MATH2089 Chap.11 Hypothesis Testing
Hypothesis Testing
Hypothesis tests compare observed evidence with a null model through a standardised statistic and a preselected alternative. This chapter fixes the tail before data inspection, interprets p values as null model probabilities for equally or more extreme results, distinguishes reject from fail to reject and connects Type I error, Type II error and power.
Two sided tests are linked directly to matching confidence intervals and effect estimates. Start by naming the population parameter. The null gives a benchmark value or relationship, and the alternative states the direction or difference the study is designed to investigate. Choose a one-sided or two-sided alternative from the scientific question before seeing the sample result.
Selecting the more favourable tail after inspection changes the error rate and is not the stated test. The test statistic measures the distance between the estimate and null value in standard-error units. Its reference distribution follows from the sampling model and assumptions. Check independence, the relevant distribution approximation and any variance or pairing assumptions before trusting a software p-value.
Also verify that the program received the intended alternative and null value. A p-value is the null-model probability of obtaining a result at least as extreme as the observed statistic in the direction defined by the alternative. It is not the probability that the null is true, the probability that the result occurred by chance, or the size of the effect.
Compare p with the selected significance level and write reject or fail to reject. Do not write accept when the evidence is insufficient. The conclusion returns to the parameter and population. Rejection supports evidence for the alternative under the model; failure to reject means the data do not provide sufficient evidence for it. Neither wording proves a universal fact.
Report the estimated effect and a confidence interval so the reader can assess plausible magnitude and practical importance. At matching levels, a two-sided test rejects a null value exactly when that value lies outside the corresponding confidence interval. Type I error is rejecting a true null and is controlled by alpha for the designed procedure. Type II error is failing to reject a false null.
Power is one minus the Type II error probability for a specified alternative. Larger samples, lower variability and larger true effects generally improve power. Power is not one permanent number without a target effect. A narrow interval around the null may rule out important effects even after failure to reject, while a wide interval may show that the study was simply insensitive.
End every test with hypotheses, statistic, tail, p-value, decision, effect estimate, conditions and a conclusion whose scope matches the data-collection design.
What this chapter covers
- 01
Null and alternative hypotheses
- 02
One sided and two sided tests
- 03
Test statistics
- 04
P values and significance
- 05
Type I and Type II errors
- 06
Power and confidence interval links
Worked example · free
Run an upper tail test
- +1State the upper tail alternative that the mean exceeds 50.
- +1Compute z=(52-50)/0.8=2.5.
- +1Compare p=0.0062 with alpha 0.05 and reject the null.
- +1Conclude that the data support a population mean above 50, with estimated increase 2.
Key terms
- Null hypothesis
- The reference parameter statement used to generate the test statistic distribution.
- P value
- A null model probability of results at least as extreme as the observation.
- Statistical power
- The probability of rejecting the null at a specified true alternative.
Hypothesis Testing FAQ
Is a p value the probability the null is true?
No. It is calculated assuming the null model and measures how extreme the observed statistic is under that assumption. It does not assign probability to the hypothesis.
Why say fail to reject?
A nonsignificant result can reflect weak evidence or low power. It does not prove equality, so fail to reject is more accurate than accept the null.
How do intervals and tests connect?
For matching two sided levels, a null parameter value outside the confidence interval is rejected. The interval additionally shows plausible effect sizes, direction and uncertainty.
Exam move
Use one fixed answer order: parameter and hypotheses, conditions, statistic, tail, p value, decision and contextual conclusion. Draw rejection regions and label both error types. Pair every two sided test with its confidence interval so significance and effect size are revised together. Practise rewriting incorrect statements such as accepting the null or treating the p value as its probability.
For each one-sided test, decide the direction before looking at a sample statistic and shade the matching tail. Build a small power table that changes sample size, variability and target effect one factor at a time. Compare a narrow interval around the null with a wide interval spanning important effects after failure to reject.
These contrasts separate evidence, effect magnitude and design sensitivity, which should all appear in a complete inference answer.
Working through Hypothesis Testing in MATH2089? Sia is AskSia’s AI Mathematics tutor — ask any MATH2089 Hypothesis Testing question and get a clear, step-by-step explanation grounded in how MATH2089 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.