Pages: 17

This chapter helps the readers to understand the basic flow of statistical modeling as a process and understand the concept of model building as a process. It describes the process of using R. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is “a formal representation of a probabilistic or statistical theory.” Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. The chapter focuses on the process of model construction. An outline is presented as a procedure to help construct mathematical models. The chapter illustrates this procedure with a few examples. It examines one of the advantages of RStudio: built in “wizards” for some operations such as data import.

Pages: 36

This chapter provides the different types of data (categorical (qualitative) and quantitative) and which displays are used for the type of data that the readers are using and displaying. It explores how to use Excel R to create “good” and useful displays of data for both categorical and quantitative data. The chapter explains how to obtain and use descriptive statistics. It also explores how to interpret key measures mean, median, mode, range, standard deviation, variance, and skewness. The three fundamental elements of bad graphical displays are data ambiguity, data distortion, and data distraction. Two common charts easily produced by spreadsheet programs that should almost always be avoided are the stacked bar chart and the pie chart. The stacked bar chart, made even worse by the use of 3D effects which makes it very difficult to estimate the values of the variables represented on the top of the bars.

Pages: 20

This chapter helps the readers to learn how to use R to obtain and interpret descriptive statistics for the center (location), spread, and shape of the distribution of a data set. It explores how to use R to obtain displays and be able to interpret them. The chapter also helps the readers to know what the measures are and which ones are used for quantitative data and which ones are better for qualitative data. A measure of the dispersion of a set of data points around their mean value. Variance is a mathematical expectation of the average squared deviations from the mean. The chapter describes asymmetry from the normal distribution in a set of statistical data. Skewness can come in the form of “negative skewness” or “positive skewness,” depending on whether data points are skewed to the left (negative skew) or to the right (positive skew) of the data average.

Pages: 49

This chapter helps the readers to know the addition rules and axioms for probability. Probability deals with experiments that yield random short-term results or outcomes yet reveal long-term predictability. The chapter introduces a tool helpful for probability problems: a tree diagram. To compute permutations and combinations, it also introduces the concept of a factorial. Tree diagrams are a useful way to delineate the outcomes of a sample space. One of the most important applications of probability and statistics is decision-making under uncertainty. The chapter demonstrates how to use the Law of Total Probability and Bayes’ Theorem to solve problems. The most difficult part of solving Bayes’ Rule problems is defining the events and determining which probability we want to know and which we do know in terms of those events.

Pages: 29

This chapter explores how to compute the mean and variance using expected values for general discrete distribution. It discusses several probability distributions that are used as models for discrete random variables. The chapter shows how to compute Bernoulli probabilities. The Bernoulli distribution is the discrete probability distribution of a random variable, which takes a binary. A Poisson process is a random process that generates occurrences over a certain period of time at a given rate. The Poisson distribution that varies over time (generally its time) is a model of the counts of these occurrences. The chapter presents a few additional discrete distributions: the hypergeometric, negative binomial, and geometric distributions. It also presents the characteristics of each, parameters of each and examples for computing probabilities in R.

Pages: 25

Time has a continuous range of values and, thus, as a continuous random variable requires a continuous probability distribution as a model. This chapter shows some modeling applications using many continuous distributions such as the uniform, the exponential distribution, and the normal distribution. It discusses a few of the distributions that the authors obtained results with R. One of the most important applications of the uniform distribution is in the generation of random numbers. That is, almost all random number generators generate random numbers on the interval. For other distributions, some transformation is applied to the uniform random numbers. Reliability theory and reliability engineering also make extensive use of the exponential distribution, to model the time until failure of products. The chapter presents methods to test things like whether data is distributed normally (hypothesis testing).

Pages: 13

This chapter demonstrates how calculus is used with continuous distributions. It also briefly mentions two flexible and important distributions for modeling, the gamma and beta distributions, if both parameters equal one, we have the uniform distribution. We can create probability distributions to model a variety of situations other than just the triangular distribution using functions. The key is that the function is never negative and the area under the curve is 1. The chapter looks at several examples of such functions. There are several named distributions that are particularly flexible for modeling. They are useful, as one need not develop a function to describe the situation assuming they produce a graph that fits the data. A PDF that is often applied to random variables that are restricted to values between 0 and 1 is the beta distribution.

