Transcript Chapter 18
Chapter 18 Probability Models Chapter 18 1 Probability Rules Four simple, logical rules which apply to how probabilities relate to each other and to real events. Review the rules at the beginning of Chapter 18. Review the examples presented in Chapter 18!! Chapter 18 2 Probability Rule A Any probability is a number between 0 and 1. A probability can be interpreted as the proportion of times that a certain event can be expected to occur. If the probability of an event is more than 1, then it will occur more than 100% of the time (Impossible!). Chapter 18 3 Probability Rule B All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one. If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent. Chapter 18 4 Probability Rule C The probability that an event does not occur is 1 minus the probability that the event does occur. As a jury member, you assess the probability that the defendant is guilty to be 0.80. Thus you must also believe the probability the defendant is not guilty is 0.20 in order to be coherent (consistent with yourself). If the probability that a flight will be on time is .70, then the probability it will be late is .30. Chapter 18 5 Probability Rule D If two events have no outcomes in common, they are said to be mutually exclusive. The probability that one or the other of two mutually exclusive events occurs is the sum of their individual probabilities. Age of woman at first child birth – under 20: 25% 24 or younger: 58% – 20-24: 33% – 25+: ? Rule C: 42% } Chapter 18 6 Avoid Being Inconsistent If the ways in which one event can occur are a subset of those in which another event can occur, then the probability of the first event cannot be higher than the probability of the one for which it is a subset. Suppose you see an elderly couple and you think the probability that they are married is 80%. Suppose you think the probability that the elderly couple is married with children is 95%. These two personal probabilities are not coherent. Why? Chapter 18 7 Avoid Being Inconsistent All Couples Unmarried Couples Married Couples Married with Children Probability of married with children must not be greater than the probability that the couple is married. Chapter 18 8 Sampling Distribution Tells what values a statistic (calculated sample value) takes and how often it takes those values in repeated sampling. Assigns probabilities to the values a statistic can take. These probabilities must satisfy Rules A-D. Probabilities are often assigned to intervals of outcomes by using areas under density curves. – often this density curve is a normal curve can use “68-95-99.7 rule” or get probabilities from Table B – sample proportions ( p̂ ’s) follow a normal curve Chapter 18 9 Sampling Distribution for Proportion Who Voted World Almanac and Book of Facts (1995), Famighetti, R. editor, Mahwah, N.J.: Funk and Wagnalls 56% of registered voters actually voted in the 1992 presidential election. In a random sample of 1600 voters, the proportion who claimed to have voted was .58. Such sample proportions ( p̂ ’s) from repeated sampling would have a normal distribution with a mean of .56 and a standard deviation of .012 . What is the probability of observing a sample proportion as large or larger than .58? Chapter 18 10 Sampling Distribution for Proportion Who Voted If we convert the observed value of .58 to a standardized score, we get standardized score = (observed value - mean) / (std dev) = (.58 - .56) / .012 = 1.67 From Table B, this is the 95.54 percentile, so the probability of observing a value as small as .58 is .9554. By Rule C (or B), the probability of observing a value as large or larger than .58 is 1-.9554 = .0446. Chapter 18 11 Key Concepts Rules for probability Avoid Inconsistencies Sampling Distributions Chapter 18 12