Confidence Intervals for Experimental Research
A confidence interval has the general form
\[\text{estimate}\pm\text{critical value}\times\text{standard error}.\]Here $z_{\alpha/2}$ and $t_{\alpha/2,df}$ denote positive upper-tail critical values. Some texts write the same quantiles as $z_{1-\alpha/2}$ and $t_{1-\alpha/2,df}$.
One mean
With known population standard deviation,
\[\bar X\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}.\]With unknown population standard deviation,
\[\bar X\pm t_{\alpha/2,n-1}\frac{s}{\sqrt n}.\]Difference in means
For independent groups,
\[(\bar X_1-\bar X_2) \pm t^*SE(\bar X_1-\bar X_2),\]where the Welch standard error is
\[SE(\bar X_1-\bar X_2)= \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}.\]For paired observations, replace the two-sample difference by $\bar D$ and use $s_D/\sqrt n$.
One proportion
The usual large-sample Wald interval is
\[\hat p\pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}.\]When counts are small or $\hat p$ is near $0$ or $1$, score or exact intervals behave better than this approximation.
Interpretation
A $100(1-\alpha)\%$ confidence procedure captures the fixed parameter in that proportion of repeated samples or assignments. After the data are observed, the computed endpoints are fixed; the frequentist confidence statement describes the procedure.
For a two-sided test using the same assumptions and standard error, a null value outside the $100(1-\alpha)\%$ interval is rejected at level $\alpha$.
Experimental treatment effects
For a randomized experiment, the same template gives
\[\hat\tau\pm z_{\alpha/2}SE(\hat\tau),\]using the design-based standard error in Neyman Repeated-Sampling Inference.