<P> which is also a 50% confidence procedure . Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every θ 1 ≠ θ (\ displaystyle \ theta _ (1) \ neq \ theta), the probability that the first procedure contains θ 1 (\ displaystyle \ theta _ (1)) is less than or equal to the probability that the second procedure contains θ 1 (\ displaystyle \ theta _ (1)). The average width of the intervals from the first procedure is less than that of the second . Hence, the first procedure is preferred under classical confidence interval theory . </P> <P> However, when X 1 − X 2 ≥ 1 / 2 (\ displaystyle X_ (1) - X_ (2) \ geq 1 / 2), intervals from the first procedure are guaranteed to contain the true value θ (\ displaystyle \ theta): Therefore, the nominal 50% confidence coefficient is unrelated to the uncertainty we should have that a specific interval contains the true value . The second procedure does not have this property . </P> <P> Moreover, when the first procedure generates a very short interval, this indicates that X 1, X 2 (\ displaystyle X_ (1), X_ (2)) are very close together and hence only offer the information in a single data point . Yet the first interval will exclude almost all reasonable values of the parameter due to its short width . The second procedure does not have this property . </P> <P> The two counter-intuitive properties of the first procedure--100% coverage when X 1, X 2 (\ displaystyle X_ (1), X_ (2)) are far apart and almost 0% coverage when X 1, X 2 (\ displaystyle X_ (1), X_ (2)) are close together--balance out to yield 50% coverage on average . However, despite the first procedure being optimal, its intervals offer neither an assessment of the precision of the estimate nor an assessment of the uncertainty one should have that the interval contains the true value . </P>

What interval about the mean includes 95 of the data in the survey