Penyampelan dan anggaran
| English | Bahasa Indonesia |
|---|---|
| sample/ˈsæmpl/ | sampel |
| population/ˌpɒpjʊˈleɪʃn/ | populasi |
| stratified/ˈstrætɪfaɪd/ | bertingkat |
| systematic/ˌsɪstəˈmætɪk/ | sistematis |
| cluster/ˈklʌstə/ | klaster |
| sampling distribution/ˈsæmplɪŋ ˌdɪstrɪˈbjuːʃn/ | distribusi sampel |
| standard error/ˈstændəd ˈerə/ | kesalahan baku |
| Central Limit Theorem/ˈsentrəl ˈlɪmɪt ˈθɪərəm/ | Teorem Had Pusat |
| confidence interval/ˈkɒnfɪdəns ˈɪntəvl/ | interval kepercayaan |
The poll that predicted the election
- A survey of 1000 people can predict the voting behaviour of 60 million. How?
- Sampling 样本 lets you draw conclusions about a whole population 总体 from a small, carefully chosen subset. The mathematics behind it is the foundation of all statistical inference.
Populations and samples
- A population is the entire group you want to study. A sample is a small group from the population.
- Good sampling needs randomness — every member must have a known chance of being selected.
- Sampling methods: simple random, stratified 分层 (proportional from each subgroup), systematic 系统 (every $k$th member), cluster 整群.
Stratified sampling. A school has 600 boys and 400 girls. A stratified sample of 50: $\dfrac{600}{1000} \times 50 = 30$ boys and $\dfrac{400}{1000} \times 50 = 20$ girls.
The sample mean is nearly normal whatever the population shape
Taburan pensampelan
X̄ ~ N(μ, σ²/n)
Mengikut Teorem Had Pusat, min sampel mengikuti lengkung normal — lebih sempit untuk sampel yang lebih besar.
Sebuah sekolah mempunyai 600 pelajar lelaki dan 400 pelajar perempuan. Sampel berlapis 50 memerlukan berapa banyak pelajar lelaki?
(600/1000) × 50 = 30 pelajar lelaki.
The sampling distribution 抽样分布 of the mean
- The sample mean $\bar{X}$ is itself a random variable:
- The standard error 标准误 of the mean is $\dfrac{\sigma}{\sqrt{n}}$ — it shrinks as $n$ grows.
Statistics studies a sample to learn about a whole population
Populasi mempunyai σ = 8. Untuk sampel n = 64, berapakah Var(X̄) = σ²/n?
Var(X̄) = σ²/n = 64/64 = 1.
Populasi mempunyai σ = 10 dan n = 25. Ralat piawai σ/√n = ?
σ/√n = 10/√25 = 10/5 = 2.
Padangkan setiap idea pensampelan dengan maknanya.
CLT menjadikan min sampel normal; sebaran mereka ialah σ²/n, memberikan selang keyakinan.
The Central Limit Theorem 中心极限定理
- By the Central Limit Theorem, for a large sample ($n \geq 30$), $\bar{X}$ is approximately normal, whatever the population's shape.
- This is one of the most powerful results in all of mathematics — it lets us use the normal distribution even when the data isn't normal.
CLT applies to the mean, not the data. The Central Limit Theorem says the distribution of $\bar{X}$ becomes normal — not the individual data points. The raw data can have any shape.
A 95% confidence interval 置信区间 reaches 1.96 standard errors each side of the sample mean
Mengikut Teorem Had Pusat, min sampel adalah kira-kira normal untuk sampel besar, tanpa mengira bentuk populasi.
CLT menjadikan X̄ kira-kira normal apabila saiz sampel bertambah.
Teorem Had Pusat mengatakan titik data individu menjadi berdistribusi normal untuk sampel besar.
CLT terpakai kepada min sampel X̄, bukan titik data individu. Data mentah boleh mempunyai mana-mana bentuk.
Confidence intervals
- A confidence interval gives a range that probably contains the true mean.
- For a normal population with known $\sigma$ (or a large sample), a 95% interval is:
- From a random sample find unbiased estimates of the mean and variance ('unbiased' = correct on average), and a confidence interval for a population proportion; random numbers help choose the sample.
Sampel n = 64 mempunyai σ = 8. Berapakah margin 1.96 × σ/√n untuk selang 95%? (2 dp)
1.96 × 8/√64 = 1.96 × 8/8 = 1.96.
You've got it
- $E(\bar{X}) = \mu$, $\text{Var}(\bar{X}) = \dfrac{\sigma^2}{n}$; CLT makes $\bar X$ ≈ normal for large $n$
- a 95% confidence interval: $\bar{x} \pm 1.96\dfrac{\sigma}{\sqrt{n}}$
- good sampling needs randomness