23 Inference Goals and Strategies

23.1 Basic Idea

Data are collected in such a way that there exists a reasonable probability model for this process that involves parameters informative about the population.

Common Goals:

1. Form point estimates the parameters
2. Quantify uncertainty on the estimates
3. Test hypotheses on the parameters

23.2 Normal Example

Suppose a simple random sample of \(n\) data points is collected so that the following model of the data is reasonable: \(X_1, X_2, \ldots, X_n\) are iid Normal(\(\mu\), \(\sigma^2\)).

The goal is to do inference on \(\mu\), the population mean.

For simplicity, assume that \(\sigma^2\) is known (e.g., \(\sigma^2 = 1\)).

23.3 Point Estimate of \(\mu\)

There are a number of ways to form an estimate of \(\mu\), but one that has several justifications is the sample mean:

\[\hat{\mu} = \overline{x} = \frac{1}{n}\sum_{i=1}^n x_i,\]

where \(x_1, x_2, \ldots, x_n\) are the observed data points.

23.4 Sampling Distribution of \(\hat{\mu}\)

If we were to repeat this study over and over, how would \(\hat{\mu}\) behave?

\[\hat{\mu} = \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\]

\[\overline{X} \sim \mbox{Normal}(\mu, \sigma^2/n)\]

How do we use this to quantify uncertainty and test hypotheses?

23.5 Pivotal Statistic

One very useful strategy is to work backwards from a pivotal statistic, which is a statistic that does not depend on any unknown paramaters.

Example:

\[\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \sim \mbox{Normal}(0,1)\]

Note that in general for a rv \(Y\) it is the case that \((Y - \operatorname{E}[Y])/\sqrt{\operatorname{Var}(Y)}\) has population mean 0 and variance 1.