MFIN6201 Chap.3 Large Samples and Statistical Inference
Large Samples and Statistical Inference
This chapter connects the law of large numbers and central limit theorem to practical inference. It develops null and alternative hypotheses, test statistics, significance levels, p-values and confidence intervals. The central exam skill is interpretive discipline: a p-value is calculated under the null, non-rejection is not acceptance, and statistical significance does not measure economic magnitude.
The law of large numbers explains why a sample average settles near its population target. The central limit theorem supplies an approximately normal reference distribution after centring and scaling, which makes tail probabilities usable even when the underlying observations are not normal. A test begins with a parameter claim, not a software star.
The statistic measures how many estimated standard errors separate the estimate from that null value. The alternative determines whether one or both tails count as more extreme. A confidence interval collects parameter values compatible with the data under the matching test. Coverage describes the long-run procedure, not a probability that the already fixed parameter moved after the interval was observed.
Large samples improve the approximation but do not cure dependence, selection or a badly defined estimand. Interpretation therefore proceeds from estimate and units to uncertainty, decision and economic meaning.
What this chapter covers
- 01
Law of large numbers
- 02
Central limit theorem
- 03
Null and alternative hypotheses
- 04
Test statistics and p-values
- 05
Significance levels
- 06
Confidence interval inversion
Worked example · free
A two-sided coefficient test
- +1Compute t=(0.018-0)/0.007=2.57.
- +1Compare the absolute statistic with 1.96.
- +1Reject zero at 5% because 2.57 exceeds 1.96.
Key terms
- Null hypothesis
- The parameter claim evaluated by a statistical test.
- P-value
- A null-based tail probability for results at least as extreme as the observation.
- Confidence interval
- A range produced by a procedure with stated repeated-sample coverage.
Large Samples and Statistical Inference FAQ
What is the difference between the LLN and CLT?
The law of large numbers supports convergence of a sample average toward the population mean. The central limit theorem describes the approximately normal shape of the centred and scaled sampling distribution, which supports large-sample tests and intervals.
Does a small p-value show a large effect?
No. A p-value combines the estimated distance from the null with uncertainty. A very precise but economically tiny effect can produce a small p-value, so the coefficient's units and confidence interval must be interpreted separately.
Why should a test and interval agree?
A two-sided confidence interval contains the parameter values that the matching two-sided test would not reject. If zero lies outside a 95% interval, the corresponding 5% two-sided test should reject a zero null, subject to the same approximation.
What does non-rejection mean?
It means the statistic is not sufficiently incompatible with the null at the selected level. It does not prove the null. Weak signal, large uncertainty, limited sample size or a genuinely small effect can produce the same decision.
When should a one-sided test be used?
Use it only when the research question fixed the direction before the estimate was inspected and the opposite direction would not count as the claimed effect. Choosing a tail after seeing the sign understates uncertainty.
Can a large sample fix a biased design?
No. More observations can shrink sampling uncertainty around the wrong target. Selection, confounding or measurement problems can persist as n grows, producing a precise estimate of a distorted relationship. Design quality and sample size solve different problems.
Why report the confidence interval as well as significance?
The interval shows the range of parameter values compatible with the procedure, including both direction and plausible magnitude. A binary significance label hides whether the estimate is precise, economically important or close to the selected threshold. Reporting both gives the reader a fuller uncertainty statement.
Exam move
Rehearse inference as a fixed sequence: write the null, standardise the estimate, compare with a critical value or p-value, state the decision, and interpret magnitude. Cross-check every two-sided decision against the corresponding confidence interval. Practise nonzero nulls so the numerator never defaults to zero by habit. Create paired cards with a test on one side and its interval on the other.
Change the significance level and tail direction to see which critical boundary changes. Translate every p-value into a sentence beginning with “assuming the null and maintained model,” which blocks the false probability-of-truth interpretation. Keep effect size and evidence strength in separate lines. Work one example where a tiny effect is precise and another where a large effect is noisy.
State “do not reject” instead of “accept,” then list the possible reasons for weak evidence. Retain full precision until the endpoints are calculated, and check that a rejected null falls outside the corresponding interval. Finally, name the sampling or large-sample approximation being used so the decision never appears assumption-free.
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