2. Excursions in Modern Mathematics, 7e: 16.6 - 2Copyright © 2010 Pearson Education, Inc. 16 Mathematics of Normal Distributions 16.1Approximately Normal Distributions of Data 16.2Normal Curves and Normal Distributions 16.3Standardizing Normal Data 16.4The 68-95-99.7 Rule 16.5Normal Curves as Models of Real- Life Data Sets 16.6Distribution of Random Events 16.7Statistical Inference

3. Excursions in Modern Mathematics, 7e: 16.6 - 3Copyright © 2010 Pearson Education, Inc. We are now ready to take up another important aspect of normal curves–their connection with random events and, through that, their critical role in margins of error of public opinion polls. Normal Curves and Random Events

4. Excursions in Modern Mathematics, 7e: 16.6 - 4Copyright © 2010 Pearson Education, Inc. John Kerrich while he was a prisoner of war during World War II, tossed a coin 10,000 times and kept records of the number of heads in groups of 100 tosses. With modern technology, we can repeat Kerrich’s experiment and take it much further. Practically any computer can imitate the tossing of a coin by means of a random- number generator. If we use this technique, we can “toss coins” in mind-boggling numbers–millions of times if we so choose. Example 16.8Coin-Tossing Experiments

5. Excursions in Modern Mathematics, 7e: 16.6 - 5Copyright © 2010 Pearson Education, Inc. We will start modestly. We will toss our make- believe coin 100 times and count the number of heads, which we will denote by X. Before we do that, let’s say a few words about X. Since we cannot predict ahead of time its exact value–we are tempted to think that it should be 50, but, in principle, it could be anything from 0 to 100–we call X a random variable. Example 16.8Coin-Tossing Experiments

6. Excursions in Modern Mathematics, 7e: 16.6 - 6Copyright © 2010 Pearson Education, Inc. The possible values of the random variable X are governed by the laws of probability: Some values of X are extremely unlikely (X = 0, X = 100) and others are much more likely (X = 50) although the likelihood of X = 50 is not as great as one would think. It also seems reasonable (assuming that the coin is fair and heads and tails are equally likely) that the likelihood of X = 49 should be the same as the likelihood X = 51, the likelihood of X = 48 should be the same as the likelihood of X = 52 and so on. Example 16.8Coin-Tossing Experiments

7. Excursions in Modern Mathematics, 7e: 16.6 - 7Copyright © 2010 Pearson Education, Inc. While all of the preceding statements are true, we still don’t have a clue as to what is going to happen when we toss the coin 100 times. One way to get a sense of the probabilities of the different values of X is to repeat the experiment many times and check the frequencies of the various outcomes. Finally, we are ready to do some experimenting! Example 16.8Coin-Tossing Experiments

8. Excursions in Modern Mathematics, 7e: 16.6 - 8Copyright © 2010 Pearson Education, Inc. Our first trial results in 46 heads out of 100 tosses (X = 46). The first 10 trials give X = 46, 49, 51, 53, 49, 52, 47, 46, 53, 49. Example 16.8Coin-Tossing Experiments

9. Excursions in Modern Mathematics, 7e: 16.6 - 9Copyright © 2010 Pearson Education, Inc. Continuing this way, we collect data for the values of X in 100 trials. Example 16.8Coin-Tossing Experiments

10. Excursions in Modern Mathematics, 7e: 16.6 - 10Copyright © 2010 Pearson Education, Inc. Then, we collect data for the values of X in 500 trials. Example 16.8Coin-Tossing Experiments

11. Excursions in Modern Mathematics, 7e: 16.6 - 11Copyright © 2010 Pearson Education, Inc. Then, we collect data for the values of X in 1000 trials. Example 16.8Coin-Tossing Experiments

12. Excursions in Modern Mathematics, 7e: 16.6 - 12Copyright © 2010 Pearson Education, Inc. Then, we collect data for the values of X in 5000 trials. Example 16.8Coin-Tossing Experiments

13. Excursions in Modern Mathematics, 7e: 16.6 - 13Copyright © 2010 Pearson Education, Inc. Then, we collect data for the values of X in 10,000 trials. Example 16.8Coin-Tossing Experiments

14. Excursions in Modern Mathematics, 7e: 16.6 - 14Copyright © 2010 Pearson Education, Inc. The bar graphs paint a pretty clear picture of what happens: As the number of trials increases, the distribution of the data becomes more and more bell shaped. At the end, we have data from 10,000 trials, and the bar graph gives an almost perfect normal distribution! What would happen if someone else decided to repeat what we did–toss an honest coin (be it by hand or by computer) 100 times, count the number of heads, and repeat this experiment a few times? Example 16.8Coin-Tossing Experiments

15. Excursions in Modern Mathematics, 7e: 16.6 - 15Copyright © 2010 Pearson Education, Inc. The first 10 trials are likely to produce results very different from ours, but as the number of trials increases, their results and our results will begin to look more and more alike. After 10,000 trials, their bar graph will be almost identical to our bar graph. In a sense, this says that doing the experiments a second time is a total waste of time–in fact, it was even a waste the first time! The outline of the final distribution could have been predicted without ever tossing a coin! Example 16.8Coin-Tossing Experiments

16. Excursions in Modern Mathematics, 7e: 16.6 - 16Copyright © 2010 Pearson Education, Inc. Knowing that the random variable X has an approximately normal distribution is, as we have seen, quite useful. The clincher would be to find out the values of the mean  and the standard deviation  of this distribution. Looking at the bar graphs, we can pretty much see where the mean is–right at 50. This is not surprising, since the axis of symmetry of the distribution has to pass through 50 as a simple consequence of the fact that the coin is honest. For now, let’s accept  = 5. Example 16.8Coin-Tossing Experiments

17. Excursions in Modern Mathematics, 7e: 16.6 - 17Copyright © 2010 Pearson Education, Inc. Let’s summarize what we now know. An honest coin is tossed 100 times. The number of heads in the 100 tosses is a random variable, which we call X. If we repeat this experiment a large number of times (say N), the random variable X will have an approximately normal distribution with mean  = 50 and standard deviation  = 5, and the larger the value of N is, the better this approximation will be. Example 16.8Coin-Tossing Experiments

18. Excursions in Modern Mathematics, 7e: 16.6 - 18Copyright © 2010 Pearson Education, Inc. The real significance of these facts is that they are true not because we took the trouble to toss a coin a million times. Even if we did not toss a coin at all, all of these statements would still be true. For a sufficiently large number of repetitions of the experiment of tossing an honest coin 100 times, the number of heads X is a random variable that has an approximately normal distribution with center  = 50 heads and standard deviation  = 5 heads. This is a mathematical, rather than an experimental, fact. Example 16.8Coin-Tossing Experiments