Common Confidence Intervals

Confidence intervals for mean ($\mu$)

Distribution	${\sigma }^2$ known?	Sample size $n$	$(1-\alpha )100\%$ Confidence interval
$N(\mu ,{\sigma }^2)$	✅	Any	$\bar{x}-z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$
Any	✅	Large	$\bar{x}-z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$
$N(\mu ,{\sigma }^2)$	❌	Small	$\bar{x}-t_{\alpha /2,n-1}\frac{s}{\sqrt{n-1}}$
Any	❌	Large	$\bar{x}-z_{\alpha /2}\frac{s}{\sqrt{n-1}}$

Confidence Intervals for Difference in Means (${\mu }_X-{\mu }_Y$)

Distribution	${\sigma }_X^2,{\sigma }_Y^2$ known?	Sample sizes	$(1-\alpha )100\%$ Confidence interval
$X\sim N({\mu }_X,{\sigma }_X^2),Y\sim N({\mu }_Y,{\sigma }_Y^2)$	✅	Any	$(\bar{X}-\bar{Y})\pm z_{\alpha /2}\sqrt{\frac{{\sigma }_X^2}{n_X}+\frac{{\sigma }_Y^2}{n_Y}}$
$X,Y$ any distribution	✅	Large	$(\bar{X}-\bar{Y})\pm z_{\alpha /2}\sqrt{\frac{{\sigma }_X^2}{n_X}+\frac{{\sigma }_Y^2}{n_Y}}$
$X\sim N({\mu }_X,{\sigma }^2),Y\sim N({\mu }_Y,{\sigma }^2)$	❌ (equal variances)	Small	$(\bar{X}-\bar{Y})\pm t_{\alpha /2,n_X+n_Y-2}{S}_p\sqrt{\frac{1}{n_X}+\frac{1}{n_Y}}$
$X\sim N({\mu }_X,{\sigma }_X^2),Y\sim N({\mu }_Y,{\sigma }_Y^2)$	❌ (unequal variances)	Small	$(\bar{X}-\bar{Y})\pm t_{\alpha /2,\nu }\sqrt{\frac{S_X^2}{n_X}+\frac{S_Y^2}{n_Y}}$
$X,Y$ any distribution	❌	Large	$(\bar{X}-\bar{Y})\pm z_{\alpha /2}\sqrt{\frac{S_X^2}{n_X}+\frac{S_Y^2}{n_Y}}$

Notes:

• $S_p=\sqrt{\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}}$ (pooled standard deviation)
• $\nu =\frac{{\left(\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}\right)}^2}{\frac{S_1^4}{n_1^2(n_1-1)}+\frac{S_2^4}{n_2^2(n_2-1)}}$ (Welch-Satterthwaite degrees of freedom)

Confidence Intervals for Difference in Proportions ($p_X-p_Y$)

Distribution	Sample sizes	$(1-\alpha )100\%$ Confidence interval
$X\sim \text{Binomial}(n_X,p_X)$ $Y\sim \text{Binomial}(n_Y,p_Y)$ (independent)	Large $n_X{\hat{p}}_X\ge 5$ $n_X(1-{\hat{p}}_X)\ge 5$ $n_Y{\hat{p}}_Y\ge 5$ $n_Y(1-{\hat{p}}_Y)\ge 5$	$\widehat{p_1}-\widehat{p_2}\pm z_{\alpha /2}\sqrt{\frac{\widehat{p_1}(1-\widehat{p_1})}{n_1}+\frac{\widehat{p_2}(1-\widehat{p_2})}{n_2}}$
$(X_i,Y_i)\sim \text{Multinomial}(1,[p_{++},p_{+-},p_{-+},p_{--}])$ (paired)	Large	$\hat{p}-z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Notes:

• ${\hat{p}}_1=\frac{X}{n_1}$, ${\hat{p}}_2=\frac{Y}{n_2}$ (sample proportions)
• For paired samples: ${\hat{p}}_d=\frac{\text{number of discordant pairs}}{n}$, $p_d$ is the proportion of discordant pairs