Introductory Statistics 2e

This opening chapter builds the vocabulary and conceptual foundation the rest of the book depends on. It starts by distinguishing population from sample and parameter from statistic, then classifies data as qualitative or quantitative (discrete vs. continuous), illustrating each distinction with worked examples like tuition surveys, backpack weights, and malpractice-lawsuit studies. It then covers how samples are actually drawn: simple random, stratified, cluster, and systematic sampling versus non-random convenience sampling, plus the difference between sampling with and without replacement and how sampling error shrinks as sample size grows. A section on frequency tables shows how to compute relative and cumulative relative frequency, and introduces the four levels of measurement (nominal, ordinal, interval, ratio), which determine what calculations are valid on a given variable. The chapter closes with experimental design: explanatory vs. response variables, treatments, control groups, placebos, blinding, and how random assignment rules out lurking variables to support causal claims, followed by a discussion of research ethics (informed consent, IRBs, and real cases of data fraud). A student should read this chapter first if they need to understand terminology (population/sample/parameter/statistic), pick or evaluate a sampling method, interpret frequency tables, classify a variable's measurement level, or design/critique an experiment for bias and validity.

This chapter is the toolbox for describing a single data set, both visually and numerically, before any formal inference is attempted. It opens with graphical methods -- stem-and-leaf plots, line graphs, bar graphs, histograms, frequency polygons, and time series graphs -- explaining when each is appropriate (e.g., histograms for large continuous data sets, time series for data collected over time) and how to choose class boundaries so no value falls on an edge. A section on misleading graphs (pie charts, mismatched axis scales) builds visual literacy. The chapter then moves to location statistics: quartiles, percentiles, and the interquartile range (IQR), including the index formula for finding any percentile and using 1.5×IQR to flag outliers. Box plots translate the five-number summary (min, Q1, median, Q3, max) into a picture of concentration and spread. Center measures follow -- mean, median, mode -- along with how to estimate a mean from grouped/frequency-table data, and how skewness governs the relative position of mean versus median. The chapter closes with spread: variance and standard deviation (sample vs. population formulas), z-scores for comparing values across different distributions, and Chebyshev's/Empirical Rules for how much data clusters near the mean. A student should read this chapter to learn how to summarize and visualize a raw data set and detect outliers before applying probability or inferential methods covered later.

This chapter builds the vocabulary and rules that later probability and inference topics in the book rely on. It opens by defining sample spaces, events, and the AND/OR/complement operations, then introduces conditional probability, P(A|B), as the probability of A once the sample space is reduced to B. From there it distinguishes two properties students routinely confuse: independence (one event's occurrence doesn't change the probability of the other, tested via P(A|B)=P(A), P(B|A)=P(B), or P(A AND B)=P(A)P(B)) and mutual exclusivity (events share no outcomes, so P(A AND B)=0), and ties independence to sampling with versus without replacement. It then packages these ideas into two formal tools, the multiplication rule and the addition rule, that let you compute joint and combined probabilities from simpler pieces. The chapter closes with two representational techniques, contingency tables and tree/Venn diagrams, that make multi-event, multi-step, or without-replacement problems tractable by organizing counts or probabilities visually. A student should read this chapter when they need to compute P(A|B), determine whether two events are independent or mutually exclusive, combine probabilities with AND/OR, or read probabilities off a two-way table or tree diagram — skills used throughout later chapters on discrete and continuous distributions.

This chapter builds the core toolkit for working with discrete random variables—outcomes that are counted rather than measured. It opens by defining the two conditions a discrete probability distribution function (PDF) must satisfy, then develops the expected value (mean) and standard deviation of a discrete random variable using expected-value tables, tying the mean's interpretation to the Law of Large Numbers. The bulk of the chapter introduces four named discrete distributions, each matched to a distinct sampling situation: the binomial (a fixed number of independent trials with two possible outcomes each), the geometric (independent trials repeated until the first success, notable for its unique memoryless property), the hypergeometric (sampling without replacement from two groups, making trials dependent), and the Poisson (counting occurrences of an event in a fixed interval at a known average rate, which can approximate the binomial under certain conditions). For each distribution the chapter gives its notation, defining conditions, mean/standard-deviation formulas, and calculator syntax for computing probabilities, illustrated through worked examples spanning business, health, sports, and everyday betting scenarios. Two hands-on labs (a playing-card draw and a three-dice game) let students compare empirical simulation results against theoretical predictions. A student should turn to this chapter to learn how to recognize which of the four distributions fits a given word problem and how to compute its mean, standard deviation, and specific probabilities—skills that recur throughout the rest of the course.

