ECMT1020 Chap.9 Hypothesis Tests and Standard Errors
Hypothesis Tests and Standard Errors
A hypothesis test measures how unusual an estimate would be if a stated null claim were true. This chapter aligns null and alternative hypotheses with one-sided or two-sided rejection regions, standardises the estimate using its standard error, and distinguishes rejection from non-rejection. It also introduces Type I error, Type II error and power as repeated-decision properties.
The raw gap between an estimate and null value gains meaning only relative to sampling uncertainty. A test statistic expresses that gap in standard-error units, while the alternative determines which directions count as evidence. The significance level controls a long-run false-rejection rate under the null, not the probability that the particular null statement is true.
Power must be evaluated at a specified alternative and changes with effect size, uncertainty, sample size and the chosen threshold.
What this chapter covers
- 01
Null and alternative hypotheses
- 02
Standard errors
- 03
Test statistics
- 04
One-sided and two-sided tests
- 05
Critical values and p-values
- 06
Testing errors and power
Worked example · free
Complete a two-sided mean test
- +1The alternative is two-sided.
- +1The estimated standard error is 5/√25=1.
- +1The null gap is 52.5-50=2.5.
- +1The t statistic is 2.5 with 24 degrees of freedom.
- +1Its absolute value exceeds 2.064.
- +1Reject the null at the stated level and report evidence that the population mean differs from 50.
Key terms
- Null hypothesis
- The population claim used as the testing reference.
- Alternative hypothesis
- The departure from the null for which evidence is sought.
- Standard error
- Estimated sampling spread used to standardise an estimator.
- Type I error
- Rejecting a null hypothesis that is true.
- Power
- The probability of rejecting the null under a specified alternative.
- Significance level
- The chosen long-run probability of rejecting the null when it is true. It fixes the size of the rejection region under the null and is not the probability that the current null claim is true.
- Type II error
- Failure to reject the null when a specified alternative is true. Its probability depends on the particular departure, standard error, sample size and decision threshold, and one minus this probability is test power.
- Null gap
- The difference between an observed estimate and its hypothesised value. Dividing this raw gap by its standard error produces the test statistic used for comparison.
Hypothesis Tests and Standard Errors FAQ
What does non-rejection mean?
It means the observed statistic did not cross the chosen threshold at the selected significance level. It does not prove the null is exactly true in the population.
How does the alternative affect the test?
It determines whether evidence is placed in an upper tail, lower tail or both tails. The decision rule must be chosen to match that direction before inspecting the estimate.
Is significance the same as importance?
No. Statistical significance compares an estimate with its uncertainty. Practical importance concerns the raw magnitude and context, which should be reported separately in the variable's original units.
Exam move
Use the same six lines for every test: parameter, hypotheses, standard error, statistic, decision rule and contextual conclusion. Practise both one-sided and two-sided alternatives with supplied critical values. After deciding, state the estimated magnitude separately and audit whether your wording says reject or do not reject rather than proving or accepting the null.
Write hypotheses in symbols and words before inspecting the estimate. Circle the directional phrase in the research question and use it to choose upper, lower or two-sided evidence. Draw the rejection region immediately so a later negative sign cannot tempt you to change the alternative. Compute the standard error on its own line, preserving the square root and units. Then calculate the raw null gap and divide.
Keeping these steps separate reveals whether an unexpected statistic comes from the estimate, null value or uncertainty. When a critical value is supplied, compare like with like: absolute magnitude for a symmetric two-sided rule or signed magnitude for the specified one-sided rule. When a p-value is supplied, compare it with the preselected significance level. Do not mix the two rules midway.
Practise paired examples with the same estimate but different standard errors to see how precision changes evidence without changing effect size. Also practise the same statistic under one-sided and two-sided alternatives so tail allocation becomes explicit. Create a language checklist for conclusions. “Reject” and “do not reject” describe the decision; they do not prove, accept or establish equality.
Name the population parameter, direction if relevant and significance level. Report the estimated raw difference separately so statistical evidence does not replace economic magnitude. For error concepts, imagine many repetitions. Type I error occurs under a true null, Type II error under a particular false null, and power is one minus the latter probability at that alternative.
Finish revision sets by diagnosing whether each mistake came from hypotheses, standard error, distribution, tail, comparison or wording, then solve a fresh item targeting that link.
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