This chapter extends hypothesis testing to comparisons between two groups, using business examples such as night-shift versus day-shift productivity and the investment returns of two different strategies. It distinguishes independent groups from matched pairs and covers testing two population means and two population proportions under each design.
Independent groups versus matched pairs
Comparing two groups requires first classifying them as independent, where sample values from one population are unrelated to those from the other, or as matched pairs, where the two samples are dependent, and the chapter uses this distinction to determine which testing method and which parameter, means or proportions, applies.
Comparing two independent population means
The comparison of two independent population means is presented as a common business question, illustrated with whether the night shift is less productive than the day shift or whether one investment strategy’s returns differ from another’s, framing the two-sample mean comparison as a natural extension of the one-sample methods covered earlier.
Comparing two proportions and paired samples
Beyond comparing means, the chapter covers hypothesis tests for two population proportions, useful for comparing rates such as production efficiency under different management styles, and for matched or paired samples, where the same subjects or units are measured under two conditions and the pairing itself must be accounted for in the test.
This chapter situates hypothesis testing within the scientific method, using the economic theory of consumer choice and the demand curve as an example of how a theory generates a testable prediction. It covers null and alternative hypotheses, Type I and Type II errors, and the distributions used depending on what is known about the population.
Hypothesis testing as part of the scientific method
The chapter frames statistical hypothesis testing as the formal mechanism behind the scientific method, where a theory or model built on stated assumptions leads to predictions, or hypotheses, that can be tested; it illustrates this with microeconomic consumer choice theory, whose assumptions predict the negative-sloped demand curve, a prediction statistics is used to test rather than simply assert.
Null and alternative hypotheses, Type I and Type II errors
Every hypothesis test begins by stating two competing hypotheses, the null and the alternative, and the chapter explains the two ways a test can go wrong: a Type I error, rejecting a true null hypothesis, and a Type II error, failing to reject a false one, along with how the outcomes of a test relate to these error types.
Choosing the right distribution for the test
Which distribution underlies a one-sample hypothesis test depends on what is known about the population, and the chapter covers testing a single population mean when the standard deviation is known, when it is unknown, and testing a single population proportion, connecting each case back to the sampling distributions developed earlier in the course.
This chapter introduces the F distribution and one-way ANOVA using business framing, such as comparing the variability of two investment portfolios or checkout times at different registers. It covers the test of two variances, the logic of comparing averages across more than two groups with single-factor ANOVA, and the F-ratio the test relies on.
Why compare variances, not just averages
The chapter opens by motivating the F distribution through situations where the question is about variability rather than the mean, such as whether two investment portfolios carry the same volatility, whether two professors grade with the same spread, or whether two checkout lines have similarly consistent service times, each requiring a formal test of two variances.
One-way ANOVA for comparing several groups
When more than two group averages must be compared, for example gas mileage across several car models or income across social backgrounds, one-way ANOVA (Analysis of Variance) provides a single hypothesis test rather than requiring many pairwise comparisons, and the chapter presents this single-factor version as the simplest form of ANOVA.
The F distribution and the F-ratio
The F distribution underlies both the test of two variances and one-way ANOVA, and the chapter explains the F-ratio as the statistic that compares variation between group means to variation within groups, noting that the method as presented relies heavily on calculator or computer computation rather than manual calculation.
This chapter uses business examples, including the number of long-distance calls employees make during peak hours, to explain discrete random variables and probability density functions. It covers random variable notation, the two defining properties of a probability density function, and works through the hypergeometric, binomial, and geometric distributions as the main discrete models covered.
Random variables and their notation
A discrete random variable takes only countable, whole-number values, and the chapter establishes the convention of an upper-case letter such as X to denote the random variable in words and a lower-case letter such as x for a specific numeric outcome, illustrated with counting the number of heads in three coin tosses.
Properties of a probability density function
A probability density function for a discrete random variable must satisfy two conditions: each individual probability lies between zero and one inclusive, and the sum of all the probabilities across every possible outcome equals one, a check used throughout the chapter to confirm that a proposed distribution is valid.
Hypergeometric, binomial, and geometric distributions
The chapter develops three named discrete distributions in turn: the hypergeometric distribution, presented as the simplest probability density function for sampling without replacement from a finite population; the binomial distribution, described as a more broadly applicable model with many business uses; and the geometric distribution, which builds on the binomial setup to model the number of trials until a first success.
This chapter explains continuous random variables through business-relevant examples such as rates of return from an investment and other measured, rather than counted, quantities. It covers the properties of continuous probability density functions, where probability corresponds to area under a curve, and introduces the uniform distribution as the first continuous distribution studied in detail.
Measured quantities versus counted quantities
A continuous random variable takes values from a measurement rather than a count, such as a rate of return, the length of a phone call, or the time a computer chip lasts, and the chapter contrasts this with the discrete random variables covered earlier, where the field of reliability and risk analysis depends heavily on continuous measures.
Probability as area under a curve
For a continuous distribution the graph is a curve and probability is represented by the area under that curve over a range of values, extending the relative-frequency idea from histograms; because a single point has zero width, the probability of any exact value is zero, and only the area over an interval carries meaning.
The uniform distribution
The uniform distribution is introduced as the simplest continuous probability distribution, where every value in an interval is equally likely and probabilities are found directly as proportions of the interval’s width, giving a first concrete example of how the area-under-the-curve principle is applied before moving on to more complex continuous distributions.