This chapter extends probability from discrete counts to continuous measurements — time, distance, money — where outcomes take any value in an interval. It opens by establishing that for a continuous random variable, probability is the area under a probability density function (pdf) between two points, not the value at a single point, and introduces the cumulative distribution function (CDF) as "area to the left." The chapter then develops two named continuous distributions in depth. The uniform distribution, X ~ U(a, b), models situations where every value in a range is equally likely (bus wait times, smiling durations, ages) and is used to practice computing probabilities, percentiles, and conditional probabilities from a rectangular pdf. The exponential distribution, X ~ Exp(m), models the waiting time until a random event (customer arrivals, equipment failure, calls) and introduces its defining memoryless property along with its close relationship to the Poisson distribution from the discrete chapter. A student should read this chapter to learn how to compute P(a < X < b) and percentiles for uniform or exponential scenarios, how to handle conditional probability questions involving elapsed time, and how the exponential and Poisson distributions connect. It ends with a hands-on random-number-generator lab, key term definitions, formula summaries, and extensive practice/homework problems for each distribution.

This chapter introduces the normal distribution, the bell-shaped continuous distribution underlying most later inferential statistics work. Section 6.1 defines the z-score, the standardized value z = (x-μ)/σ that converts any X~N(μ,σ) observation into the standard normal Z~N(0,1), and uses it to state the Empirical (68-95-99.7) Rule for how data cluster around the mean. Section 6.2 shows how to actually compute probabilities and percentiles for a normal variable using calculator functions (normalcdf for areas/probabilities, invNorm for percentiles and critical values) or standard normal tables, working through worked examples on exam scores, screen time, smartphone-user ages, and orange diameters, including finding quartiles and the IQR. Sections 6.3 and 6.4 are hands-on stats labs where students collect their own data (race lap times, pinkie-finger lengths) and compare the empirical histogram to a fitted theoretical normal model. The chapter closes with key term definitions, a formula review, and a large practice/homework problem set (with solutions) covering z-scores, empirical-rule ranges, and normalcdf/invNorm-style probability and percentile problems across health, sports, testing, and consumer-behavior contexts. A student should read this chapter when they need to compute a probability or percentile for normally distributed data, interpret a z-score, or understand why 68/95/99.7% figures appear in a problem.

This chapter explains the central limit theorem (CLT), the result that makes most of inferential statistics possible: for a population with any distribution, known or unknown, the distribution of sample means (and separately, sample sums) becomes approximately normal as sample size increases. Section 7.1 covers the CLT for sample means, giving X̄ ~ N(μ, σ/√n) and worked examples computing probabilities and percentiles for averages using normalcdf/invNorm-style calculations. Section 7.2 parallels this for sums, ΣX ~ N(nμ, √n·σ). Section 7.3 is the practical payoff: it teaches students to recognize which formula a question calls for (mean, sum, or neither — individual values never use the CLT), connects the CLT to the law of large numbers, and introduces the normal approximation to the binomial distribution with the continuity correction. Sections 7.4 and 7.5 are hands-on "stats lab" activities (pocket change, cookie recipes) that let students empirically observe the CLT's convergence to normality as n grows. The chapter closes with key terms, formula review, and extensive practice/homework problems with solutions. A student should read this chapter to learn how to compute probabilities and percentiles for sample averages and totals, and to understand why those differ from individual-value probabilities — it directly sets up Chapter 8's confidence intervals.

This chapter teaches how to build and interpret confidence intervals for a single population mean or a single population proportion — the core tool for estimating an unknown parameter from one sample. Section 8.1 covers the case where the population standard deviation σ is known, using the normal distribution and the point-estimate ± error-bound (EBM) format. Section 8.2 covers the far more common real-world case where σ is unknown, introducing the Student's t-distribution (degrees of freedom = n − 1) as the correct substitute for the normal distribution when the sample standard deviation s is used instead. Section 8.3 extends the same logic to proportions (binomial data), building intervals around the sample proportion p′, and adds the "plus four" adjustment for small samples. Throughout, the chapter emphasizes how confidence level and sample size trade off against interval width, and how to work backward from a stated interval to recover the sample mean or error bound, plus how to solve for the sample size needed to hit a target margin of error. Three hands-on Stats Labs (home costs, place of birth, women's heights) let students build intervals from real or classroom data. A student should read this chapter to learn exactly which distribution (z or t) and which formula to use for a given estimation problem, and how to correctly state what a confidence interval does and does not mean.

This chapter teaches the complete mechanics of testing a claim about a single population mean or a single population proportion. It opens by formalizing null (H0) and alternative (Ha) hypotheses, then covers the four possible outcomes of a test — correct decisions plus Type I and Type II errors — and their probabilities α and β, including the tradeoff with the Power of the Test. It explains how to pick the right sampling distribution: a z-test when the population standard deviation is known, a Student's t-test when it isn't, and a normal approximation for proportions when np and nq both exceed five. The heart of the chapter is the p-value framework: computing the probability of a result as extreme as the sample's under H0, comparing it to a preset significance level α, and using the simple rule "if α exceeds the p-value, reject H0" to reach a decision and write a plain-language conclusion. Roughly a dozen fully worked examples (swim times, bench-press weights, cell-phone cancer risk, blood-type rates, bride ages) walk through every test type end to end, including TI-83/84 calculator steps, so a student can pattern-match their own problem to a template. This is the foundational chapter for anyone needing to run or interpret a one-sample z-test, t-test, or proportion test and decide whether sample evidence is strong enough to overturn an assumed population value.

