By Peter Freeman
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The goal of this poster is to demonstrate that the construction of confidence intervals for a population parameter is a remarkably straightforward mathematical exercise, much more straightforward than typical mathematical statistics textbooks would lead one to believe, what with their pivotal quantities and their reliance on statistical tables. The following has been implemented at an R1 research university in the calculus-based mathematical statistics courses for statistics majors, which approximately 160 students take per year. Assume that we are given n independent and identically distributed data sampled from some distribution with parameter theta, and that we use these data to form a statistic Y whose observed value is y_obs. Let q be a quantile (e.g., 0.025 or 0.975 for a two-sided 95% confidence interval). An interval bound theta_b is simply the solution of the equation F_Y(y_obs | theta_b) - q = 0, where F_Y is the cumulative distribution function for the statistic Y. As we show through a series of examples, the key to this approach is that we can use it in any situation where we can write down the function F_Y: if we cannot solve for theta_b by hand, then we can easily solve for it using a numerical root-finding function, such as R's uniroot(). Given its generality, this approach to confidence interval construction is the only one that students ever need to learn!