Confidence Intervals for the Mean: To Bootstrap or Not to Bootstrap.
Maria E. Calzada, Holly Gardner · Mathematics and computer education · 2011
ABSTRACT The results of a simulation conducted by a research team involving undergraduate and high school students indicate that when data is symmetric the student's t confidence interval for a mean is superior to the studied nonparametric bootstrap confidence intervals. When data is skewed and for sample sizes n ≥ 10, the results suggest the superiority of the bootstrap t confidence interval, underscoring the utility of this procedure. We recommend introduction of bootstrapping at the undergraduate level for students who already have an understanding of confidence intervals. Our simulations may be replicated by readers and their students to aid in their understanding of the procedures and to confirm our results. (ProQuest: ... denotes formulae omitted.) INTRODUCTION In the summer of 2009 the authors participated in Loyola University's SCORE (Summer Collaborative Outreach and Research Experience) program, funded by the Louisiana Board of Regents1. SCORE'S ultimate goal is to increase the number of Louisiana students pursuing Science Technology Engineering and Mathematics (STEM) careers. To that end, SCORE creates research groups consisting of a faculty mentor, undergraduate students, high school students and high school teachers. It is hoped that the experiences in these teams will encourage the high school and undergraduate students to pursue science careers. The authors participated in a research group that involved two high school students, one undergraduate (second author) and a faculty member (first author)2. It should be noted that the research team was small and the program ran in the summer. Therefore, the team may not be representative of possible participants during a regular semester. It was the responsibility of the faculty mentor to identify a suitable research question to be pursued by the group. The topic selected was a Monte Carlo simulation study of student's t and non-parametric bootstrap confidence intervals for estimating a mean. This topic was selected since the bootstrap, a primarily computational method, has received a great deal of attention in statistical research literature, but only recently has begun to appear in materials used at the undergraduate level. In fact, a cursory review of introductory statistics books at www.amazon.com reveals that most undergraduate statistics texts omit the bootstrap idea altogether. The texts which include topics in bootstrapping and resampling techniques do so in optional sections or extended chapters, often skipped by undergraduate instructors. Bootstrap methods have the added appeal that they do not require an extensive mathematical background and are easily programmable. The participants in our research team used student versions of Matlab 20093 to write and run the bootstrap simulations, which was an added benefit of participating in the SCORE program. In the next section we briefly review the student's t confidence interval. Then the bootstrap idea and the three different bootstrap confidence intervals utilized are discussed, including the simulation plan pursued by the research team. Finally, we present the results of our simulations and give recommendations for practical application. THE STUDENT'S / CONFIDENCE INTERVAL Constructing a (1- Qt)1 00% student's t confidence interval for an unknown mean µ is routine for undergraduate statistics students and practitioners alike. The well-known formula for constructing such a confidence interval is given by where 3c and S are the sample mean and standard deviation, respectively, n is the sample size, and ... is the appropriate critical value from the student's t distribution with n-1 degrees of freedom. The development of this confidence interval relies on the normality assumption, that is, the data used to produce this confidence interval is assumed to come from a distribution which is approximately normal (no statistical procedure can overcome bad data). …