The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.
Since the development of carbon nanotubes (CNTs) in 1991, they have received much attention with improved mechanical, thermal, and electrical properties of their composites compared to common polymer composites. The CNTs are currently used to increase the modulus of common thermoplastics and thermosets, including urethanes and epoxies. The CNTs are difficult to disperse within any media because of limited chemical reactivity and potential agglomeration in their ''as grown'' state. This study evaluated the effect of incorporating bundled and unbundled CNTs at different concentrations into Polyurethane/CNT/woven fiber reinforced composites. Optical microscopy and atomic force microscopy (AFM) characterized the dispersion of CNTs within the polymer matrix in injection molded CNT/polyurethane composites. Polyurethane/ CNT/woven fiber reinforced composite plaques were prepared and then characterized by mechanical compression testing. Optical microscopy and AFM qualitatively determined a decreased agglomerate size resulting in improved mechanical properties.Results of this study show significant differences in yield stress, stress at failure, and modulus of elasticity within the various treatments. No significant differences were found for yield strain, strain at failure, and toughness. However, the conservativeness of the statistical model warrants further investigation for strain at failure and toughness with possible interaction effects of CNT concentration for each composite.
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