We take a Student process that is based on independent copies of a random variable X and has trajectories in the function space D[0,1]. As a consequence of a functional central limit theorem for this process, with X in the domain of attraction of the normal law, we consider convergence in distribution of several functionals of this process and derive respective asymptotic confidence intervals for the mean of X. We explore the expected lengths and finite-sample coverage probabilities of these confidence intervals and the one obtained from the asymptotic normality of the Student t-statistic, thus concluding some alternatives to the latter confidence interval that are shorter and/or have at least as high coverage probabilities.

Thesis (MA)- University of Stellenbosch, 2004. / ENGLISH ABSTRACT: The aim of the present study was to investigate the psychometric properties of the Family Sense of Coherence Questionnaire (FSoC) for use among Northern Sothos. Antonovsky and Sourani developed the FSoC-questionnaire to measure the sense of coherence of families. The family sense of coherence is based on the individual sense of coherence construct, which was developed by Antonovsky. According to Antonovsky, families with a strong sense of coherence will function better than families with a weak sense of coherence. Therefore family functioning was used as criteria to validate the external validity of the FSoC-questionnaire. Olson, Russell and Sprenkle developed the Family Attachment and Changeability Index (FACI-8) to measure family functioning according to the Circumplex model. For the purpose of this study the two questionnaires were translated to Sepedi (the language of the Northern Sotho). The reliability of the two questionnaires was assessed by means of Cronbach-alpha coefficients. The construct validity of the two questionnaires was assessed by means of JOreskog's Goodness of Fit Index (GFI). The statistical analysis was done with Statistica 6 software. The results drawn from the present study indicates that the FSoC-questionnaire is not a reliable and valid instrument to measure families' sense of coherence among Northern Sothos. Significant high correlations with the FACI-8 however, indicated that the FSoC might be considered for use among Northern Sothos, if only the total scores are taken into consideration. The FACI-8 however, proved to be a reliable and valid instrument to measure family functioning according to the Circumplex model among Northern Sothos. The conclusions drawn from the present study are that the FACI-8 is a suitable instrument for use among the Northern Sothos, whereas the FSoC-questionnaire is not. / AFRIKAANSE OPSOMMING: Die doel van die huidige ondersoek was om die psigometriese eienskappe van die Gesinskoherensie-vraelys (FSoC) wat deur Antonovsky en Sourani ontwikkel was, vir gebruik onder Noord-Sothos te bepaal. Om hierdie doeleindes te bereik, was die Gesinsfunksioneringsvraelys (FACI-8) wat deur McCubbin, Thompson en Elver ontwikkel was, as 'n kriterium gebruik om die eksterne geldigheid van die FSoC-vraelys te valideer. Die FACI-8 meet gesinsfunksionering volgens die Kontekstuele-model van gesinsfunksionering, wat deur Olson, Russell en Sprenkle ontwikkel was. Volgens Antonovsky sal gesinne wat oor 'n sterk ontwikkelde koherensiesin beskik, beter funksioneer as gesinne wat oor 'n swak ontwikkelde koherensiesin beskik. Gevolglik was gesinsfunksionering as 'n maatstaf van gesinskoherensie gebruik. Die vraelyste moes eers na Sepedi (die taal van die Noorcl-Sotho) vertaal word, alvorens die vraelyste afgeneem kon word. Die betroubaarheid van die vraelyste was deur middel van Cronbach-alfa koêffisiênte bereken. Die konstrukgeldigheid van die vraelyste was deur middel van JOreskog se Passingstoets bepaal. Die statistiese verwerkings was deur middel van die Statistica 6 rekenaarprogram behartig. Die resultate van die huidige ondersoek dui daarop dat die FSoC-vraelys met omsigtigheid benader moet word vir gebruik onder Noord-Sothos. Die FSoCvraelys het nie bevredigende betroubaarheids- en geldigheidsindekse gelewer nie. Die FSoC-vraelys het wel eksterne geldigheid getoon, wanneer dit met die FACI-8 gevalideer was. Die FACI-8 het betroubare en geldige metings van gesinsfunksionering volgens die Kontekstuele-model onder Noord-Sothos in die huidige ondersoek gelewer. Die gevolgtrekkings wat op grond van die huidige ondersoek gemaak word, is dat die FSoC-vraelys nie bruikbaar is onder Noorcl-Sothos nie, terwyl die FACI-8 wel gebruik kan word om gesinsfunksionering onder Noord-Sothos te meet.