Why Be Normal? · 为什么要正态？
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| inference/ˈɪnfərəns/ | 推断 | tuī duàn |
| normal model/ˈnɔːml ˈmɒdl/ | 正态模型 | zhèng tài mó xíng |
A poll cannot inspect every library user
- A fictional random sample of 200 library users contains 110 who support later opening. The observed proportion is $\hat p=110/200=0.55$.
- Inference 推断 uses this sample to learn about the population proportion $p$. The estimate can vary if we take another sample.
Choose the question before the calculation
- A confidence interval estimates a range for $p$. A significance test assesses a specified claim such as $p=0.50$.
- The normal model 正态模型 describes an approximate sampling distribution. It does not turn a sample percentage into certain knowledge of the population.
Inference about a proportion mainly does what?
Inference connects a sample to a population parameter and reports sampling uncertainty under the study assumptions.
The two main tools of inference for a proportion are...
Intervals estimate a range; tests judge a claim.
Check how the sample was collected
- Confirm random sampling and define the population. For a sample without replacement, check that 200 is at most 10% of the population size.
- If users volunteered through one social-media group, a narrow calculated interval does not repair selection bias. Large counts cannot replace a sound design.
Use the counts for the chosen procedure
- For an interval, observed counts are $200(0.55)=110$ and $200(0.45)=90$. Both exceed ten.
- For a test of $p=0.50$, null expected counts are $200(0.50)=100$ and $200(0.50)=100$. Use the null value for this test check.
An interval estimates p; a test checks p = 0.50. The same 110/200 sample needs a different count check for each procedure.
For n = 200 and p̂ = 0.55, compute np̂ (the count of successes).
200 × 0.55 = 110, which is ≥ 10.
Match each normal-procedure check to the counts used here.
The interval estimates variability using the observed proportion; this test checks its null model.
Normal approximation is not a guarantee
- The large-count rules support a useful approximation under the study assumptions. They do not guarantee an exactly normal distribution or an unbiased response.
- For 4 successes in 20 observations, the observed success count fails the interval rule. Do not approve the normal interval merely because there are 20 observations.
Random sampling, independence and adequate counts answer separate questions. Passing one does not pass the others.
For a normal interval, having at least ten observed successes and ten failures supports the normal approximation; design and independence checks are also needed.
Observed counts support the interval approximation, but they do not guarantee normality or remove sampling bias.
A voluntary online poll has 1,000 responses. Which statements are justified?
Response count does not establish a random design or repair biased selection.
Report what the method can support
- For the library sample, state the population, the estimate and the uncertainty procedure. Include the random-design and count checks.
- An interval and a test answer different questions. Choose their conditions and formulas separately, even when they use the same sample.
For the library sample, state the population, the estimate and the uncertainty procedure. Include the random-design and count checks.
You can safely report an interval or test without checking any conditions.
Conditions (esp. large counts) must hold for the procedure to be valid.
Arrange this inference workflow.
The question selects the procedure; the assumptions must be checked before reporting its result.