Confidence intervals and one-sample-based hypothesis testing for Cronbach’s $$\alpha $$ coefficient for interval-valued data

Abstract The definition of the well-known Cronbach $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> coefficient has been recently extended to measure the internal consistency of interval-valued scale-based items in questionnaires. Although some of its main properties have already been studied, no inferential technique has yet been proposed to analyze its population value. In this paper, large sample theory and bootstrap procedures are used to obtain one-sample-based confidence intervals and perform hypothesis testing for the population Cronbach $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> coefficient for interval-valued data. The asymptotic correctness and consistency of the suggested methods are theoretically derived by analyzing the limit distribution of the sample and bootstrap analogue estimators of such extended reliability index. In addition, some simulation studies are conducted to empirically investigate the performance of the proposed confidence intervals and hypothesis tests, which is also illustrated through a real-life example.