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:
- Form point estimates the parameters
- Quantify uncertainty on the estimates
- 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.