Confidence interval

Description:

• From a Sample
• Based on Distribution of sample mean

For mean of known standard deviation:

• A confidence interval for a Population mean $\mu$, is an interval of values between two limits, together with a percentage indicating our confidence that $\mu$ lies in that interval
  • For $Z=\frac{{\bar{X}}_n-\mu }{\sigma /\sqrt{n}}$
    • By Central limit theorem
  • $P(-a\le Z\le a)=\alpha \to P(Z\le a)=1-\frac{\alpha }{2}+\alpha =z_{\frac{\alpha }{2}}$
    • $\alpha$ is the area under graph between 2 limits while $z_{\alpha /2}$ is the upper limit
  • $CI=\{{\bar{x}}_n-z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}\le \mu \le {\bar{x}}_n+z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}\}$
• The width is then $2\times z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}$

For mean of unknown sd:

• Use t distribution

For population proportional:

• True population proportion $p$ is unknown
• $\bar{p}=\frac{X}{n}$
  • where $X$ is binomial random variable denotes the number of successes in the sample
  • $X\sim B(n,p).E[X]=np$ and ${\sigma }_X=\sqrt{npq}$
• $E[\bar{p}]=p$ and $\sigma (\bar{p})=\sqrt{\frac{pq}{n}}$
• By Central limit theorem, $\bar{p}\sim N(p,\frac{pq}{n})$
  • For $np>5$ and $nq>5$
• $\alpha \text{CI}=\bar{p}\pm z_{\alpha /2}\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$

For difference of two variables:

• Difference random variable of 2 variables
• For known $\sigma$
  • $CI(\alpha )={\bar{x}}_1-{\bar{x}}_2\pm z_{\alpha /2}\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}$
• For unknown $\sigma$
  • $CI(\alpha )={\bar{x}}_1-{\bar{x}}_2\pm t_{\alpha /2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$

$$\begin{gathered} (x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ ) \end{gathered}$$