Laura M. Chihara

• Professor at Carleton College

This article describes a collaborative project across three institutions to develop, implement, and evaluate a series of tutorials and case studies that highlight fundamental tools of data science—such as visualization, data manipulation, and database usage—that instructors at a wide-range of institutions can incorporate into existing statistics co...

Classical hypothesis testing is used to calculate how often pure random chance would give an effect as large as that observed in the data, in the absence of any real effect. A test statistic is a numerical function of the data whose value determines the result of the test. Permutation testing actually offers considerably more freedom than that; the...

For a symmetric distribution, the sample mean and sample median are both plug‐in estimators for the middle of the distribution, but they give different answers. This chapter examines these choices, discusses what makes a good estimator, introduces general procedures for estimating parameters, and discusses some practical issues. The general methods...

In statistics, Bayes offers a way to combine prior information with information provided by the data. The starting point for Bayesian methods is Bayes' theorem. The Bayesian approach contrasts to the frequentist approach. Bayesian methods are not subject to the same multiple testing issues as frequentist approaches, and they offer a logical self‐co...

Exploratory data analysis (EDA) is an approach to examining and describing data to gain insight, discover structure, and detect anomalies and outliers. This chapter helps the reader to learn many of the basic techniques and tools for gaining insight into data. Statistical software packages can easily do the calculations needed for the basic plots a...

This chapter discusses more classical approaches to constructing interval estimates using theoretical models for the sampling distribution. It derives confidence intervals for means and difference of means by considering the sampling distribution of a statistic. Desirable properties of confidence intervals include accurate coverage, short length, t...

Statistics is the art and science of collecting and analyzing data and understanding the nature of variability. In analyzing data, it is needed to determine whether the data represent a population or a sample. There are many kinds of random samples. Strictly speaking, a “random sample” is any sample obtained using a random procedure. When conductin...

This chapter begins by considering the fundamental bootstrap principle – plugging in an estimate for the population in place of the population and looking at the new possibilities for what to plug in. The bootstrap distributions are much improved for the median; they are now continuous, and the spreads are closer to the spread of the sampling distr...

Sampling distribution is the distribution of a statistic that summarizes a data set and represents how the statistic varies across many random data sets. A histogram of one set of observations drawn from a population does not represent a sampling distribution. A histogram of permutation means, each from one sample, does represent a sampling distrib...

This chapter utilizes mathematical derivations and approximations to find reference distributions from common families of distributions. In permutation testing, the reference distribution is the permutation distribution obtained by permuting the data. There are two possible errors in a hypothesis test: type I errors and type II errors. A type I err...

This chapter discusses the methods to handle contingency tables with more than two rows and columns and other categorical data. It explains three approaches – permutation testing, chi‐square distributions, and exact calculations. The chi‐square test statistic has an approximate chi‐square distribution if the null hypothesis is true. The chi‐square...

This chapter compares the means of three or more populations. It starts off with an approach that relies on a theoretical model for the sampling distribution of a test statistic, and ends with a permutation test approach. The analysis of variance (ANOVA) F‐test is for comparing means of populations. The big problem with non‐normality in t‐tests is...

The bootstrap is a procedure that uses the given sample to create a new distribution, called the bootstrap distribution, that approximates the sampling distribution for the sample mean. The bootstrap principle is to plug in an estimate for the population and then mimic the real‐life sampling procedure and statistic calculation. The bootstrap sampli...

This chapter describes a method to find a mathematical equation that explains the relationship between change in height and the change in diameter. Since correlation is not affected by linear transformations, the correlation may be expressed in terms of the correlation of standardized variables, which in turn equals the covariance of the standardiz...

Team-based learning (TBL) is a pedagogical strategy that uses groups of students working together in teams to learn course material. The main learning objective in TBL is to provide students the opportunity to practice course concepts during class-time. A key feature is multiple-choice quizzes that students take individually and then re-take as a t...

It is widely known that prolonged exposure to high levels of traffic noise has several health effects. While scholarship in environmental justice has explored the environmental equity hypothesis in a wide range of areas, whether the spatial distribution of traffic noise is equitable among different racial and socioeconomic groups has rarely been ex...

Black spruce (Picea mariana) is the most abundant tree species in the boreal biome, but little is known about how climate warming may change recruitment in peatlands, especially those affected by permafrost thaw. We used results from a seven-year study in northern Manitoba, Canada, to address the following questions: (1) What is the relative import...

We suggest here a new method of the estimation of missing entries in a gene expression matrix, which is done simultaneously—i.e., the estimation of one missing entry influences the estimation of other entries. Our method is closely related to the methods and techniques used for solving inverse eigenvalue problems.

Let be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matricesT such that (PT)3 =I. Moreover, all of the solutions for the classical infinite families of such schemes (including the Hamming scheme) are classified.

The eigenvalues of the classical association schemes are given by certain Askey-Wilson polynomials (or limiting cases). Kernel polynomials of the Askey-Wilson polynomials are also Askey-Wilson polynomials. We determine when both the Askey-Wilson polynomials and their kernels are the eigenvalues for known association schemes.

The zeros of generalized Krawtchouk polynomials are studied. Some interlacing theorems for the zeros are given. A new infinite family of integral zeros is given, and it is conjectured that these comprise most of the non-trivial zeros. The integral zeros for two families of q-Krawtchouk polynomials are classified.

Let {P n (x)} be a sequence of polynomials orthogonal with respect to a positive measure on [0,∞), and let {K n (x)} denote the corresponding sequence of kernel polynomials with k-parameter 0. The authors study the sequence of polynomials {R n (x)} defined by R 2n (x)=P n (x 2 )+ϑ 2n xK n-1 (x 2 ), R 2n+1 (x)=xK n (x 2 )+ϑ 2n+1 P n (x 2 ) for some...

In a symmetric association scheme that is (P and Q)-polynomial, the P and Q eigenmatrices are given by balanced ${}_4 \phi _3 $ Askey–Wilson polynomials. In this paper, the parameters of the Askey–Wilson polynomial are classified so that its zeros are not contained in its spectrum. These results, together with theorems of Biggs and Delsarte, imply...

Special cases of the Askey-Wilson polynomials are the eigenmatrices of the classical association schemes. Three constructions on the schemes — multiple polynomial structures, bipartite halves, and antipodal quotients — give quadratic transformations for the polynomials. It is shown that these transformations essentially follow from a quadratic tran...