A Confidence Interval That Is Consistent and More Informative Than a Sequential Equivalence Test for a Normal Mean

Abstract

This note shows how to construct a 1 − β level symmetric confidence interval for a normal mean θ that is consistent with and more informative than a size β sequential equivalence test of H ₀ :|θ| > δ against the alternative H_a :|θ | ≤ δ, where δ>0 is a prespecified limit of equivalence. If the null hypothesis H ₀ is rejected and so equivalence can be declared by the test, then the confidence interval is contained in the interval [− δ, δ]. If the null hypothesis H₀ is not rejected and so equivalence cannot be declared by the test, then the confidence interval contains the interval [− δ, δ]. Therefore the test and the confidence interval are consistent in rejection or non-rejection of H ₀. But the confidence interval provides extra information on the magnitude of θ, and is therefore more informative than the test.

Keywords:

• Confidence intervals
• Equivalence
• Ordering of observations
• Sequential tests

Acknowledgment

The author would like to thank a referee for useful comments.

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