Let P D .X; / be a lattice, and suppose that for all x; y 2 X, we have x ^ y D x _ y. Prove that P contains at most one element

Chapter 3 Producing Data – MAT 221 3.1 & 3.2 - Sources of Data & Design of Experiments Anecdotal Data – unusual cases that we draw conclusions from past experiences - May not me representative of any larger group of cases Available Data – past data we produced that may help us Population – the entire group of individuals we are studying Parameter – the part of the population we are studying and have data for Statistic – a number describing a characteristic of a sample Experimental units – the individuals in an experiment - Called Subjects if they are human - Treatment or factor: the “something” we do to a subject that’s response gets measured Observational Study: Simply observing and recording data of individuals without influencing responses - Cannot establish cause and effect relationships Experimental Study – Deliberately giving individuals a sort of treatment and recording their responses Control – a situation where no treatment is given; serves as reference mark/basis Placebo – a fake treatment to test that the results are from the actual treatment and not the subjects belief that they are being treated Ronald Fisher (1890 – 1962) – randomized comparative experiments; fertilizer Principles of Experimental Design: 1. Control the effects of lurking variables 2. Randomize 3. Replicate treatment on enough subjects to reduce chance of variation in results Biased – systematically favoring certain outcomes - Random assignment is the best way to avoid B