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Review • ordinary random variation. • statistical significance When the difference between the results for the two treatments is so large that it would be rare to see such a difference by ordinary random variation, we say that the results are statistically significant. Other Random Sampling Designs (other than Simple Random Sampling) • Cluster Random Sampling • Stratified Random Sampling Retrospective Observational Studies (backward looking) Prospective Observational Studies (forward looking) Multifactor Experiments Matched Pairs Designs Blocking Designs Probability • Law of Large Numbers As the number of trials increases, the proportion of occurrences of any given outcome approaches a particular number “in the long run”. (We need to assume that the outcome of any one trial does not depend on the outcome of any other trial.) • Probability With a randomized experiment or a random sample or other random phenomenon (such as a simulation), the probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations. • Independent Trials Different trials of a random phenomenon are independent if the outcome of any one trial is not affected by the outcome of any other trial. How to Find Probabilities? • In practice, the sample proportion estimates the actual probability. • Sample Space For a random phenomenon, the sample space is the set of all possible outcomes. • Event An event is a subset of the sample space. Probabilities for a Sample Space • The probability of each individual outcome is between 0 and 1. • The total of all the indicidual probabilities equals 1. Probability of an Event • The probability of an event A, denoted by P(A), is obtained by adding the probabilities of the individual outcomes in the event. • When all the possible outcomes are equally likely, P(A) = number of outcomes in event A number of outcomes in the sample space In practice, equally likely outcomes are unusual.