This chapter introduces the chi-square distribution through examples such as whether a coffee machine dispenses a consistent amount each time, framing it as the tool behind three major hypothesis tests. It covers the facts and notation of the chi-square distribution, the test of a single variance, and the goodness-of-fit test.
Three applications of one distribution
The chapter introduces the chi-square distribution as the basis for three distinct hypothesis tests: the goodness-of-fit test, which checks whether data fit a particular distribution, such as evenly distributed lottery numbers; the test of independence, which checks whether two categorical variables such as movie preference and age group are related; and the test of a single variance, illustrated with whether a coffee machine’s dispensed amount is consistent.
Facts about the chi-square distribution
The chi-square distribution is nonsymmetrical and skewed to the right, with a different curve for each value of its degrees of freedom, and the chapter notes that its test statistic is always greater than or equal to zero and that, for a sufficiently large number of degrees of freedom, the chi-square curve begins to approximate the normal distribution.
Testing a single variance and goodness of fit
Where earlier chapters focused on means and proportions, the chi-square test of a single variance shifts attention to variability itself, while the goodness-of-fit test determines whether an observed distribution of categorical data is consistent with a claimed or expected distribution, both worked through with the degrees of freedom appropriate to each specific use.
This chapter presents the Central Limit Theorem as a rigorously demonstrated theorem rather than a theory, comparable in certainty to the Pythagorean theorem. It explains why means are so central to statistics, how the theorem describes the distribution of sample means regardless of the original population’s shape, and what sample size counts as large enough.
Why the theorem matters
The chapter argues that means are central to statistics because they provide both a middle ground for comparison and are easy to calculate, and it stresses that the Central Limit Theorem is a genuine theorem, mathematically demonstrated with the same rigor as the Pythagorean theorem, not merely a proposed way of looking at the world.
Sample means and the normal distribution
The theorem concerns drawing finite samples of size n from a population with a known mean and known standard deviation; if n is large enough, the distribution of the resulting sample means will tend to follow an approximately normal distribution regardless of the shape of the original population’s distribution, a result the chapter calls astounding precisely because it does not require knowing that original shape.
How large a sample is large enough
The chapter addresses the practical question of what sample size counts as large enough for the theorem to apply well, generally citing a sample size of at least 30 unless the underlying population is already known to be normal, and noting that a population further from normal requires a larger sample before the sample-mean distribution becomes reliably normal.
This opening chapter introduces the basic vocabulary of statistics and probability, noting that fields from economics and business to law and computer science all require at least one statistics course. It defines descriptive and inferential statistics, and covers how data are gathered and what distinguishes reliable data from unreliable data.
Why statistics matters across fields
The chapter motivates the subject by pointing out that statistical information appears constantly in newspapers, television, and the internet, covering topics from crime to real estate, and that fields as varied as economics, business, psychology, education, biology, law, computer science, and police science all require statistical literacy to interpret this information thoughtfully.
Descriptive versus inferential statistics
The science of statistics is defined as the collection, analysis, interpretation, and presentation of data, and the chapter splits the subject into descriptive statistics, which organizes and summarizes data through graphs and numerical summaries, and inferential statistics, which uses probability to draw and quantify confidence in conclusions about a larger population from sample data.
Gathering data and judging its quality
Because later statistical inference is only as good as the data behind it, the chapter covers how data are gathered, the basic ideas of sampling methods, and what separates good data from bad, giving students in the business track a foundation for evaluating survey results and other data before analyzing them further.
This chapter introduces the vocabulary and systematic methods of probability, from the everyday intuition used when weighing whether to study for an exam to the formal terminology of experiments, outcomes, and events. It covers sample spaces, the notation for events and their probabilities, and distinguishes independent from mutually exclusive events.
Everyday intuition and formal terminology
The chapter starts from the observation that decisions ranging from a politician’s campaign strategy to a doctor’s choice of treatment already rely on an intuitive sense of probability, then formalizes this by defining an experiment as a planned operation carried out under controlled conditions and a chance experiment as one whose result is not predetermined.
Sample spaces, outcomes, and events
A result of an experiment is called an outcome, and the sample space, denoted S, is the set of all possible outcomes, which can be represented by listing them, by a tree diagram, or by a Venn diagram; an event is any combination of outcomes and is written with an upper-case letter, with its probability denoted P of that letter.
Independent and mutually exclusive events
The chapter draws a careful distinction between events that are independent, where the occurrence of one does not affect the probability of the other, and events that are mutually exclusive, which cannot occur together, a distinction the chapter stresses is not the same relationship despite the two terms sometimes being confused.
This chapter grounds linear regression and correlation in economic model-building, using the theory of consumer choice and the demand curve as an example of how assumptions lead to a testable relationship between variables. It covers the correlation coefficient, testing its significance, linear equations, and the regression equation itself.
Models, theories, and testable relationships
The chapter opens by connecting regression to the broader idea of a model, a theorized cause-and-effect relationship, illustrated with the economic model of consumer choice, where assumptions about preferences and utility maximization generate the prediction embodied in the demand curve, an example of how a theory produces a relationship that can then be tested statistically.
The correlation coefficient and its significance
The correlation coefficient r measures the strength and direction of a linear relationship between two numeric variables, and the chapter covers how to calculate and interpret it, followed by a hypothesis test for whether the observed correlation is statistically significant or could plausibly have arisen from an uncorrelated population.
Linear equations and the regression equation
Building on the algebra of a linear equation, the chapter develops the regression equation as the statistical technique for finding the line of best fit through a set of paired data, letting an analyst predict one variable, such as pay for a repair job, from another, such as an initial fee plus an hourly rate.
