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UT Dallas CS 6313 - Chapter_9_4_5-4_9

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Slide 1OverviewOverviewOverviewZ-testZ-testT-testProbability and Statistics in Computer Science and Software Engineering Chapter 9: Statistical Inference1OverviewLast time, we introduced hypothesis testingWe saw that testing statistical hypotheses involved two hypotheses: The Null and AlternativeThe Null hypothesis is stated as an equality for a test statisticWe saw that we can have two-sided alternatives, and one-sided, left/right-tail alternatives, depending on how the alternative hypothesis is statedWe introduced the concepts of Type I and Type II errorsType I - we reject the true Null hypothesisType II – We accept a false Null hypothesisType I is usually considered more dangerous and undesired2OverviewWe saw that hypothesis testing essentially came down to three steps:1. Form the test statistic, a quantity to be computed from the data that has a known distribution if the Null hypothesis is true2. We create acceptance and rejection regions, based upon the level of the test (). This is similar to the confidence intervals we were developing earlier3. Calculate the test statistic based on some sample data, and see what region it falls intoThe significance of the test was the probability of a Type I error - The power of the test was the probability that we reject the Null hypothesis given that the alternative is true – it is the probability that we avoid a Type II error•23OverviewIn this lecture, we will explore some common testsThe Z-test is based upon the standard normal distributionWe will see how to apply this to the means and proportions estimates we developed earlier in the chapterIn particular, we will see how to develop hypotheses about means and proportions and then develop tests of desired powersAs before, we will also see what to do if the standard deviation is unknown …This will lead to T-tests, which rely upon Student’s t-distribution4Z-testIf we know that the estimator is normally distributed (or approximately normal), and we know and (or we have a good approximation), then we can form the following test statistic which will have a standard normal distribution:Note that if the estimator is unbiased, Here would be the parameter value that is assumed by the Null hypothesis•25Z-testFor a sample mean, we know , where is the variance of the population. We assume the population is either known or can be well estimated (by , for example, if the sample size is large)For proportions, that variance of the estimator is unknown, since it depends on the (unknown) proportion p. However, it can be approximated by If n is large, this estimator is approximately normally distributedSimilar formulas held for the difference between two means or proportions•26T-testIf we know that the estimator is normally distributed but we do not have a good estimate for (for example, the sample size is too small), then we can form the following test statistic which will have a Student’s t-distribution:Again, if the estimator is unbiased, The same type of analysis can be used to accept or reject the Null


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UT Dallas CS 6313 - Chapter_9_4_5-4_9

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