Basic Statistics - (08) Sampling Distribution

1. Inferential Statistics

Methods to draw conclusion about a population based on data coming from samples.

• Sampling methods

  1. Simple Random Sample

     

     Each subject has the same choice to be chosen.

  2. Random Multi-stage Cluster Sample

  

  • unknow sampling frame

  1. Stratified Random Sample

     

     • More representative in every stratum
     • Must know sampling frame

2. Sampling Distribution and Central Limit Theorem

• Sampling Distribution

  With infinite number of samples,Distribution of sample means is bell-shaped with a mean equal to the population mean.

  

• Central Limit Theorem

  • the sampling distribution of sample mean x_bar is approximately normal (provided that n is sufficiently large).

  • even if the variable is not normally distributed in the population.(n>30)

    

• Sample mean

  • \[\mu_{\overline{x}} = \mu\]

  

• Sample deviation

  • \[\sigma_{\overline{x}} = \frac{\sigma} {\sqrt{n}}\]
  • 总体方差与样本方差正相关。Larger variation in population larger variation in samples
    • 样本数量与样本方差负相关。Larger n lower variation in samples(Central limit theorem!!!)

3. Population distribution & Sample distribution & Sampling distribution

• Differences of 3 distributions

  

  • sample distribution是一次取样分布。
  • sampling distribution (theoretical distribution)是无限次取样分布，符合中央极限定理，平均值无限接近总体平均值。
  • population distribution 总体分布。

• Z-score for sample mean

  

  这个问题问的是取样平均值在某个区间的概率，应该利用sampling distribution来进行计算。

  

4. Binominal Sampling Distribution

• Sampling Distribution of Sample Proportion

  

  • Sample space

    • \[n * \pi >= 15\]
    • \[n(1-\pi) >= 15\]

      n为每次sample取样，样本个数。所以，一次取样样本量小的话，sampling distribution将不准确。

• mean

  • \[\mu_p =\pi\]

  pi为事件成功概率。

  • deviation

    • \[\sigma_{p}^2 =\frac{\pi(1-\pi)}\]
    • 注：二项分布总体方差: \(\sigma^2 = p(1-p)\)

5. Example

6. Conclusion

无论是二项分布还是正态分布：

1. 无限次样本量的平均值（sampling mean）都无限接近总体平均值（population mean）。
2. 由于中央极限定理，无限次样本会趋中，所以方差需要除以样本个数，以降低趋中趋势。