Pages: 11

This chapter is somewhat of a “bridge” from probability to statistics. The central limit theorem (CLT) acts as that bridge. In statistics, it is critical to remember the difference between a population and a sample. The CLT is an important tool for identifying a “sampling distribution.” The chapter explores how CLT applies to means and totals. The CLT (when applicable) will be used extensively in many applications to follow in the book involving statistical inference: confidence intervals and hypothesis testing. The probability of the sample average in the interval is quite high while the probability for a single batch is low. The variance of the sample average is much smaller, so the probability of a sample mean near the population mean of 4 is fairly high.

Pages: 18

This chapter explores interval estimates known as confidence intervals for a variety of situations. It considers confidence intervals using two assumptions: the population distribution is normal or the sample size is large enough and the standard deviation is known. The distribution is more spread and will therefore make the confidence interval wider and account for the variability due to estimating the standard deviation from the sample. The chapter helps the readers to apply interval estimates to population means and proportions. It explores how to use formulas for sample size for margins of error. The chapter also helps the readers to apply interval estimates to variances and standard deviations.

Pages: 18

This chapter explores how the authors look at a hypothesis test. It provides a framework for hypothesis testing. The chapter begins with tests for a mean when the authors have either a large sample, or they know the sample is from a population with a normal distribution, and assuming they know the variance in the population. The analogy to “beyond a reasonable doubt” in hypothesis testing is what is known as the significance level. R and most statistical software can essentially do the “work” for us with built-in commands for various hypothesis tests. Hypothesis testing for a proportion involves similar changes to that of a mean that the authors saw for confidence intervals. Less commonly, there is a need to test the variance. Methods for confidence intervals and hypothesis tests are similar conceptually for doing so.

Pages: 14

Inference on two population means, σ known.

Pages: 16

This chapter shows we can use elementary probability to generate models that can be used to determine the reliability of equipment and systems of equipment. It shows how methods from elementary probability can be used to model military reliability problems. The modeling approach presented is useful in helping students simultaneously improve their understanding of both the military problems addressed and the mathematics behind these problems. The models presented also motivate students to appreciate the power of mathematics and its relevance to today’s world. The chapter describes the application of redundant and standby redundant systems. It explores how to solve problems involving large systems.

Pages: 37

This chapter explores how to obtain and interpret a linear regression model. The correlation is one of the most common and most useful statistics. A correlation is a single number that describes the degree of linear relationship between two variables. The chapter describes the relationship between age and five-year growth by examining the plot. In looking at the data and/or the scatterplot, not all the five-year growths are the same. Therefore, there is some variation in the response variable. The hope is that the least-squares regression line will fit between the data points in a manner that will “explain” quite a bit of that variation. The chapter briefly describes curve-fitting criterion: least squares or linear regression. After fitting a specified model, the difference between the observed and predicted values can be calculated by using the array manipulations.

Pages: 29

Analysis in data science and digital business requires analysis of the data and, in many cases, the use of regression techniques. This chapter discusses some simple regression and advanced regression techniques that have been used often in the analysis of data for business, industry, and government. It helps the readers to know the advanced regression methods for nonlinear, logistic, and sine regression. The chapter shows linear regression, polynomial regression, and then exponential regression in order to obtain a useful model. It focuses on the binary case which is often handled with logistic regression models. Decision-makers need to see and believe concepts concerning correlation, linear relationships, and nonlinear (or no) relationship. R treats sinusoidal regression models as part of the larger family of nonlinear least-squares regression models. The chapter analyzes the interrelation of dehumanization’s effects on the outcome of conflict.

Pages: 18

Analysis of Variance (ANOVA) is one of the most basic yet powerful statistical models that we can utilize. While it is commonly used for categorical data, because ANOVA is a type of linear model, it can be modified to include continuous data. Although ANOVA is relatively simple as compared to many statistical models, there are still some things we should consider. This chapter explores both the features and functions of ANOVA as handled by R. ANOVA analyzes the variance in the data to look for differences. It does this by considering two sources of variance, the between-group variance and the within-group variance. The primary purpose of a two-way ANOVA is to understand if there is an interaction between the two independent variables on the dependent variable.

Pages: 13

Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. This chapter introduces single-factor as well as multifactor analysis. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV), often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. MANCOVA also has some use in regression models, where its main function is to fit the regressions in both categorical and interval independents.