Confidence Interval

An interval, with a certain level of confidence (specified in percentage), in which the true population parameter is likely to lie. Calculated based on sample data. Quantifies uncertainty in estimation.

A confidence interval at level $1 − \alpha$ means that if you repeatedly sampled and built intervals, about $(1 − \alpha) \times 100$ of them would contain the true parameter. Usually 90%, 95%, 99% are used.

Formula

When $\sigma$ is known: $$

$$\bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$$

When $\sigma$ is unknown (t-distribution is used): $$

$$\bar{X} \pm t_{\alpha/2,, n-1} \frac{s}{\sqrt{n}}$$

Properties

• As sample size (n) increases, confidence interval becomes narrower
• Higher confidence level and higher population variance causes wider confidence interval

Standard Errors

Parameter	Standard Error
$\bar{x}$	$\frac{s}{\sqrt{n}}$
$\bar{p}$	$\frac{1}{\sqrt{n}} \sqrt{ p(1-p) }$
$\bar{x_1} - \bar{x_2}$	$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
$\bar{p_1} - \bar{p_2}$	$\sqrt{\frac{p_1 (1 - p_1)}{n_1} + \frac{p_2 (1 - p_2)}{n_2}}$