Which distribution do confidence and prediction intervals follow when analyzing the relationship between two quantitative variables?

Confidence and prediction intervals are particularly relevant when assessing the relationship between two quantitative variables, typically within the context of regression analysis. The t distribution is significant in these cases, especially when dealing with small sample sizes or estimating means.

When constructing confidence intervals for the mean response in regression analysis, the t distribution is used because it accounts for the added uncertainty associated with estimating the population standard deviation from a sample. This is critical because the sample size could influence the width of the interval—the smaller the sample, the greater the uncertainty, which is adequately addressed by the t distribution.

For prediction intervals, which estimate where a new observation will fall based on the model, the t distribution is also applicable. This is due to the variability in regression predictions, as it reflects both the error in estimating the mean response and additional variability in the individual observations.

In contrast, other distributions like the normal distribution are typically used when the sample size is large enough for the Central Limit Theorem to apply, while the binomial distribution pertains to counting successes in a fixed number of trials, and the chi-square distribution is primarily used for distributions of variance and categorical data analysis. This understanding of the t distribution's role in creating intervals is crucial for making accurate statistical inferences in regression settings.