Given a random sample of size n\documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} with mean X¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{X} $$\end{document} and standard deviation s\documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document} from a symmetric distribution F(x;μ,σ)=F0((x-μ)/σ)\documentclass[12pt]{minimal}
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\begin{document}$$F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) $$\end{document} with F0\documentclass[12pt]{minimal}
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\begin{document}$$F_0$$\end{document} known, and X∼F(x;μ,σ)\documentclass[12pt]{minimal}
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\begin{document}$$X \sim F(x;\; \mu , \sigma )$$\end{document} independent of the sample, we show how to construct an expansion an′=∑i=0∞cin-i\documentclass[12pt]{minimal}
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\begin{document}$$ a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} $$\end{document} such that X¯-san′<X<X¯+san′\documentclass[12pt]{minimal}
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\begin{document}$$\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } $$\end{document} with a given probability β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document}. The practical value of this result is illustrated by simulation and using a real data set.
