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CCJS200 Chapter 6 Probability Probability Distributions and an Introduction to Hypothesis Testing Probability Probability of an event Event A is defined as the number of times A can occur over the total number of events or trials P A of times A can occur Total of possible events or trials Bounding rule of probabilities is that the probability of any event can never be less than zero not greater than 1 0 0 P A 1 0 Complement of an event the probability of an event not occurring Odds equal to the ratio of the probability that the event will occur to the probability that it will not occur Odds P A 1 P A Mutually Exclusive Events events that cannot occur at the same time there is no intersection of mutually exclusive events so their joint probability is equal to zero Restricted Addition Rule of Probabilities if two events are mutually exclusive the probability of event A occurring or event B occurring is equal to the sum of their separate probabilities P A or B P A P B Non mutually exclusive events events that can occur simultaneously The joint probability of non mutually exclusive events is therefore greater than zero General addition rule of probability If two events are not mutually exclusive the probability of event A occurring or event B occurring is equal to the sum of their separate probabilities minus their joint probability P A or B P A P B P A and B Joint Probability simultaneous occurrence of two events Restricted multiplication rule of probabilities if two events are independent of each other the probability of event A occurring and event B occurring is equal to the product of their separate probabilities P A and B P A x P B General Multiplication rule of probabilities if two events are not independent of each other the probability of event A occurring and event B occurring is equal to the product of the unconditional probability of event A and the conditional probability of event B given A P A and B P A x P B given A Independent Events when the unconditional probability of A is equal to the probability of A is equal to the conditional probability of A given B When two events are independent knowledge of one event does not help predict the probability of the other event occurring P A P A given B All probability rules listed on page 221 Probability Distributions Probability distribution a theoretical distribution of what we should observe Binomial Distribution The probability distribution based on a Bernouilli Null hypothesis implication that nothing is going on Research Alternative Hypothesis the hypothesis about an alternative process outcome Directional Alternative Hypothesis state a specific direction in our alternative with respect to the null hypothesis less than that assumed in the null Non directional Alternative Hypothesis state that the expected outcome is different from that assumed in the null hypothesis not that it is greater or less just different In the process of hypothesis testing only the null hypothesis is tested Type 1 error occurs when we reject a null hypothesis that is really true This level of risk is called Alpha level or level of significance Type 2 error occurs when we fail to reject a null hypothesis that is really false Critical Region defines the entire class of outcomes that will lead us to reject the null hypothesis If the event we observe falls into the critical region our decision will be to reject the null hypothesis Normal Distribution probability distribution for continuous events and look like a smooth curve unlike the probability distribution which looks like a series of steps Sampling Distribution the connection between sample information and population characteristics involves a sampling distribution Standard error Steps of a Hypothesis Test Make an assumption about the null and alternative hypothesis Determine what probability distribution you will use to find the probability of observing the event recorded in your sample data Define what you mean by a very unlikely event by selecting a level of Calculate the probability of observing your sample data under the null Make a decision about the null hypothesis reject or fail to reject and significance hypothesis interpret your results We will have more faith in our sample mean as an accurate or precise estimate of the population mean if we have a large sample size There is a certain amount of error in using a known sample mean to estimate an unknown population mean since there is variation in the means from sample to sample Standard Error Standard Error of the mean reflects the amount of error due to sampling variation the standard deviation of the sample distribution Central Limit Theorem if an infinite number of random samples of size n are drawn from any population with mean and standard deviation then as the sample size becomes large the sampling distribution of sample means will approach normality with mean and standard deviation even if the population distribution is not normally distributed

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