Slide 1OverviewOverviewP-ValuesVariancePROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapter 9: Statistical Inference1OVERVIEWWe’ve seen how to construct confidence intervals, given a level of significance and a computed statistic …We compute the estimator and the margin for the interval, which depends upon whether we are using a Z-distribution or a t-distributionWe worked examples for sample means and proportions – we used these to create confidence intervals for the population means and proportionsWe saw similar tools for analyzing the difference between means and proportions of two populationsWe studied hypothesis testing, which involved creating a null hypothesis and then checking a sample statistic to determine whether to accept or reject the Null hypothesisIn our cases, this involved assuming a population parameter equaled some value•;2OVERVIEWWhen conducting a hypothesis test, we formulate an alternative hypothesis, and (depending upon ) use tables to find critical values that determine accept and reject regionsWe saw left-tail, right-tail, and two-sided testsWe created a test statistic (again, either Z- or t- distributed) based upon sample data, and used this draw a conclusion about the Null hypothesisIf the test statistic fell into the reject region, we rejected the Null hypothesis; otherwise, we accepted itFinally, we looked at how to compute P-values; in some sense, this allowed us to find the lowest that could be associated with the test statistic we observed•;3P-VALUESP-values can be thought of as the probability that we will observe a value as extreme or more extreme than the test statistic we computedSuppose we have a Null hypothesis and we compute a test statistic that we know has a standard normal distribution. Then we can check …Right-tail alternative (). Here the P-value is Left-tail alternative (). Here the P-value is Two-sided alternative (). Here the P-value is It helps to think of these as areas under the PDF curve …These P-values can be found from the Standard Normal Distribution table A4, given The same ideas can be used for statistics that are t-distributed, but it is a little trickier …•;4VARIANCEIn this lecture, we will spend a little more time computing P-values, and we will see how to do this for the t-distribution table A5Then we will consider a new statistic – the sample varianceWe’ll see that using this statistic will require us to explore new distributions, namely the and F-distributions …We’ll look at how to compute confidence intervals, do hypothesis tests, and calculate P-values for this statistic – all using the -
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