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.