Introduction

We have modeled uncertainty using the tools of probability theory. Problems in probability theory may require a fair level of mathematical sophistication, and often students are led to believe that the involved calculations are the real difficulty. However, this is not the correct view; the real issue is that whatever we do in probability theory assumes a lot of knowledge. When dealing with a random variable, we need a function describing its whole probability distribution, like CDF, PMF, or PDF; in the multivariate case, the full joint distribution might be required, which can be a tricky object to specify. More often than not, this knowledge is not available and it must be somehow inferred from available data, if we are lucky enough to have them.

It is quite instructive to compare the definition of expected value in probability theory, assuming a continuous distribution, and sample mean in descriptive statistics:

The first expression looks definitely more intimidating than the second one, which is an innocent-looking average. Yet, the real trouble is getting to know the PDF f_X(x); calculating the integral is just a technicality. We relate these two concepts, as the sample mean can be used to estimate the expected value, when we do not know the underlying probability density. Indeed, the sample mean does not look troublesome, yet it is: If we select the sample randomly, the sample mean is a random variable, possibly with a large variance. We need to clarify the difference between a parameter, like the expected value μ = E[X], which is a number that we want to estimate, an estimator, like the sample variance , which is a random variable, and an estimate, which is a specific realization of that random variable for a sample. This raises many issues:

• How reliable is the estimate we get?
• How can we state something about estimation errors?
• How can we determine a suitable sample size?
• How can we check the truth of a hypothesis about the unknown expected value?

To find an answer to these and many other questions, we need the tools of inferential statistics.

• The first few sections provide the reader with the essentials of point and interval estimation of a parameter and hypothesis testing; these procedures have been automated in many software packages, but any business student and practitioner should have a reasonable background in order to apply these techniques with a minimum of critical sense.
• Then, we dig a bit deeper into issues such as stochastic convergence, the law of large numbers, and parameter estimation; readers who are just interested in the essentials of statistical inference can safely skip these sections, which are a bit more challenging and aimed at bridging the gap with advanced books on inferential statistics.

It is also worth pointing out that, given the aims we have taken an orthodox approach to statistics, as this is definitely simpler and corresponds to what is generally taught in business classes around the world. However, We stress that this is just one possible approach, and we provide readers with a glimpse of Bayesian statistics.

In Section 9.1 we clarify the meaning of terms like “random sample” and “statistic,” laying down the foundations for the remainder. Then, in Sections 9.2 and 9.3, we cover two classical topics, confidence intervals and hypothesis testing, within the framework of the basic problem of inferential statistics: estimating the expected value of a probability distribution. These three sections provide readers with the essential knowledge that anyone involved in business management should have, if anything, to understand essential issues and difficulties in analyzing data.

Then, we broaden our perspective a bit, while keeping the required mathematics to a rather basic level. In Section 9.4 we consider estimating other parameters of interest, like variance, probabilities, correlation, skewness, and kurtosis; we also consider the comparison of two populations in terms of their means. In all of these techniques, we assume that the probability distribution of the underlying population is known qualitatively, and we want to estimate some quantitative parameters. We will mostly refer to the simple case of a normal population. However, assuming that we are dealing with the parameters of a given distribution is itself a hypothesis that should be tested. Can we be sure that the underlying distribution is normal? Nonparametric statistics can be used, among other things, to check the fit of an assumed probability distribution against empirical data. We outline some basic tools, like the chi-square test, in Section 9.5. The basics of analysis of variance (ANOVA), in Section 9.6, and Monte Carlo simulation, in Section 9.7. It should be mentioned that both of these topics would require a whole book for a thorough coverage, but we can provide readers with an essential understanding of why they are useful and what kind of knowledge is required for their sensible use.

The last three sections, as we mentioned, are more challenging and can be skipped at first reading. Their aim is to bridge the gap between the elementary “cookbook” treatment that any introduction to business statistics offers, and the more advanced and mathematically demanding references. In Section 9.8 we outline the essential concepts of stochastic convergence; they are needed for an understanding of the law of the large numbers and also provide us with a justification of many estimation concepts that find wide application in statistical modeling and econometrics. We consider a more general framework for parameter estimation in Section 9.9, where we discuss desirable properties of estimators, as well as general strategies to obtain them, like the method of moments and maximum likelihood. By a similar token, we outline a more general approach to hypothesis testing in Section 9.10, dealing with a few issues that are skipped in the very elementary treatment of Section 9.3.