Why Be Normal?
| English | Chinese | Pinyin |
|---|---|---|
| inference | 推断 | tuī duàn |
| normal model | 正态模型 | zhèng tài mó xíng |
Why the bell curve, again?
- Unit 5 showed that $\hat{p}$ has a roughly normal sampling distribution.
- Inference uses that normal model 正态模型 to reason from a sample back to the population.
- Because we know the shape, we can attach numbers to our uncertainty.
- Unit 6 turns the sampling distribution into estimates and decisions.
From sampling distribution to inference
- A sampling distribution tells us how a statistic behaves around the parameter.
- Inference 推断 runs it backwards: from one observed statistic, estimate the parameter.
- Two tools: confidence intervals (estimate a range) and significance tests (judge a claim).
- Both are built on the normal model for $\hat{p}$.
Inference quantifies uncertainty
- We never know $p$ exactly from a sample — there's always uncertainty.
- Inference doesn't remove uncertainty; it measures it honestly.
- A confidence interval reports "how sure" with a range and a level.
- A test reports "how surprising" with a probability.
The large counts link
- The normal-based procedures are valid only when the sampling distribution is roughly normal.
- That's guaranteed by the large counts condition: $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$.
- If counts are too small, the normal model — and the inference — can fail.
- Checking conditions is step one of every procedure in this unit.
Inference measures uncertainty; it doesn't eliminate it. A confidence interval or p-value is only trustworthy when its conditions hold — especially the large counts condition that makes the normal model valid. Reporting an interval or test without checking conditions is the most common way inference goes wrong.
A poll finds $\hat{p} = 0.55$ support in a sample of $n = 200$.
- Large counts: $n\hat{p} = 110 \ge 10$ and $n(1-\hat{p}) = 90 \ge 10$. ✓
- So the normal model applies, and we can build an interval or run a test.
- Inference will report how far the true $p$ might plausibly be from $0.55$.
Inference reasons from a sample statistic back to a population parameter using the normal model for the sampling distribution. It quantifies uncertainty — via confidence intervals and tests — rather than removing it. Its validity rests on conditions, especially the large counts condition that justifies normality.
The normal model behind inference
Inference reasons from this normal sampling distribution back to p.
Inference about a proportion mainly does what?
Inference measures uncertainty honestly — it doesn't remove it.
The large counts condition (np̂ ≥ 10 and n(1−p̂) ≥ 10) justifies using a normal model.
Large counts make the sampling distribution roughly normal.
The two main tools of inference for a proportion are...
Intervals estimate a range; tests judge a claim.
You can safely report an interval or test without checking any conditions.
Conditions (esp. large counts) must hold for the procedure to be valid.
For n = 200 and p̂ = 0.55, compute np̂ (the count of successes).
200 × 0.55 = 110, which is ≥ 10.