statisticalrethinkin..

84 4. LINEAR MODELSFIGURE 4.1. e Ptolemaic Universe, in which complex motion of the planetsin the night sky was explained by orbits within orbits, called epicycles.e model is incredibly wrong, yet makes quite good predictions.sense that once you learn to build and interpret linear regression models, you can quicklymove on to other types of regression which are less normal.4.1. Why normal distributions are normalSuppose you and a thousand of your closest friends line up on the half-way line of asoccer field (football pitch). Each of you has a coin in your hand. At the sound of the whistle,you begin flipping the coins. Each time a coin comes up heads, that person moves one steptowards the lehand goal. Each time a coin comes up tails, that person moves one steptowards the righthand goal. Each person flips the coin 16 times, follows the implied moves,and then stands still. Now we measure the distance of each person from the half-way line.Can you predict what proportion of the one-thousand people who are standing on the halfwayline? How about the proportion 5 yards le of the line?It’s hard to say where any individual person will end up, but you can say with great confidencewhat the collection of positions will be. e distances will be distributed in approximatelynormal, or Gaussian, fashion. is is true even though the underlying distributionis binomial. It does this because there are so many more possible ways to realize a sequenceof le-right steps that sums to zero. ere are slightly fewer ways to realize a sequence thatends up one step le or right of zero. And so on, with the number of possible sequencesdeclining in the characteristic bell curve of the normal distribution.4.1.1. Normal by addition. Let’s see this result, by simulating this experiment in R. To showthat there’s nothing special about the underlying coin flip, assume instead that each step isdifferent from all the others, a random distance between zero and one yard. us a coin is