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STAXXXX_Notes_ExamTwo - Statistics Exam Two Notes Events...

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Statistics Exam Two Notes Events, Sample Spaces and Probability This section introduces the basic concept of the probability of an event. Three different methods for finding probability values will be presented. The most important objective of this section is to learn how to interpret probability values. The Rare Event Rule for Inferential Statistics If under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. An example of the Rare Event Rule would be as follows: Say that you assume that a college graduate will have a starting salary of 75k or more, but a random survey of 32 recent graduates indicates that the starting salaries were around 35k. If your assumption is actually true the probability that a sample of 32 recent grads would have an average salary of only 35k would be extremely small, so we must conclude your assumption was wrong (in actuality, we need to know what the standard deviation is before we could decide how probable the above sample results would be, but we will get to that later). Before we get to probability, there are some terms we need to discuss: An experiment is an act of observation that leads to a single outcome that cannot be predicted with certainty. An event is a specific collection of sample points. For example: Event A : Observe an even number. A sample point (or simple event) is the most basic outcome of an experiment. For example, getting a four on a single roll of a die. Sample point ~ outcome
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A sample space (denoted S) is the collection of all possible outcomes of an experiment. For example: a roll of a single die: S: {1, 2, 3, 4, 5, 6} Example 28: List the different possible families that can occur when a couple has three children… Example 29: List the possible outcomes for three flips of a fair coin… Solution: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Here are some commonly used notational conventions: P - denotes a probability. A, B , and C - denote specific events. P ( A ) - denotes the probability of event A occurring. The probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A. In other words: ( ) n P A N = , n = # of times you observed event A (number of ways A can happen), N = number of observations (Number of total possibilities). You might have noticed that the statements in the parenthesis in the above definition seem to define a second definition of probability. That is because there are two ways to think about probability. I have those two ways defined below: Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P ( A ) is estimated as follows: Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then number of ways A can occur ( )
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STAXXXX_Notes_ExamTwo - Statistics Exam Two Notes Events...

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