This chapter extends one-sample hypothesis testing to situations comparing two groups, which is the most common real-world testing scenario (comparing two products, two demographics, two treatments, before/after). It walks through four distinct test setups, each keyed to a different combination of research question and data structure: independent means with unknown population standard deviations (Welch's t-test, the default case), independent means with known population standard deviations (a z-test, rarely applicable but instructive), independent proportions (a pooled z-test), and matched/paired samples (reducing two dependent samples to one sample of differences and running a one-sample t-test). It also introduces Cohen's d as a companion measure of effect size, since statistical significance alone doesn't convey how large a difference actually is. Each section follows the same worked-example structure: state hypotheses, pick the distribution, compute the test statistic and p-value, decide, and write a conclusion sentence, with parallel TI-83/84 calculator instructions. A student should read this chapter to learn how to identify which two-sample test applies to their data (the recurring diagnostic questions are: are the two samples independent or paired, are you comparing means or proportions, and are population standard deviations known), and to get the exact formulas and worked p-value examples for each case.

This chapter teaches four hypothesis-testing procedures built on the chi-square distribution, all of which compare observed data to expected values under a null hypothesis. The goodness-of-fit test asks whether a single population's data match a claimed or theoretical distribution (uniform, given percentages, etc.). The test of independence uses a two-way contingency table to ask whether two categorical factors are related. The test for homogeneity asks whether two or more separate populations share the same distribution across one categorical variable. All three of these are right-tailed and require expected cell counts of at least five; a comparison section lays out exactly which wording and setup signals each test. The chapter closes with the test of a single variance (or standard deviation), a structurally different chi-square test that can be left-, right-, or two-tailed and checks a claim about population variability rather than distribution shape. Two hands-on labs let students apply goodness-of-fit and independence tests to real data they collect. A student trying to decide whether this chapter answers their question should look here for: contingency-table analysis, categorical survey/frequency comparisons, or any claim phrased in terms of variance or standard deviation rather than the mean.

This chapter introduces bivariate data analysis: how to describe and test a linear relationship between an independent variable x and a dependent variable y. It starts with linear equations (y = a + bx) and scatter plots, which let a student visually judge whether a linear model is even appropriate before doing any calculation. It then builds the least-squares regression line, showing how a and b are derived by minimizing the sum of squared residuals, and introduces residuals plots as a diagnostic. The correlation coefficient r and the coefficient of determination r² quantify how strong the linear relationship is and how much of y's variation x accounts for. A central section covers the hypothesis test of the significance of the correlation coefficient (H0: ρ = 0), using either a p-value/t-test or a table of critical values, which determines whether the sample relationship is strong enough to trust for population-level prediction. The chapter then addresses prediction itself, distinguishing safe interpolation from unreliable extrapolation, and closes with outliers and influential points, showing how to flag and evaluate them using the standard deviation of residuals. Worked examples throughout (tutoring fees, exam scores, dive times, CPI over time) tie each formula to interpretation in context. A student should read this chapter when they need to fit and justify a straight-line model relating two numeric variables and decide whether that model can be used to predict.

This chapter extends hypothesis testing about means from two groups to three or more, using the F distribution and one-way Analysis of Variance (ANOVA). It opens with the mechanics of the F-ratio: partitioning total variation into between-group and within-group sums of squares, converting these into mean squares (MSbetween, MSwithin) using numerator degrees of freedom k-1 and denominator degrees of freedom n-k, then combining them into the test statistic F = MSbetween/MSwithin, organized in a standard ANOVA source table. Several fully worked examples (diet plans, tomato mulching conditions, sorority grades, bean plant heights) walk through building this table by hand, including a simplified formula for when all groups are the same size. A section on the shape and behavior of the F distribution explains why the one-way ANOVA test is always right-tailed and how the curve's shape changes with different numerator/denominator degrees-of-freedom pairs. The chapter then pivots to a second, related use of the F distribution: testing whether two independent normal populations have equal variances, using F = s1²/s2², with an explicit warning that this variance test is unreliable when normality doesn't hold. A hands-on lab closes the chapter, having students collect their own price data and run a complete one-way ANOVA. Students who need to compare means across more than two groups, or compare the spread of two datasets, will find the full method, formulas, worked examples, and TI-83/84 calculator steps here.