Slide 1overviewoverviewEstimation of standard errorsConfidence intervalsConfidence intervalsConfidence intervalsPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapter 9: Statistical Inference1OVERVIEWWe’ve been exploring methods that allow us to do parameter estimation on a set of sample data …This allows us to estimate the parameters of the underlying population distributionThe method of moments allowed us approximate the population moment with a sample moment We also saw a centralized version of this …•22OVERVIEWIf we have an idea of the what the underlying population distribution is, and the parameters that describe the distribution can be found from the moments, we can use the sample moments to estimate the parametersWe also saw the method of maximum likelihood …This approach found parameters of the (assumed) underlying distribution that made it most likely that the sample data we collected would appear in a sample3ESTIMATION OF STANDARD ERRORSThe question now becomes: How good are the estimates that we get from the methods we saw? Can we estimate the standard error in these approximations?Recall from chapter 8 (section 8.2.5) that we defined the standard error of an estimator to be its standard deviation, namely .We will see how to estimate the standard error of estimators if we know something about the population distribution•24CONFIDENCE INTERVALSWhen we estimate a parameter with an estimator , we know that most likely ….This is because of the sampling error – we are not taking a census of the entire population, just a sample – and so there is some probability that our estimator is not accurateThe question becomes: How far off can the estimator be? What is the probability that it is close to the actual value?To answer this, we will use confidence intervals, which provide some level of confidence in the estimate given the observed data•25CONFIDENCE INTERVALSIn this lecture we will develop a general method for constructing a confidence interval …We will then see how to apply this to construct a confidence interval for the population meanWe will further expand upon this method to analyze two populations, and determine if they have different meansTo do this, we will construct a confidence interval for the difference of the means6CONFIDENCE INTERVALS1. Find an estimator of that is unbiased2. Check that the estimator has a Normal distribution3. Compute the standard deviation of the estimator, 4. Given the confidence level , find the quantiles from the table of Standard Normal distribution5. Apply the formula for the confidence interval shown in
View Full Document