Many studies collect longitudinal data but do not use them efficiently. We propose a technique for reducing a set of longitudinal data into four common summary statistics together with time. These statistics can be used in place of the raw data in further analyses, for example in a regression analysis. Instead of arbitrarily choosing the summary statistics, we derive them mathematically, thus justifying their use. This simplifies the subsequent analyses, and also improves the interpretation of the results. An example from the Framingham Heart Study illustrates the procedure.
This article describes a one semester hour mathematics senior seminar cours e at Olivet Nazarene University. The course has four main requirements: a resum e packet, a short article presentation, a poster, and a proj ect presentation. We describe how these four assignments fit together to provide a development of the mathematical maturity of our students. Additionally we discuss the strengths and weaknesses of the course.
What is the likelihood P(n, T) that at least two people in a gathering of n people are born within a given time T of each other? In particular, for T = 24 hours what is the smallest n for which P(n, T) is at least 50%? The well-known classic version of the birthday problem asks for the smallest number of people needed to give a better than 50% chance of at least one birthday match with the assumptions that the birthdays are independently selected from a discrete uniform distribution over 365 days. Using the calculus tool of the limit, we refine two characterisations for P(n, T) and show that they give consistent results with each other and with the classic birthday problem as well. At first thought P(n, 24 hours) should answer the classic birthday question. Yet consider this experiment: suppose Andrea was born at noon on June 1, then potential matching birth times (for the other n - 1 people) with her birth time extend from noon on May 31 to noon on June 2, a period of 48 hours! What this period suggests is that P(n, 24hours) exceeds the classic answer. But, by how much? Read on.
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