Estimation and the Normal Distribution
In this chapter, we introduce and make use of a probability distribution that often arises in practice, albeit indirectly, the “normal” or “Gaussian” distribution. An observation that is the sum of a large number of factors, each of which makes only a small contribution to the total, will have a normal distribution. Thus, the mean of a large number of independent observations on the elements of a homogeneous population will have a normal distribution.
You also will learn in this chapter about the desirable properties of both point and interval estimates and how to apply this knowledge to estimate the parameters of a normal distribution. You’ll be provided with the R functions you need test hypotheses about the parameters.
4.1 POINT ESTIMATES
We are often required to estimate some property of a population based on a random representative sample from that population. For example, we might want to estimate the population mean or the population variance.
A desirable estimator will be consistent, that is, as the sample grows larger, an estimate based on the sample will get closer and closer to the true value of the population parameter. The sample mean is a consistent estimator of the population mean. The sample variance is a consistent estimator of the population variance.
When we make decisions based on an estimate h of a population parameter θ, we may be subject to losses based on some function L of the difference between our estimate ...