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, togegether 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 Proportion:

• 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 $\sigma$
  • $CI(\alpha )={\bar{x}}_1-{\bar{x}}_2\pm t_{\alpha /2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$
  • With $df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}{)}^2}{\frac{1}{n_1-1}(\frac{s_1^2}{n_1}{)}^2+\frac{1}{n_2-1}(\frac{s_2^2}{n_2}{)}^2}$
    • Round down for more conversative interval estimation

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