Suppose we have an unknown population parameter, such as a population mean \(\mu\) or a population proportion \(p\), which we'd like to estimate. For example, suppose we are interested in estimating:
- \(p\) = the (unknown) proportion of American college students, 18-24, who have a smart phone
- \(\mu\) = the (unknown) mean number of days it takes Alzheimer's patients to achieve certain milestones
In either case, we can't possibly survey the entire population. That is, we can't survey all American college students between the ages of 18 and 24. Nor can we survey all patients with Alzheimer's disease. So, of course, we do what comes naturally and take a random sample from the population, and use the resulting data to estimate the value of the population parameter. Of course, we want the estimate to be "good" in some way.
In this lesson, we'll learn two methods, namely the method of maximum likelihood and the method of moments, for deriving formulas for "good" point estimates for population parameters. We'll also learn one way of assessing whether a point estimate is "good." We'll do that by defining what a means for an estimate to be unbiased.
- To learn how to find a maximum likelihood estimator of a population parameter.
- To learn how to find a method of moments estimator of a population parameter.
- To learn how to check to see if an estimator is unbiased for a particular parameter.
- To understand the steps involved in each of the proofs in the lesson.
- To be able to apply the methods learned in the lesson to new problems.