothesis test?

The data is assumed to come from a bivariate normal distribution.

What does the significance level represent in terms of the null hypothesis?

The probability of incorrectly rejecting a true H0 (a Type I error).

Why must hypothesis test conclusions be stated in context?

Marks are awarded for relating the statistical decision back to the original problem, with appropriate uncertainty (e.g. "evidence that the mean journey time has changed").