Section 16.2 A Review
The second part of the book dealt with statistical inference; the science of making general statement on an entire population on the basis of data from a sample. The basis for the statements are theoretical models that produce the sampling distribution. Procedures for making the inference are evaluated based on their properties in the context of this sampling distribution. Procedures with desirable properties are applied to the data. One may attach to the output of this application summaries that describe these theoretical properties.
In particular, we dealt with two forms of making inference. One form was estimation and the other was hypothesis testing. The goal in estimation is to determine the value of a parameter in the population. Point estimates or confidence intervals may be used in order to fulfill this goal. The properties of point estimators may be assessed using the mean square error (MSE) and the properties of the confidence interval may be assessed using the confidence level.
The target in hypotheses testing is to decide between two competing hypothesis. These hypotheses are formulated in terms of population parameters. The decision rule is called a statistical test and is constructed with the aid of a test statistic and a rejection region. The default hypothesis among the two, is rejected if the test statistic falls in the rejection region. The major property a test must possess is a bound on the probability of a Type I error, the probability of erroneously rejecting the null hypothesis. This restriction is called the significance level of the test. A test may also be assessed in terms of its statistical power, the probability of rightfully rejecting the null hypothesis.
Estimation and testing were applied in the context of single measurements and for the investigation of the relations between a pair of measurements. For single measurements we considered both numeric variables and factors. For numeric variables one may attempt to conduct inference on the expectation and/or the variance. For factors we considered the estimation of the probability of obtaining a level, or, more generally, the probability of the occurrence of an event.
We introduced statistical models that may be used to describe the relations between variables. One of the variables was designated as the response. The other variable, the explanatory variable, is identified as a variable which may affect the distribution of the response. Specifically, we considered numeric variables and factors that have two levels. If the explanatory variable is a factor with two levels then the analysis reduces to the comparison of two sub-populations, each one associated with a level. If the explanatory variable is numeric then a regression model may be applied, either linear or logistic regression, depending on the type of the response.
The foundations of statistical inference are the assumption that we make in the form of statistical models. These models attempt to reflect reality. However, one is advised to apply healthy skepticism when using the models. First, one should be aware what the assumptions are. Then one should ask oneself how reasonable are these assumption in the context of the specific analysis. Finally, one should check as much as one can the validity of the assumptions in light of the information at hand. It is useful to plot the data and compare the plot to the assumptions of the model.
