On the Asymptotic Theory of Fixed-Width Sequential Confidence Intervals for the Mean

Let x1, x2, • • • be a sequence of independent observations from some population. We want to find a confidence interval of prescribed width 2d and prescribed coverage probability a for the unknown mean µ of the population. If the variance σ2 of the population is known, and if d is small compared to σ 2, this can be done as follows. For any n ⪴ 1 define \({x_n} = {n^{ - 1}}\sum\nolimits_1^n {Xi,{I_n}} = \left[ {{x_n} - d,{x_n} + d} \right],\) and choose a to satisfy \(\left( {2\pi } \right){ - ^{{1 \over 2}}}{\int_{ - a}^a e ^{ - u2/2}}du = a\)