Basic Statistics(10)-Confidence Interval

Basic Statistics - (10) Confidence Interval

Estimation
- a point estimate
- an interval estimate is a range of numbers which, most likely, contains the actual population value.

It is impossible to compute the confidence interval because we usually don’t know the value of the population standard deviation.
- \[\overline{x} \pm 1.96*\sigma_{\overline{x}}\]
- \[\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}\]
- 总数标准差未知。
T Distribution
- The T distribution (also called Student’s T Distribution) is a family of distributions that look almost identical to the normal distribution curve, only a bit shorter and fatter. The t distribution is used instead of the normal distribution when you have small samples (for more on this, see: t-score vs. z-score). The larger the sample size, the more the t distribution looks like the normal distribution. In fact, for sample sizes larger than 20 (e.g. more degrees of freedom), the distribution is almost exactly like the normal distribution.
- T table
- 2 assumptions
  - randomization sampling
  - approximately normal population distribution
    - be wary of extreme outliers

It is impossible to compute the confidence interval because we usually don’t know the value of the population standard deviation.
\[p \pm 1.96*\sigma_p\]
\[\sigma_p = \sqrt{\frac{\pi(1-\pi)}{n}}\]
- the population standard deviation pi is unknown.
公式
- 二项样本分布：stick to standard normal distribution
- 1 assumption
  - at least 15 successes and 15 failures. \(\begin{align*} np &\geq15 \\ n(1-p) &\geq 15 \end{align*}\)

We have to compromise between confidence and precision.

As one gets better, the other gets worse.

Mean
- \[n = \frac{\sigma^2 * z^2}{m^2}\]
- m: margin of error 容错区间。
- \sigma: educated guess
Proportion
- \[n= \frac{p(1-p) z^2}{m^2}\]
- safe approach最大数值