2.4 Confidence intervals

In addition to producing point estimates, one of the fundamental objectives of a survey is to quantify the uncertainty associated with those estimates. In this context, confidence intervals make it possible to construct a range of plausible values for the population parameter of interest, explicitly incorporating the variability introduced by the sample design. The \((1-\alpha)100\%\) confidence interval for the population total \(Y\) is calculated as:

\[ \hat{\theta} \pm t_{1-\alpha/2, df} \times \sqrt{\widehat{Var}(\hat{\theta})} \]

where \(\hat{\theta}\) corresponds to the sampling estimator of the population parameter \(\theta\); \(\widehat{Var}(\hat{\theta})\) represents the variance estimator under the complex survey design; and \(t_{1-\alpha/2, df}\) is the quantile of Student’s \(t\) distribution with \(df\) degrees of freedom. In complex surveys, the degrees of freedom are usually related to the number of primary sampling units (UPM) and the strata considered in the design. A common approximation is to calculate them as:

\[ df = m - H \]

where \(m\) represents the total number of UPM observed in the sample and \(H\) the number of strata. This approximation reflects the effective amount of independent information available to estimate sampling variability. As the degrees of freedom increase, Student’s \(t\) distribution converges to the standard normal distribution. This property explains why normal approximations are often used to report confidence intervals, especially in large-scale household surveys. However, when the number of clusters is small or there are strata with few UPM, the normal approximation may underestimate uncertainty and produce excessively narrow intervals.