Statistical Hypothesis Testing
Formulating null and alternative hypotheses, selecting one- or two-tailed tests, calculating exact significance and interpreting conclusions in context.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
A hypothesis test evaluates observed evidence under a null model at a preselected significance level.
Exam cue: Define the parameter and write both hypotheses before using data.
Concept 2
The alternative hypothesis determines whether a test is lower-tailed, upper-tailed or two-tailed.
Exam cue: Use the alternative hypothesis to choose tails and allocate the significance level.
Concept 3
A rejection region or p-value supports a contextual decision about the null hypothesis, not proof that either hypothesis is true.
Exam cue: State the decision and contextual evidence without claiming certainty.
Risk pitfalls and guardrails
Choosing the tail after seeing the sample result.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Using a probability that does not include outcomes at least as extreme as the observation.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Writing that the null hypothesis has been proved or accepted.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Null Hypothesis
The null hypothesis is the model assessed as the starting assumption.
Alternative Hypothesis
The alternative hypothesis states the direction or difference being investigated.
Significance Level
The significance level is the chosen probability threshold for rejecting the null model.
Critical Region
A critical region contains results sufficiently extreme to reject the null hypothesis.
p-value
A p-value is the null-model probability of the observed result or something at least as extreme.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
A manufacturer claims a success probability p=0.8. Which is the null hypothesis?
A researcher suspects a new process increases p above 0.8. Which alternative is appropriate?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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