Section 9.2 Key Terms
The first part of the book deals with descriptive statistics and with probability. In descriptive statistics one investigates the characteristics of the data by using graphical tools and numerical summaries. The frame of reference is the observed data. In probability, on the other hand, one extends the frame of reference to include all data sets that could have potentially emerged, with the observed data as one among many.
The second part of the book deals with inferential statistics. The aim of statistical inference is to gain insight regarding the population parameters from the observed data. The method for obtaining such insight involves the application of formal computations to the data. The interpretation of the outcome of these formal computations is carried out in the probabilistic context, in which one considers the application of these formal computations to all potential data sets. The justification for using the specific form of computation on the observed data stems from the examination of the probabilistic properties of the formal computations.
Typically, the formal computations will involve statistics, which are functions of the data. The assessment of the probabilistic properties of the computations will result from the sampling distribution of these statistics.
An example of a problem that requires statistical inference is the estimation of a parameter of the population using the observed data. Point estimation attempts to obtain the best guess to the value of that parameter. An estimator is a statistic that produces such a guess. One may prefer an estimator whose sampling distribution is more concentrated about the population parameter value over another estimator whose sampling distribution is less so. Hence, the justification for selecting a specific statistic as an estimator is a consequence of the probabilistic characteristics of this statistic in the context of the sampling distribution.
For example, a car manufacture may be interested in the fuel consumption of a new type of car. In order to do so the manufacturer may apply a standard test cycle to a sample of 10 new cars of the given type and measure their fuel consumptions. The parameter of interest is the average fuel consumption among all cars of the given type. The average consumption of the 10 cars is a point estimate of the parameter of interest.
An alternative approach for the estimation of a parameter is to construct an interval that is most likely to contain the population parameter. Such an interval, which is computed on the basis of the data, is called a confidence interval. The sampling probability that the confidence interval will indeed contain the parameter value is called the confidence level. Confidence intervals are constructed so as to have a prescribed confidence level.
A different problem in statistical inference is hypothesis testing. The scientific paradigm involves the proposal of new theories and hypothesis that presumably provide a better description for the laws of Nature. On the basis of these hypothesis one may propose predictions that can be examined empirically. If the empirical evidence is consistent with the predictions of the new hypothesis but not with those of the old theory then the old theory is rejected in favor of the new one. Otherwise, the established theory maintains its status. Statistical hypothesis testing is a formal method for determining which of the two hypothesis should prevail that uses this paradigm.
Each of the two hypothesis, the old and the new, predicts a different distribution for the empirical measurements. In order to decide which of the distributions is more in tune with the data a statistic is computed. This statistic is called the test statistic. A threshold is set and, depending on where the test statistic falls with respect to this threshold, the decision is made whether or not to reject the old theory in favor of the new one.
This decision rule is not error proof, since the test statistic may fall by chance on the wrong side of the threshold. Nonetheless, by the examination of the sampling distribution of the test statistic one is able to assess the probability of making an error. In particular, the probability of erroneously rejecting the currently accepted theory (the old one) is called the significance level of the test. Indeed, the threshold is selected in order to assure a small enough significance level.
Returning to the car manufacturer. Assume that the car in question is manufactured in two different factories. One may want to examine the hypothesis that the car’s fuel consumption is the same for both factories. If 5 of the tested cars were manufactured in one factory and the other 5 in the other factory then the test may be based on the absolute value of the difference between the average consumption of the first 5 and the average consumption of the other 5.
The method of testing hypothesis is also applied in other practical settings where it is required to make decisions. For example, before a new treatment to a medical condition is approved for marketing by the appropriate authorities it must undergo a process of objective testing through clinical trials. In these trials the new treatment is administered to some patients while other obtain the (currently) standard treatment. Statistical tests are applied in order to compare the two groups of patient. The new treatment is released to the market only if it is shown to be beneficial with statistical significance and it is shown to have no unacceptable side effects.
In subsequent chapters we will discuss in more details the computation of point estimation, the construction of confidence intervals, and the application of hypothesis testing. The discussion will be initiated in the context of a single measurement but will later be extended to settings that involve comparison of measurements.
An example of such analysis is the analysis of clinical trials where the response of the patients treated with the new procedure is compared to the response of patients that were treated with the conventional treatment. This comparison involves the same measurement taken for two sub-samples. The tools of statistical inference — hypothesis testing, point estimation and the construction of confidence intervals — may be used in order to carry out this comparison.
Other comparisons may involve two measurements taken for the entire sample. An important tool for the investigation of the relations between two measurements, or variables, is regression. Models of regression describe the change in the distribution in one variable as a function of the other variable. Again, point estimation, confidence intervals, and hypothesis testing can be carried out in order to examine regression models. The variable whose distribution is the target of investigation is called the response. The other variable that may affect that distribution is called the explanatory variable.
